42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017), MFCS 2017, August 21-25, 2017, Aalborg, Denmark
MFCS 2017
August 21-25, 2017
Aalborg, Denmark
Mathematical Foundations of Computer Science
MFCS
https://dblp.org/db/conf/mfcs
Leibniz International Proceedings in Informatics
LIPIcs
https://www.dagstuhl.de/dagpub/1868-8969
https://dblp.org/db/series/lipics
1868-8969
Kim G.
Larsen
Kim G. Larsen
Hans L.
Bodlaender
Hans L. Bodlaender
Jean-Francois
Raskin
Jean-Francois Raskin
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
83
2017
978-3-95977-046-0
https://www.dagstuhl.de/dagpub/978-3-95977-046-0
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter
Table of Contents
Preface
Conference Organization
0:i-0:xvi
Front Matter
Kim G.
Larsen
Kim G. Larsen
Hans L.
Bodlaender
Hans L. Bodlaender
Jean-Francois
Raskin
Jean-Francois Raskin
10.4230/LIPIcs.MFCS.2017.0
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Does Looking Inside a Circuit Help?
The Black-Box Hypothesisstates that any property of Boolean functions decided efficiently (e.g., in BPP) with inputs represented by circuits can also be decided efficiently in the black-box setting, where an algorithm is given an oracle access to the input function and an upper bound on its circuit size. If this hypothesis is true, then P neq NP. We focus on the consequences of the hypothesis being false, showing that (under general conditions on the structure of a counterexample) it implies a non-trivial algorithm for CSAT. More specifically, we show that if there is a property F of boolean functions such that F has high sensitivity on some input function f of subexponential circuit complexity (which is a sufficient condition for F being a counterexample to the Black-Box Hypothesis), then CSAT is solvable by a subexponential-size circuit family. Moreover, if such a counterexample F is symmetric, then CSAT is in Ppoly. These results provide some evidence towards the conjecture (made in this paper) that the Black-Box Hypothesis is false if and only if CSAT is easy.
Black-Box Hypothesis
Rice's theorem
circuit complexity
SAT
sensitivity of boolean functions
decision tree complexity
1:1-1:13
Regular Paper
Russell
Impagliazzo
Russell Impagliazzo
Valentine
Kabanets
Valentine Kabanets
Antonina
Kolokolova
Antonina Kolokolova
Pierre
McKenzie
Pierre McKenzie
Shadab
Romani
Shadab Romani
10.4230/LIPIcs.MFCS.2017.1
Leonard Adleman. Two theorems on random polynomial time. In Proceedings of the Nineteenth Annual IEEE Symposium on Foundations of Computer Science, pages 75-83, 1978.
Eric Allender, Dhiraj Holden, and Valentine Kabanets. The minimum oracle circuit size problem. Computational Complexity, 26(2):469-496, 2017. URL: http://dx.doi.org/10.1007/s00037-016-0124-0.
http://dx.doi.org/10.1007/s00037-016-0124-0
Andris Ambainis and Jevgēnijs Vihrovs. Size of sets with small sensitivity: A generalization of Simon’s lemma. In International Conference on Theory and Applications of Models of Computation, pages 122-133. Springer International Publishing, 2015.
Laci Babai, Lance Fortnow, Noam Nisan, and Avi Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity, 3:307-318, 1993.
Boaz Barak, Oded Goldreich, Russell Impagliazzo, Steven Rudich, Amit Sahai, Salil Vadhan, and Ke Yang. On the (im) possibility of obfuscating programs. Journal of the ACM (JACM), 59(2):6, 2012.
Bernd Borchert and Frank Stephan. Looking for an analogue of Rice’s theorem in circuit complexity theory. Math. Log. Q., 46(4):489-504, 2000. URL: http://dx.doi.org/10.1002/1521-3870(200010)46:4<489::AID-MALQ489>3.0.CO;2-F.
http://dx.doi.org/10.1002/1521-3870(200010)46:4<489::AID-MALQ489>3.0.CO;2-F
Harry Buhrman and Ronald de Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science, 288(1):21–43, Oct 2002.
Ashok K Chandra, Larry Stockmeyer, and Uzi Vishkin. Constant depth reducibility. SIAM Journal on Computing, 13(2):423-439, 1984.
Pooya Hatami, Raghav Kulkarni, and Denis Pankratov. Variations on the sensitivity conjecture. Theory of Computing, Graduate Surveys, 2:1-27, 2011.
Lane A. Hemaspaandra and Jörg Rothe. A second step towards complexity-theoretic analogs of Rice’s theorem. Theor. Comput. Sci., 244(1-2):205-217, 2000.
Lane A. Hemaspaandra and Mayur Thakur. Lower bounds and the hardness of counting properties. Theor. Comput. Sci., 326(1-3):1-28, 2004.
Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, Pierre McKenzie, and Shadab Romani. Does looking inside a circuit help? Electronic Colloquium on Computational Complexity, 17(109), 2017.
Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1774.
http://dx.doi.org/10.1006/jcss.2001.1774
Russell Impagliazzo and Avi Wigderson. P=BPP if E requires exponential circuits: Derandomizing the XOR Lemma. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 220-229, 1997.
Noam Nisan. CREW PRAMs and decision trees. SIAM Journal on Computing, 20(6):999-1007, 1991.
Noam Nisan and Avi Wigderson. Hardness vs. randomness. Journal of Computer and System Sciences, 49:149-167, 1994.
Ramamohan Paturi and Pavel Pudlák. On the complexity of circuit satisfiability. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 241-250, 2010.
Hans-Ulrich Simon. A tight ω (loglog n)-bound on the time for parallel RAM’s to compute nondegenerated boolean functions. In Foundations of Computation Theory, pages 439-444. Springer, 1983.
Leslie Valiant and Vijay Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85-93, 1986.
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The Power of Programs over Monoids in DA
The program-over-monoid model of computation originates with Barrington's proof that it captures the complexity class NC^1. Here we make progress in understanding the subtleties of the model. First, we identify a new tameness condition on a class of monoids that entails a natural characterization of the regular languages recognizable by programs over monoids from the class. Second, we prove that the class known as DA satisfies tameness and hence that the regular languages recognized by programs over monoids in DA are precisely those recognizable in the classical sense by morphisms from QDA. Third, we show by contrast that the well studied class of monoids called J is not tame and we exhibit a regular language, recognized by a program over a monoid from J, yet not recognizable classically by morphisms from the class QJ. Finally, we exhibit a program-length-based hierarchy within the class of languages recognized by programs over monoids from DA.
Programs over monoids
DA
lower-bounds
2:1-2:20
Regular Paper
Nathan
Grosshans
Nathan Grosshans
Pierre
McKenzie
Pierre McKenzie
Luc
Segoufin
Luc Segoufin
10.4230/LIPIcs.MFCS.2017.2
Miklós Ajtai. Σ₁¹-formulae on finite structures. In Ann. Pure and Appl. Logic, volume 24, pages 1-48, 1983.
Jorge Almeida. A syntactical proof of locality of DA. Int. J. of Algebra and Computation (IJAC), 6(2):165-178, 1996.
David A. Mix Barrington. Bounded-width polynomial-size branching programs recognize exactly those languages in NC¹. J. Comput. Syst. Sci., 38(1):150-164, 1989.
David A. Mix Barrington, Kevin J. Compton, Howard Straubing, and Denis Thérien. Regular languages in NC¹. J. Comput. Syst. Sci., 44(3):478-499, 1992.
David A. Mix Barrington, Howard Straubing, and Denis Thérien. Non-uniform automata over groups. Inf. Comput., 89(2):109-132, 1990.
David A. Mix Barrington and Denis Thérien. Finite monoids and the fine structure of NC^1. J. ACM, 35(4):941-952, 1988.
Laura Chaubard, Jean-Éric Pin, and Howard Straubing. Actions, wreath products of C-varieties and concatenation product. Theoretical Computer Science, 356(1-2):73-89, 2006.
Luc Dartois. Méthodes algébriques pour la théorie des automates. PhD thesis, Université Paris Diderot, Paris, 2014.
Luc Dartois and Charles Paperman. Adding modular predicates. CoRR, abs/1401.6576, 2014. URL: http://arxiv.org/abs/1401.6576.
http://arxiv.org/abs/1401.6576
Samuel Eilenberg. Automata, Languages, and Machines, volume A. Academic Press, New York, 1974.
Samuel Eilenberg. Automata, Languages, and Machines, volume B. Academic Press, New York, 1976.
Merrick L. Furst, James B. Saxe, and Michael Sipser. Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 17(1):13-27, 1984.
Ricard Gavaldà and Denis Thérien. Algebraic characterizations of small classes of Boolean functions. In Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science (STACS), volume 2607 of Lecture Notes in Computer Science, pages 331-342. Springer, 2003.
Johan Håstad. Almost optimal lower bounds for small depth circuits. In Proceedings of the 18th Annual ACM Symposium on Theory of Computing (STOC), pages 6-20. ACM, 1986.
Kenneth Krohn and John L. Rhodes. Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines. In Trans. Amer. Math. Soc., volume 116, pages 450-464, 1965.
Alexis Maciel, Pierre Péladeau, and Denis Thérien. Programs over semigroups of dot-depth one. Theor. Comput. Sci., 245(1):135-148, 2000.
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Pierre McKenzie, Pierre Péladeau, and Denis Thérien. NC¹: The automata-theoretic viewpoint. Computational Complexity, 1:330-359, 1991.
Charles Paperman. Circuits booléens, prédicats modulaires et langages réguliers. PhD thesis, Université Paris Diderot, Paris, 2014.
Pierre Péladeau. Classes de circuits booléens et variétés de monoïdes. PhD thesis, Université Pierre-et-Marie-Curie (Paris-VI), Paris, France, 1990.
Pierre Péladeau, Howard Straubing, and Denis Thérien. Finite semigroup varieties defined by programs. Theor. Comput. Sci., 180(1-2):325-339, 1997.
Jean-Éric Pin. Varieties of formal languages. North Oxford, London and Plenum, New-York, 1986. (Traduction de Variétés de langages formels).
Jean-Éric Pin and Howard Straubing. Some results on 𝒞-varieties. RAIRO-Theor. Inf. Appl., 39(1):239-262, 2005.
Alexander A. Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Mathematical Notes, 41(4):333-338, 1987.
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Pascal Tesson. Computational Complexity Questions Related to Finite Monoids and Semigroups. PhD thesis, McGill University, Montreal, Canada, 2003.
Pascal Tesson and Denis Thérien. The computing power of programs over finite monoids. J. Autom. Lang. Comb., 7(2):247-258, November 2001.
Pascal Tesson and Denis Thérien. Diamonds are forever: the variety DA. Semigroups, algorithms, automata and languages, 1:475-500, 2002.
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Regular Language Distance and Entropy
This paper addresses the problem of determining the distance between two regular languages. It will show how to expand Jaccard distance, which works on finite sets, to potentially-infinite regular languages.
The entropy of a regular language plays a large role in the extension. Much of the paper is spent investigating the entropy of a regular language. This includes addressing issues that have required previous authors to rely on the upper limit of Shannon's traditional formulation of channel capacity, because its limit does not always exist. The paper also includes proposing a new limit based formulation for the entropy of a regular language and proves that formulation to both exist and be equivalent to Shannon's original formulation (when it exists). Additionally, the proposed formulation is shown to equal an analogous but formally quite different notion of topological entropy from Symbolic Dynamics -- consequently also showing Shannon's original formulation to be
equivalent to topological entropy.
Surprisingly, the natural Jaccard-like entropy distance is trivial in most cases. Instead, the entropy sum distance metric is suggested, and shown to be granular in certain situations.
regular languages
channel capacity
entropy
Jaccard
symbolic dynamics
3:1-3:14
Regular Paper
Austin J.
Parker
Austin J. Parker
Kelly B.
Yancey
Kelly B. Yancey
Matthew P.
Yancey
Matthew P. Yancey
10.4230/LIPIcs.MFCS.2017.3
F. Blanchard, E. Glasner, S. Kolyada, and A. Maass. On Li-Yorke pairs. J. Reine Angew. Math., 547:51-68, 2002.
M. Bodirsky, T. Gärtner, T. von Oertzen, and J. Schwinghammer. Efficiently computing the density of regular languages. In LATIN 2004: Theoretical informatics, volume 2976 of Lecture Notes in Comput. Sci., pages 262-270. Springer, Berlin, 2004. URL: http://dx.doi.org/10.1007/978-3-540-24698-5_30.
http://dx.doi.org/10.1007/978-3-540-24698-5_30
T. Ceccherini-Silberstein, A. Machì, and F. Scarabotti. On the entropy of regular languages. Theoretical computer science, 307(1):93-102, 2003.
C. Chan, M. Garofalakis, and R. Rastogi. Re-tree: an efficient index structure for regular expressions. The VLDB Journal—The International Journal on Very Large Data Bases, 12(2):102-119, 2003.
C. Chang. Algorithm for the complexity of finite automata. 31st Workshop on Combinatorial Mathematics and Computation Theory, pages 216-220, 2014.
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C. Cortes, M. Mohri, A. Rastogi, and M. Riley. Efficient computation of the relative entropy of probabilistic automata. In LATIN 2006: Theoretical informatics, volume 3887 of Lecture Notes in Comput. Sci., pages 323-336. Springer, Berlin, 2006. URL: http://dx.doi.org/10.1007/11682462_32.
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C. Cui, Z. Dang, T. Fischer, and O. Ibarra. Similarity in languages and programs. Theoretical Computer Science, 498:58-75, 2013.
C. Cui, Z. Dang, T. Fischer, and O. Ibarra. Information rate of some classes of non-regular languages: an automata-theoretic approach (extended abstract). In Mathematical foundations of computer science 2014. Part I, volume 86343 of Lecture notes in Comput. Sci., pages 232-243. Springer, Heidelberg, 2014.
Jürgen Dassow, Gema M. Martín, and Francisco J. Vico. A similarity measure for cyclic unary regular languages. Fundam. Inform., 96(1-2):71-88, 2009. URL: http://dx.doi.org/10.3233/FI-2009-168.
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The Complexity of Boolean Surjective General-Valued CSPs
Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with the objective function given as a sum of fixed-arity functions; the values are rational numbers or infinity.
In Boolean surjective VCSPs variables take on labels from D={0,1} and an optimal assignment is required to use both labels from D. A classic example is the global min-cut problem in graphs. Building on the work of Uppman, we establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs. The newly discovered tractable case has an interesting structure related to projections of downsets and upsets. Our work generalises the dichotomy for {0,infinity}-valued constraint languages corresponding to CSPs) obtained by Creignou and Hebrard, and the dichotomy for {0,1}-valued constraint languages (corresponding to Min-CSPs) obtained by Uppman.
constraint satisfaction problems
surjective CSP
valued CSP
min-cut
polymorphisms
multimorphisms
4:1-4:14
Regular Paper
Peter
Fulla
Peter Fulla
Stanislav
Zivny
Stanislav Zivny
10.4230/LIPIcs.MFCS.2017.4
Libor Barto. Constraint satisfaction problem and universal algebra. ACM SIGLOG News, 1(2):14-24, 2014. URL: http://dx.doi.org/10.1145/2677161.2677165.
http://dx.doi.org/10.1145/2677161.2677165
Libor Barto and Marcin Kozik. Constraint Satisfaction Problems Solvable by Local Consistency Methods. Journal of the ACM, 61(1), 2014. Article No. 3. URL: http://dx.doi.org/10.1145/2556646.
http://dx.doi.org/10.1145/2556646
Libor Barto, Andrei Krokhin, and Ross Willard. Polymorphisms, and how to use them. In Andrei Krokhin and Stanislav Živný, editors, The Constraint Satisfaction Problem: Complexity and Approximability, volume 7 of Dagstuhl Follow-Ups, pages 1-44. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2017. URL: http://dx.doi.org/10.4230/DFU.Vol7.15301.1.
http://dx.doi.org/10.4230/DFU.Vol7.15301.1
Manuel Bodirsky, Jan Kára, and Barnaby Martin. The complexity of surjective homomorphism problems - a survey. Discrete Applied Mathematics, 160(12):1680-1690, 2012. URL: http://dx.doi.org/10.1016/j.dam.2012.03.029.
http://dx.doi.org/10.1016/j.dam.2012.03.029
Andrei Bulatov. A dichotomy theorem for constraint satisfaction problems on a 3-element set. Journal of the ACM, 53(1):66-120, 2006. URL: http://dx.doi.org/10.1145/1120582.1120584.
http://dx.doi.org/10.1145/1120582.1120584
Andrei Bulatov, Andrei Krokhin, and Peter Jeavons. Classifying the Complexity of Constraints using Finite Algebras. SIAM Journal on Computing, 34(3):720-742, 2005. URL: http://dx.doi.org/10.1137/S0097539700376676.
http://dx.doi.org/10.1137/S0097539700376676
Hubie Chen. An algebraic hardness criterion for surjective constraint satisfaction. Algebra universalis, 72(4):393-401, 2014. URL: http://dx.doi.org/10.1007/s00012-014-0308-x.
http://dx.doi.org/10.1007/s00012-014-0308-x
David A. Cohen, Martin C. Cooper, Peter G. Jeavons, and Andrei A. Krokhin. The Complexity of Soft Constraint Satisfaction. Artificial Intelligence, 170(11):983-1016, 2006. URL: http://dx.doi.org/10.1016/j.artint.2006.04.002.
http://dx.doi.org/10.1016/j.artint.2006.04.002
Nadia Creignou. A dichotomy theorem for maximum generalized satisfiability problems. Journal of Computer and System Sciences, 51(3):511-522, 1995. URL: http://dx.doi.org/10.1006/jcss.1995.1087.
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Nadia Creignou and Jean-Jacques Hébrard. On generating all solutions of generalized satisfiability problems. ITA, 31(6):499-511, 1997.
Tomás Feder and Moshe Y. Vardi. The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory. SIAM Journal on Computing, 28(1):57-104, 1998. URL: http://dx.doi.org/10.1137/S0097539794266766.
http://dx.doi.org/10.1137/S0097539794266766
Jiří Fiala and Jan Kratochvíl. Locally constrained graph homomorphisms - structure, complexity, and applications. Computer Science Review, 2(2):97-111, 2008. URL: http://dx.doi.org/10.1016/j.cosrev.2008.06.001.
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Jiří Fiala and Daniël Paulusma. A complete complexity classification of the role assignment problem. Theoretical Computer Science, 349(1):67-81, 2005. URL: http://dx.doi.org/10.1016/j.tcs.2005.09.029.
http://dx.doi.org/10.1016/j.tcs.2005.09.029
Peter Fulla and Stanislav Živný. The complexity of Boolean surjective general-valued CSPs. CoRR, abs/1702.04679, 2017. URL: http://arxiv.org/abs/1702.04679.
http://arxiv.org/abs/1702.04679
Anna Huber, Andrei Krokhin, and Robert Powell. Skew bisubmodularity and valued CSPs. SIAM Journal on Computing, 43(3):1064-1084, 2014. URL: http://dx.doi.org/10.1137/120893549.
http://dx.doi.org/10.1137/120893549
Peter Jonsson, Mikael Klasson, and Andrei A. Krokhin. The approximability of three-valued MAX CSP. SIAM Journal on Computing, 35(6):1329-1349, 2006. URL: http://dx.doi.org/10.1137/S009753970444644X.
http://dx.doi.org/10.1137/S009753970444644X
David R. Karger. Global Min-cuts in RNC, and Other Ramifications of a Simple Min-Cut Algorithm. In Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'93), pages 21-30, 1993.
Vladimir Kolmogorov, Andrei A. Krokhin, and Michal Rolínek. The complexity of general-valued CSPs. In Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS'15). IEEE Computer Society, 2015.
Marcin Kozik and Joanna Ochremiak. Algebraic properties of valued constraint satisfaction problem. In Proceedings of the 42nd International Colloquium on Automata, Languages and Programming (ICALP'15), volume 9134 of Lecture Notes in Computer Science, pages 846-858. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_69.
http://dx.doi.org/10.1007/978-3-662-47672-7_69
Barnaby Martin and Daniël Paulusma. The computational complexity of disconnected cut and 2K₂-partition. Journal of Combinatorial Theory, Series B, 111:17-37, 2015. URL: http://dx.doi.org/10.1016/j.jctb.2014.09.002.
http://dx.doi.org/10.1016/j.jctb.2014.09.002
Thomas J. Schaefer. The Complexity of Satisfiability Problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing (STOC'78), pages 216-226. ACM, 1978. URL: http://dx.doi.org/10.1145/800133.804350.
http://dx.doi.org/10.1145/800133.804350
Alexander Schrijver. Combinatorial Optimization: Polyhedra and Efficiency, volume 24 of Algorithms and Combinatorics. Springer, 2003.
Mechthild Stoer and Frank Wagner. A simple min-cut algorithm. Journal of the ACM, 44(4):585-591, 1997. URL: http://dx.doi.org/10.1145/263867.263872.
http://dx.doi.org/10.1145/263867.263872
Hannes Uppman. Max-Sur-CSP on Two Elements. In Proceedings of the 18th International Conference on Principles and Practice of Constraint Programming (CP'12), volume 7514 of Lecture Notes in Computer Science, pages 38-54. Springer, 2012. URL: http://dx.doi.org/10.1007/978-3-642-33558-7_6.
http://dx.doi.org/10.1007/978-3-642-33558-7_6
Vijay V. Vazirani and Mihalis Yannakakis. Suboptimal Cuts: Their Enumeration, Weight and Number (Extended Abstract). In Proceedings of the 19th International Colloquium on Automata, Languages and Programming (ICALP'92), pages 366-377. Springer-Verlag, 1992. URL: http://dl.acm.org/citation.cfm?id=646246.684856.
http://dl.acm.org/citation.cfm?id=646246.684856
Narayan Vikas. Algorithms for partition of some class of graphs under compaction and vertex-compaction. Algorithmica, 67(2):180-206, 2013. URL: http://dx.doi.org/10.1007/s00453-012-9720-9.
http://dx.doi.org/10.1007/s00453-012-9720-9
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On the Expressive Power of Quasiperiodic SFT
In this paper we study the shifts, which are the shift-invariant and topologically closed sets of configurations over a finite alphabet in Z^d. The minimal shifts are those shifts in which all configurations contain exactly the same patterns. Two classes of shifts play a prominent role in symbolic dynamics, in language theory and in the theory of computability: the shifts of finite type (obtained by forbidding a finite number of finite patterns) and the effective shifts (obtained by forbidding a computably enumerable set of finite patterns).
We prove that every effective minimal shift can be represented as a factor of a projective subdynamics on a minimal shift of finite type in a bigger (by 1) dimension. This result transfers to the class of minimal shifts a theorem by M.Hochman known for the class of all effective shifts and thus answers an open question by E. Jeandel. We prove a similar result for quasiperiodic shifts and also show that there exists a quasiperiodic shift of finite type for which Kolmogorov complexity of all patterns of size n\times n is \Omega(n).
minimal SFT
tilings
quasiperiodicityIn this paper we study the shifts
which are the shift-invariant and topologically closed sets of configurations
5:1-5:14
Regular Paper
Bruno
Durand
Bruno Durand
Andrei
Romashchenko
Andrei Romashchenko
10.4230/LIPIcs.MFCS.2017.5
Nathalie Aubrun and Mathieu Sablik. Simulation of effective subshifts by two-dimensional subshifts of finite type. Acta Applicandae Mathematicae, 128(1):35-63, 2013.
Sergey V. Avgustinovich, Dmitrii G. Fon-Der-Flaass, and Anna E. Frid. Arithmetical complexity of infinite words. In 3rd Int. Colloq. on Words, Languages and Combinatorics, pages 51-62, 2003.
Alexis Ballier and Emmanuel Jeandel. Computing (or not) quasiperiodicity functions of tilings. In 2nd Symposium on Cellular Automata Journées Automates Cellulaires (JAC 2010), pages 54-64, 2010.
Bruno Durand. Tilings and quasiperiodicity. Theoretical Computer Science, 221(1):61-75, 1999.
Bruno Durand, Leonid Levin, and Alexander Shen. Complex tilings. The Journal of Symbolic Logic, 73(2):593-613, 2008.
Bruno Durand, Andrei Romashchenko, and Alexander Shen. Fixed-point tile sets and their applications. Journal of Computer and System Sciences, 78(3):731-764, 2012.
Brunourand Durand and Andrei Romashchenko. Quasiperiodicity and non-computability in tilings. In Proc. International Symposium on Mathematical Foundations of Computer Science (MFCS 2015), pages 218-230, 2015.
Peter Gács. Reliable computation with cellular automata. Journal of Computer and System Sciences, 32(1):15-78, 1986.
Gustav Hedlund and Marston Morse. Symbolic dynamics. American Journal of Mathematics, 60(4):815-866, 1938.
Michael Hochman. On the dynamics and recursive properties of multidimensional symbolic systems. Inventiones mathematicae, 176(1):131-167, 2009.
Michael Hochman and Pascal Vanier. A note on turing degree spectra of minimal shifts. In The 12th International Computer Science Symposium in Russia, pages 154-161, 2017.
Emmanuel Jeandel. Personal communication. private communication, 2015.
Emmanuel Jeandel and Pascal Vanier. Turing degrees of multidimensional sfts. Theoretical Computer Science, 505:81-92, 2013.
Andrey Rumyantsev and Maxim Ushakov. Forbidden substrings, kolmogorov complexity and almost periodic sequences. In Annual Symposium on Theoretical Aspects of Computer Science, pages 396-407, 2006.
Pavel V. Salimov. On uniform recurrence of a direct product. Discrete Mathematics and Theoretical Computer Science, 12(4), 2010.
Linda Brown Westrick. Seas of squares with sizes from a Π⁰₁ set. arXiv preprint arXiv:1609.07411, 2016.
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Parameterized Algorithms for Partitioning Graphs into Highly Connected Clusters
Clustering is a well-known and important problem with numerous applications. The graph-based model is one of the typical cluster models. In the graph model generally clusters are defined as cliques. However, such approach might be too restrictive as in some applications, not all objects from the same cluster must be connected. That is why different types of cliques relaxations often considered as clusters.
In our work, we consider a problem of partitioning graph into clusters and a problem of isolating cluster of a special type where by cluster we mean highly connected subgraph. Initially, such clusterization was proposed by Hartuv and Shamir. And their HCS clustering algorithm was extensively applied in practice. It was used to cluster cDNA fingerprints, to find complexes in protein-protein interaction data, to group protein sequences hierarchically into superfamily and family clusters, to find families of regulatory RNA structures. The HCS algorithm partitions graph in highly connected subgraphs. However, it is achieved by deletion of not necessarily the minimum number of edges. In our work, we try to minimize the number of edge deletions. We consider problems from the parameterized point of view where the main parameter is a number of allowed edge deletions. The presented algorithms significantly improve previous known running times for the Highly Connected Deletion (improved from \cOs\left(81^k\right) to \cOs\left(3^k\right)), Isolated Highly Connected Subgraph (from \cOs(4^k) to \cOs\left(k^{\cO\left(k^{\sfrac{2}{3}}\right)}\right) ), Seeded Highly Connected Edge Deletion (from \cOs\left(16^{k^{\sfrac{3}{4}}}\right) to \cOs\left(k^{\sqrt{k}}\right)) problems. Furthermore, we present a subexponential algorithm for Highly Connected Deletion problem if the number of clusters is bounded. Overall our work contains three subexponential algorithms which is unusual as very recently there were known very few problems admitting subexponential algorithms.
clustering
parameterized complexity
highly connected
6:1-6:14
Regular Paper
Ivan
Bliznets
Ivan Bliznets
Nikolai
Karpov
Nikolai Karpov
10.4230/LIPIcs.MFCS.2017.6
Balabhaskar Balasundaram, Sergiy Butenko, and Illya V Hicks. Clique relaxations in social network analysis: The maximum k-plex problem. Operations Research, 59(1):133-142, 2011.
Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Fourier meets möbius: fast subset convolution. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pages 67-74, 2007. URL: http://dx.doi.org/10.1145/1250790.1250801.
http://dx.doi.org/10.1145/1250790.1250801
Gary Chartrand. A graph-theoretic approach to a communications problem. SIAM Journal on Applied Mathematics, 14(4):778-781, 1966.
Wladimir de Azevedo Pribitkin. Simple upper bounds for partition functions. The Ramanujan Journal, 18(1):113-119, 2009. URL: http://dx.doi.org/10.1007/s11139-007-9022-z.
http://dx.doi.org/10.1007/s11139-007-9022-z
Fedor V. Fomin, Stefan Kratsch, Marcin Pilipczuk, Michal Pilipczuk, and Yngve Villanger. Tight bounds for parameterized complexity of cluster editing with a small number of clusters. J. Comput. Syst. Sci., 80(7):1430-1447, 2014. URL: http://dx.doi.org/10.1016/j.jcss.2014.04.015.
http://dx.doi.org/10.1016/j.jcss.2014.04.015
Fedor V. Fomin and Yngve Villanger. Treewidth computation and extremal combinatorics. Combinatorica, 32(3):289-308, 2012. URL: http://dx.doi.org/10.1007/s00493-012-2536-z.
http://dx.doi.org/10.1007/s00493-012-2536-z
Jiong Guo, Iyad A Kanj, Christian Komusiewicz, and Johannes Uhlmann. Editing graphs into disjoint unions of dense clusters. Algorithmica, 61(4):949-970, 2011.
Erez Hartuv, Armin O Schmitt, Jörg Lange, Sebastian Meier-Ewert, Hans Lehrach, and Ron Shamir. An algorithm for clustering cdna fingerprints. Genomics, 66(3):249-256, 2000.
Erez Hartuv and Ron Shamir. A clustering algorithm based on graph connectivity. Inf. Process. Lett., 76(4-6):175-181, 2000. URL: http://dx.doi.org/10.1016/S0020-0190(00)00142-3.
http://dx.doi.org/10.1016/S0020-0190(00)00142-3
Wayne Hayes, Kai Sun, and Nataša Pržulj. Graphlet-based measures are suitable for biological network comparison. Bioinformatics, 29(4):483-491, 2013.
Falk Hüffner, Christian Komusiewicz, Adrian Liebtrau, and Rolf Niedermeier. Partitioning biological networks into connected clusters with maximum edge coverage. IEEE/ACM Trans. Comput. Biology Bioinform., 11(3):455-467, 2014. URL: http://dx.doi.org/10.1109/TCBB.2013.177.
http://dx.doi.org/10.1109/TCBB.2013.177
Falk Hüffner, Christian Komusiewicz, and Manuel Sorge. Finding highly connected subgraphs. In SOFSEM 2015: Theory and Practice of Computer Science - 41st International Conference on Current Trends in Theory and Practice of Computer Science, Pec pod Sněžkou, Czech Republic, January 24-29, 2015. Proceedings, pages 254-265, 2015. URL: http://dx.doi.org/10.1007/978-3-662-46078-8_21.
http://dx.doi.org/10.1007/978-3-662-46078-8_21
Antje Krause, Jens Stoye, and Martin Vingron. Large scale hierarchical clustering of protein sequences. BMC bioinformatics, 6(1):15, 2005.
Hannes Moser, Rolf Niedermeier, and Manuel Sorge. Algorithms and experiments for clique relaxations—finding maximum s-plexes. In International Symposium on Experimental Algorithms, pages 233-244. Springer, 2009.
Brian J Parker, Ida Moltke, Adam Roth, Stefan Washietl, Jiayu Wen, Manolis Kellis, Ronald Breaker, and Jakob Skou Pedersen. New families of human regulatory rna structures identified by comparative analysis of vertebrate genomes. Genome research, 21(11):1929-1943, 2011.
Jeffrey Pattillo, Alexander Veremyev, Sergiy Butenko, and Vladimir Boginski. On the maximum quasi-clique problem. Discrete Applied Mathematics, 161(1):244-257, 2013.
Jeffrey Pattillo, Nataly Youssef, and Sergiy Butenko. On clique relaxation models in network analysis. European Journal of Operational Research, 226(1):9-18, 2013.
Jeffrey Pattillo, Nataly Youssef, and Sergiy Butenko. On clique relaxation models in network analysis. European Journal of Operational Research, 226(1):9-18, 2013. URL: http://dx.doi.org/10.1016/j.ejor.2012.10.021.
http://dx.doi.org/10.1016/j.ejor.2012.10.021
Alexander Schäfer. Exact algorithms for s-club finding and related problems. PhD thesis, Friedrich-Schiller-University Jena, 2009.
Shahram Shahinpour and Sergiy Butenko. Distance-based clique relaxations in networks: s-clique and s-club. In Models, algorithms, and technologies for network analysis, pages 149-174. Springer, 2013.
Haiyuan Yu, Alberto Paccanaro, Valery Trifonov, and Mark Gerstein. Predicting interactions in protein networks by completing defective cliques. Bioinformatics, 22(7):823-829, 2006.
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Hypercube LSH for Approximate near Neighbors
A celebrated technique for finding near neighbors for the angular distance involves using a set of random hyperplanes to partition the space into hash regions [Charikar, STOC 2002]. Experiments later showed that using a set of orthogonal hyperplanes, thereby partitioning the space into the Voronoi regions induced by a hypercube, leads to even better results [Terasawa and Tanaka, WADS 2007]. However, no theoretical explanation for this improvement was ever given, and it remained unclear how the resulting hypercube hash method scales in high dimensions.
In this work, we provide explicit asymptotics for the collision probabilities when using hypercubes to partition the space. For instance, two near-orthogonal vectors are expected to collide with probability (1/pi)^d in dimension d, compared to (1/2)^d when using random hyperplanes. Vectors at angle pi/3 collide with probability (sqrt[3]/pi)^d, compared to (2/3)^d for random hyperplanes, and near-parallel vectors collide with similar asymptotic probabilities in both cases.
For c-approximate nearest neighbor searching, this translates to a decrease in the exponent rho of locality-sensitive hashing (LSH) methods of a factor up to log2(pi) ~ 1.652 compared to hyperplane LSH. For c = 2, we obtain rho ~ 0.302 for hypercube LSH, improving upon the rho ~ 0.377 for hyperplane LSH. We further describe how to use hypercube LSH in practice, and we consider an example application in the area of lattice algorithms.
(approximate) near neighbors
locality-sensitive hashing
large deviations
dimensionality reduction
lattice algorithms
7:1-7:20
Regular Paper
Thijs
Laarhoven
Thijs Laarhoven
10.4230/LIPIcs.MFCS.2017.7
SVP challenge, 2015. URL: http://latticechallenge.org/svp-challenge/.
http://latticechallenge.org/svp-challenge/
Milton Abramowitz and Irene A. Stegun. Handbook of Mathematical Formulas. Dover Publications, 1972. URL: http://people.math.sfu.ca/~cbm/aands/toc.htm.
http://people.math.sfu.ca/~cbm/aands/toc.htm
Dimitris Achlioptas. Database-friendly random projections. In PODS, pages 274-281, 2001. URL: http://dx.doi.org/10.1145/375551.375608.
http://dx.doi.org/10.1145/375551.375608
Alexandr Andoni. Nearest Neighbor Search: the Old, the New, and the Impossible. PhD thesis, Massachusetts Institute of Technology, 2009. URL: http://hdl.handle.net/1721.1/55090.
http://hdl.handle.net/1721.1/55090
Alexandr Andoni, Piotr Indyk, Thijs Laarhoven, Ilya Razenshteyn, and Ludwig Schmidt. Practical and optimal LSH for angular distance. In NIPS, pages 1225-1233, 2015. URL: https://papers.nips.cc/paper/5893-practical-and-optimal-lsh-for-angular-distance.
https://papers.nips.cc/paper/5893-practical-and-optimal-lsh-for-angular-distance
Alexandr Andoni, Piotr Indyk, Huy Lê Nguyên, and Ilya Razenshteyn. Beyond locality-sensitive hashing. In SODA, pages 1018-1028, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.76.
http://dx.doi.org/10.1137/1.9781611973402.76
Alexandr Andoni, Thijs Laarhoven, Ilya Razenshteyn, and Erik Waingarten. Optimal hashing-based time-space trade-offs for approximate near neighbors. In SODA, pages 47-66, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.4.
http://dx.doi.org/10.1137/1.9781611974782.4
Alexandr Andoni and Ilya Razenshteyn. Optimal data-dependent hashing for approximate near neighbors. In STOC, pages 793-801, 2015. URL: http://dx.doi.org/10.1145/2746539.2746553.
http://dx.doi.org/10.1145/2746539.2746553
Anja Becker, Léo Ducas, Nicolas Gama, and Thijs Laarhoven. New directions in nearest neighbor searching with applications to lattice sieving. In SODA, pages 10-24, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch2.
http://dx.doi.org/10.1137/1.9781611974331.ch2
Anja Becker and Thijs Laarhoven. Efficient (ideal) lattice sieving using cross-polytope LSH. In AFRICACRYPT, pages 3-23, 2016. URL: http://dx.doi.org/10.1007/978-3-319-31517-1_1.
http://dx.doi.org/10.1007/978-3-319-31517-1_1
Christopher M. Bishop. Pattern Recognition and Machine Learning (Information Science and Statistics). Springer-Verlag, 2006.
Moses S. Charikar. Similarity estimation techniques from rounding algorithms. In STOC, pages 380-388, 2002. URL: http://dx.doi.org/10.1145/509907.509965.
http://dx.doi.org/10.1145/509907.509965
Tobias Christiani. A framework for similarity search with space-time tradeoffs using locality-sensitive filtering. In SODA, pages 31-46, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.3.
http://dx.doi.org/10.1137/1.9781611974782.3
Amir Dembo and Ofer Zeitouni. Large deviations techniques and applications (2nd edition). Springer, 2010. URL: http://dx.doi.org/10.1007/978-3-642-03311-7.
http://dx.doi.org/10.1007/978-3-642-03311-7
Moshe Dubiner. Bucketing coding and information theory for the statistical high-dimensional nearest-neighbor problem. IEEE Transactions on Information Theory, 56(8):4166-4179, Aug 2010. URL: http://dx.doi.org/10.1109/TIT.2010.2050814.
http://dx.doi.org/10.1109/TIT.2010.2050814
Richard O. Duda, Peter E. Hart, and David G. Stork. Pattern Classification (2nd Edition). Wiley, 2000.
Kave Eshghi and Shyamsundar Rajaram. Locality sensitive hash functions based on concomitant rank order statistics. In KDD, pages 221-229, 2008. URL: http://dx.doi.org/10.1145/1401890.1401921.
http://dx.doi.org/10.1145/1401890.1401921
Piotr Indyk and Rajeev Motwani. Approximate nearest neighbors: Towards removing the curse of dimensionality. In STOC, pages 604-613, 1998. URL: http://dx.doi.org/10.1145/276698.276876.
http://dx.doi.org/10.1145/276698.276876
Tiefeng Jiang. How many entries of a typical orthogonal matrix can be approximated by independent normals? The Annals of Probability, 34(4):1497-1529, 2006. URL: http://dx.doi.org/10.1214/009117906000000205.
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William B. Johnson and Joram Lindenstrauss. Extensions of Lipschitz mappings into a Hilbert space. Contemporary Mathematics, 26(1):189-206, 1984. URL: http://dx.doi.org/10.1090/conm/026/737400.
http://dx.doi.org/10.1090/conm/026/737400
Christopher Kennedy and Rachel Ward. Fast cross-polytope locality-sensitive hashing. In ITCS, 2017. URL: https://arxiv.org/abs/1602.06922.
https://arxiv.org/abs/1602.06922
Thorsten Kleinjung. Private communication, 2014.
Thijs Laarhoven. Sieving for shortest vectors in lattices using angular locality-sensitive hashing. In CRYPTO, pages 3-22, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47989-6_1.
http://dx.doi.org/10.1007/978-3-662-47989-6_1
Thijs Laarhoven and Benne de Weger. Faster sieving for shortest lattice vectors using spherical locality-sensitive hashing. In LATINCRYPT, pages 101-118, 2015. URL: http://dx.doi.org/10.1007/978-3-319-22174-8_6.
http://dx.doi.org/10.1007/978-3-319-22174-8_6
Artur Mariano. Private communication., 2016.
Artur Mariano and Christian Bischof. Enhancing the scalability and memory usage of HashSieve on multi-core CPUs. In PDP, pages 545-552, 2016. URL: http://dx.doi.org/10.1109/PDP.2016.31.
http://dx.doi.org/10.1109/PDP.2016.31
Artur Mariano, Thijs Laarhoven, and Christian Bischof. Parallel (probable) lock-free HashSieve: a practical sieving algorithm for the SVP. In ICPP, pages 590-599, 2015. URL: https://eprint.iacr.org/2015/041.
https://eprint.iacr.org/2015/041
Artur Mariano, Thijs Laarhoven, and Christian Bischof. A parallel variant of LDSieve for the SVP on lattices. PDP, 2017.
Alexander May and Ilya Ozerov. On computing nearest neighbors with applications to decoding of binary linear codes. In EUROCRYPT, pages 203-228, 2015. URL: http://dx.doi.org/10.1007/978-3-662-46800-5_9.
http://dx.doi.org/10.1007/978-3-662-46800-5_9
Rajeev Motwani, Assaf Naor, and Rina Panigrahy. Lower bounds on locality sensitive hashing. SIAM Journal of Discrete Mathematics, 21(4):930-935, 2007. URL: http://dx.doi.org/10.1137/050646858.
http://dx.doi.org/10.1137/050646858
Phong Q. Nguyên and Thomas Vidick. Sieve algorithms for the shortest vector problem are practical. Journal of Mathematical Cryptology, 2(2):181-207, 2008. URL: http://dx.doi.org/10.1515/JMC.2008.009.
http://dx.doi.org/10.1515/JMC.2008.009
Ryan O'Donnell, Yi Wu, and Yuan Zhou. Optimal lower bounds for locality sensitive hashing (except when q is tiny). In ICS, pages 276-283, 2011. URL: http://conference.itcs.tsinghua.edu.cn/ICS2011/content/papers/2.html.
http://conference.itcs.tsinghua.edu.cn/ICS2011/content/papers/2.html
Ludwig Schmidt, Matthew Sharifi, and Ignacio Lopez-Moreno. Large-scale speaker identification. In ICASSP, pages 1650-1654, 2014. URL: http://dx.doi.org/10.1109/ICASSP.2014.6853878.
http://dx.doi.org/10.1109/ICASSP.2014.6853878
Gregory Shakhnarovich, Trevor Darrell, and Piotr Indyk. Nearest-Neighbor Methods in Learning and Vision: Theory and Practice. MIT Press, 2005. URL: http://ttic.uchicago.edu/~gregory/annbook/book.html.
http://ttic.uchicago.edu/~gregory/annbook/book.html
Narayanan Sundaram, Aizana Turmukhametova, Nadathur Satish, Todd Mostak, Piotr Indyk, Samuel Madden, and Pradeep Dubey. Streaming similarity search over one billion tweets using parallel locality-sensitive hashing. VLDB, 6(14):1930-1941, 2013. URL: http://dx.doi.org/10.14778/2556549.2556574.
http://dx.doi.org/10.14778/2556549.2556574
Kengo Terasawa and Yuzuru Tanaka. Spherical LSH for approximate nearest neighbor search on unit hypersphere. In WADS, pages 27-38, 2007. URL: http://dx.doi.org/10.1007/978-3-540-73951-7_4.
http://dx.doi.org/10.1007/978-3-540-73951-7_4
Kengo Terasawa and Yuzuru Tanaka. Approximate nearest neighbor search for a dataset of normalized vectors. In IEICE Transactions on Information and Systems, volume 92, pages 1609-1619, 2009. URL: http://search.ieice.org/bin/summary.php?id=e92-d_9_1609.
http://search.ieice.org/bin/summary.php?id=e92-d_9_1609
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Generalized Predecessor Existence Problems for Boolean Finite Dynamical Systems
A Boolean Finite Synchronous Dynamical System (BFDS, for short) consists of a finite number of objects that each maintains a boolean state, where after individually receiving state assignments, the objects update their state with respect to object-specific time-independent boolean functions synchronously in discrete time steps.
The present paper studies the computational complexity of determining, given a boolean finite synchronous dynamical system,
a configuration, which is a boolean vector representing the states
of the objects, and a positive integer t, whether there exists another configuration from which the given configuration can be reached in t steps. It was previously shown that this problem, which we call the t-Predecessor Problem, is NP-complete even for t = 1
if the update function of an object is either the conjunction of
arbitrary fan-in or the disjunction of arbitrary fan-in.
This paper studies the computational complexity of the t-Predecessor Problem for a variety of sets of permissible update functions as well as for polynomially bounded t. It also studies the t-Garden-Of-Eden Problem, a variant of the t-Predecessor Problem that asks whether a configuration has a t-predecessor, which itself has no predecessor. The paper obtains complexity theoretical characterizations of all but one of these problems.
Computational complexity
dynamical systems
Garden of Eden
predecessor
8:1-8:13
Regular Paper
Akinori
Kawachi
Akinori Kawachi
Mitsunori
Ogihara
Mitsunori Ogihara
Kei
Uchizawa
Kei Uchizawa
10.4230/LIPIcs.MFCS.2017.8
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https://creativecommons.org/licenses/by/3.0/legalcode
Dividing Splittable Goods Evenly and With Limited Fragmentation
A splittable good provided in n pieces shall be divided as evenly as possible among m agents, where every agent can take shares of at most F pieces. We call F the fragmentation. For F=1 we can solve the max-min and min-max problems in linear time. The case F=2 has neat formulations and structural characterizations in terms of weighted graphs. Here we focus on perfectly balanced solutions. While the problem is strongly NP-hard in general, it can be solved in linear time if m>=n-1, and a solution always exists in this case. Moreover, case F=2 is fixed-parameter tractable in the parameter 2m-n. The results also give rise to various open problems.
packing
load balancing
weighted graph
linear-time algorithm
parameterized algorithm
9:1-9:13
Regular Paper
Peter
Damaschke
Peter Damaschke
10.4230/LIPIcs.MFCS.2017.9
Manuel Blum, Robert W. Floyd, Vaughan R. Pratt, Ronald L. Rivest, and Robert Endre Tarjan. Time bounds for selection. J. Comput. Syst. Sci., 7(4):448-461, 1973. URL: http://dx.doi.org/10.1016/S0022-0000(73)80033-9.
http://dx.doi.org/10.1016/S0022-0000(73)80033-9
M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979.
Dorit S. Hochbaum and David B. Shmoys. A polynomial approximation scheme for scheduling on uniform processors: Using the dual approximation approach. SIAM J. Comput., 17(3):539-551, 1988. URL: http://dx.doi.org/10.1137/0217033.
http://dx.doi.org/10.1137/0217033
Ellis Horowitz and Sartaj Sahni. Exact and approximate algorithms for scheduling nonidentical processors. J. ACM, 23(2):317-327, 1976. URL: http://dx.doi.org/10.1145/321941.321951.
http://dx.doi.org/10.1145/321941.321951
Hans Kellerer, Ulrich Pferschy, and David Pisinger. Knapsack problems. Springer, 2004.
John Martinovic and Guntram Scheithauer. Integer rounding and modified integer rounding for the skiving stock problem. Discrete Optimization, 21:118-130, 2016. URL: http://dx.doi.org/10.1016/j.disopt.2016.06.004.
http://dx.doi.org/10.1016/j.disopt.2016.06.004
Rolf Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford Univ. Press, 2006.
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Small-Space LCE Data Structure with Constant-Time Queries
The longest common extension (LCE) problem is to preprocess a given string w of length n so that the length of the longest common prefix between suffixes of w that start at any two given positions is answered quickly. In this paper, we present a data structure of O(z \tau^2 + \frac{n}{\tau}) words of space which answers LCE queries in O(1) time and can be built in O(n \log \sigma) time, where 1 \leq \tau \leq \sqrt{n} is a parameter, z is the size of the Lempel-Ziv 77 factorization of w and \sigma is the alphabet size. The proposed LCE data structure not access the input string w when answering queries, and thus w can be deleted after preprocessing. On top of this main result, we obtain further results using (variants of) our LCE data structure, which include the following:
- For highly repetitive strings where the z\tau^2 term is dominated by \frac{n}{\tau}, we obtain a constant-time and sub-linear space LCE query data structure.
- Even when the input string is not well compressible via Lempel-Ziv 77 factorization, we still can obtain a constant-time and sub-linear space LCE data structure for suitable \tau and for \sigma \leq 2^{o(\log n)}.
- The time-space trade-off lower bounds for the LCE problem by Bille et al. [J. Discrete Algorithms, 25:42-50, 2014] and by Kosolobov [CoRR, abs/1611.02891, 2016] do not apply in some cases with our LCE data structure.
longest common extension
truncated suffix trees
t-covers
10:1-10:15
Regular Paper
Yuka
Tanimura
Yuka Tanimura
Takaaki
Nishimoto
Takaaki Nishimoto
Hideo
Bannai
Hideo Bannai
Shunsuke
Inenaga
Shunsuke Inenaga
Masayuki
Takeda
Masayuki Takeda
10.4230/LIPIcs.MFCS.2017.10
Hideo Bannai, Tomohiro I, Shunsuke Inenaga, Yuto Nakashima, Masayuki Takeda, and Kazuya Tsuruta. A new characterization of maximal repetitions by Lyndon trees. In Proc. SODA 2015, pages 562-571, 2015.
Hideo Bannai, Shunsuke Inenaga, and Dominik Köppl. Computing all distinct squares in linear time for integer alphabets. CoRR, abs/1610.03421, 2016.
Michael A. Bender and Martin Farach-Colton. The LCA problem revisited. In Proc. Latin 2000, pages 88-94, 2000.
Michael A. Bender and Martin Farach-Colton. The level ancestor problem simplified. Theor. Comput. Sci., 321(1):5-12, 2004.
Omer Berkman and Uzi Vishkin. Finding level-ancestors in trees. Journal of Computer and System Sciences, 48(2):214-230, 1994.
Philip Bille, Anders Roy Christiansen, Patrick Hagge Cording, and Inge Li Gørtz. Finger search in grammar-compressed strings. In Proc. FSTTCS 2016, pages 36:1-36:16, 2016.
Philip Bille, Inge Li Gørtz, Patrick Hagge Cording, Benjamin Sach, Hjalte Wedel Vildhøj, and Søren Vind. Fingerprints in compressed strings. J. Comput. Syst. Sci., 86:171-180, 2017.
Philip Bille, Inge Li Gørtz, Mathias Bæk Tejs Knudsen, Moshe Lewenstein, and Hjalte Wedel Vildhøj. Longest common extensions in sublinear space. In Proc. CPM 2015, pages 65-76, 2015.
Philip Bille, Inge Li Gørtz, Benjamin Sach, and Hjalte Wedel Vildhøj. Time-space trade-offs for longest common extensions. J. Discrete Algorithms, 25:42-50, 2014.
Gerth Stølting Brodal, Pooya Davoodi, and S. Srinivasa Rao. On space efficient two dimensional range minimum data structures. Algorithmica, 63(4):815-830, 2012.
Gerth Stølting Brodal, Rune B. Lyngsø, Christian N. S. Pedersen, and Jens Stoye. Finding maximal pairs with bounded gap. In Proc. CPM 1999, pages 134-149, 1999.
Stefan Burkhardt and Juha Kärkkäinen. Fast lightweight suffix array construction and checking. In Proc. CPM 2003, pages 55-69, 2003.
Bastien Cazaux, Thierry Lecroq, and Eric Rivals. Construction of a de Bruijn graph for assembly from a truncated suffix tree. In LATA 2015, pages 109-120, 2015.
Maxime Crochemore, Roman Kolpakov, and Gregory Kucherov. Optimal bounds for computing α-gapped repeats. In Proc. LATA 2016, pages 245-255, 2016.
Johannes Fischer, Travis Gagie, Pawel Gawrychowski, and Tomasz Kociumaka. Approximating LZ77 via small-space multiple-pattern matching. CoRR, abs/1504.06647, 2015.
Michael L. Fredman and Dan E. Willard. Surpassing the information theoretic bound with fusion trees. J. Comput. Syst. Sci., 47(3):424-436, 1993.
Zvi Galil and Raffaele Giancarlo. Improved string matching with k mismatches. ACM SIGACT News, 17:52-54, 1986.
Pawel Gawrychowski, Tomohiro I, Shunsuke Inenaga, Dominik Köppl, and Florin Manea. Efficiently finding all maximal α-gapped repeats. In Proc. STACS 2016, pages 39:1-39:14, 2016.
Pawel Gawrychowski, Tomasz Kociumaka, Wojciech Rytter, and Tomasz Walen. Faster longest common extension queries in strings over general alphabets. In Proc. CPM 2016, pages 5:1-5:13, 2016.
Sara Geizhals and Dina Sokol. Finding maximal 2-dimensional palindromes. In Proc. CPM 2016, pages 19:1-19:12, 2016.
Dan Gusfield. Algorithms on Strings, Trees, and Sequences. Cambridge University Press, 1997.
Dan Gusfield and Jens Stoye. Linear time algorithms for finding and representing all the tandem repeats in a string. J. Comput. Syst. Sci., 69(4):525-546, 2004.
Dov Harel and Robert Endre Tarjan. Fast algorithms for finding nearest common ancestors. SIAM Journal on Computing, 13(2):338-355, 1984.
Tomohiro I. Longest common extensions with recompression. In Proc. CPM 2017, 2017. To appear.
Shunsuke Inenaga. A faster longest common extension algorithm on compressed strings and its applications. In Proc. PSC 2015, pages 1-4, 2015.
Juha Kärkkäinen. Repetition-based text indexes. Ph.D. thesis, University of Helsinki, Department of Computer Science, 1999.
Juha Kärkkäinen, Peter Sanders, and Stefan Burkhardt. Linear work suffix array construction. J. ACM, 53(6):918-936, 2006.
Toru Kasai, Gunho Lee, Hiroki Arimura, Setsuo Arikawa, and Kunsoo Park. Linear-time longest-common-prefix computation in suffix arrays and its applications. In Proc. CPM 2001, pages 181-192, 2001.
Roman Kolpakov and Gregory Kucherov. Searching for gapped palindromes. Theor. Comput. Sci., 410(51):5365-5373, 2009.
Roman M. Kolpakov and Gregory Kucherov. Finding maximal repetitions in a word in linear time. In Proc. FOCS 1999, pages 596-604, 1999.
Roman M. Kolpakov and Gregory Kucherov. Finding repeats with fixed gap. In Proc. SPIRE 2000, pages 162-168, 2000.
Dominik Köppl and Kunihiko Sadakane. Lempel-Ziv computation in compressed space (LZ-CICS). In Proc. DCC 2016, pages 3-12, 2016.
Dmitry Kosolobov. Tight lower bounds for the longest common extension problem. CoRR, abs/1611.02891, 2016.
Gad M. Landau, Eugene W. Myers, and Jeanette P. Schmidt. Incremental string comparison. SIAM J. Comput., 27(2):557-582, 1998.
Gad M. Landau and Uzi Vishkin. Efficient string matching with k mismatches. Theor. Comput. Sci., 43:239-249, 1986.
Mamoru Maekawa. A square root N algorithm for mutual exclusion in decentralized systems. ACM Trans. Comput. Syst., 3(2):145-159, 1985.
Udi Manber and Gene Myers. Suffix arrays: A new method for on-line string searches. SIAM J. Comput., 22(5):935-948, 1993.
Joong Chae Na, Alberto Apostolico, Costas S. Iliopoulos, and Kunsoo Park. Truncated suffix trees and their application to data compression. Theor. Comput. Sci., 1-3(304):87-101, 2003. URL: http://dx.doi.org/10.1016/S0304-3975(03)00053-7.
http://dx.doi.org/10.1016/S0304-3975(03)00053-7
Shintaro Narisada, Diptarama, Kazuyuki Narisawa, Shunsuke Inenaga, and Ayumi Shinohara. Computing longest single-arm-gapped palindromes in a string. In Proc. SOFSEM 2017, pages 375-386, 2017.
Takaaki Nishimoto, Tomohiro I, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Dynamic index and LZ factorization in compressed space. In Proc. PSC 2016, pages 158-170, 2016.
Takaaki Nishimoto, Tomohiro I, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Fully dynamic data structure for LCE queries in compressed space. In MFCS 2016, pages 72:1-72:15, 2016.
Nicola Prezza. In-place longest common extensions. CoRR, abs/1608.05100, 2016.
Simon J. Puglisi and Andrew Turpin. Space-time tradeoffs for longest-common-prefix array computation. In Proc. ISAAC 2008, pages 124-135, 2008.
Wojciech Rytter. Application of Lempel-Ziv factorization to the approximation of grammar-based compression. Theor. Comput. Sci., 302(1-3):211-222, 2003.
Yuka Tanimura, Tomohiro I, Hideo Bannai, Shunsuke Inenaga, Simon J. Puglisi, and Masayuki Takeda. Deterministic sub-linear space LCE data structures with efficient construction. In Proc. CPM 2016, pages 1:1-1:10, 2016.
Luciana Vitale, Alvaro Martín, and Gadiel Seroussi. Space-efficient representation of truncated suffix trees, with applications to Markov order estimation. Theor. Comput. Sci., 595:34-45, 2015.
P. Weiner. Linear pattern-matching algorithms. In Proc. of 14th IEEE Ann. Symp. on Switching and Automata Theory, pages 1-11, 1973.
Jacob Ziv and Abraham Lempel. A universal algorithm for sequential data compression. IEEE Trans. Information Theory, 23(3):337-343, 1977.
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ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics
The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language - i.e. the ability to derive any true equation - is a crucial question. In the quest of a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles.
We introduce a generalised supplementarity - called cyclotomic supplementarity - which consists in merging n subdiagrams at once, when the n angles divide the circle into equal parts. We show that when n is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new axioms for diagrammatic quantum reasoning.
We exhibit another new simple axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics. We end up with a new axiomatisation of an extended ZX-Calculus, including an axiom schema for the cyclotomic supplementarity.
Categorical Quantum Mechanincs
ZX-Calculus
Completeness
Cyclotomic Supplmentarity
Clifford+T
11:1-11:13
Regular Paper
Emmanuel
Jeandel
Emmanuel Jeandel
Simon
Perdrix
Simon Perdrix
Renaud
Vilmart
Renaud Vilmart
Quanlong
Wang
Quanlong Wang
10.4230/LIPIcs.MFCS.2017.11
Miriam Backens. The zx-calculus is complete for stabilizer quantum mechanics. New Journal of Physics, 16(9):093021, 2014. URL: http://dx.doi.org/10.1088/1367-2630/16/9/093021.
http://dx.doi.org/10.1088/1367-2630/16/9/093021
Miriam Backens. The zx-calculus is complete for the single-qubit clifford+t group. Electronic Proceedings in Theoretical Computer Science, 2014. URL: http://dx.doi.org/10.4204/EPTCS.172.21.
http://dx.doi.org/10.4204/EPTCS.172.21
Miriam Backens. Making the stabilizer zx-calculus complete for scalars. Electronic Proceedings in Theoretical Computer Science, 2015. URL: http://dx.doi.org/10.4204/EPTCS.195.2.
http://dx.doi.org/10.4204/EPTCS.195.2
Miriam Backens and Ali Nabi Duman. A complete graphical calculus for spekkens' toy bit theory. Foundations of Physics, pages 1-34, 2014. URL: http://dx.doi.org/10.1007/s10701-015-9957-7.
http://dx.doi.org/10.1007/s10701-015-9957-7
Miriam Backens, Simon Perdrix, and Quanlong Wang. A simplified stabilizer zx-calculus. Electronic Proceedings in Theoretical Computer Science, 2016. URL: http://dx.doi.org/10.4204/EPTCS.236.1.
http://dx.doi.org/10.4204/EPTCS.236.1
Categorical quantum mechanics: Zx-completeness. URL: http://cqm.wikidot.com/zx-completeness.
http://cqm.wikidot.com/zx-completeness
Bob Coecke. Axiomatic description of mixed states from selinger’s cpm-construction. Electron. Notes Theor. Comput. Sci., 210:3-13, July 2008. URL: http://dx.doi.org/10.1016/j.entcs.2008.04.014.
http://dx.doi.org/10.1016/j.entcs.2008.04.014
Bob Coecke and Ross Duncan. Interacting quantum observables: categorical algebra and diagrammatics. New Journal of Physics, 13(4):043016, 2011. URL: http://dx.doi.org/10.1088/1367-2630/13/4/043016.
http://dx.doi.org/10.1088/1367-2630/13/4/043016
Bob Coecke and Bill Edwards. Three qubit entanglement within graphical z/x-calculus. Electronic Proceedings in Theoretical Computer Science, 52:22-33, 2011. URL: http://dx.doi.org/10.4204/EPTCS.52.3.
http://dx.doi.org/10.4204/EPTCS.52.3
Bob Coecke and Simon Perdrix. Environment and classical channels in categorical quantum mechanics. Logical Methods in Computer Science, Volume 8, Issue 4, November 2012. URL: http://dx.doi.org/10.2168/LMCS-8(4:14)2012.
http://dx.doi.org/10.2168/LMCS-8(4:14)2012
Ross Duncan and Simon Perdrix. Graphs states and the necessity of euler decomposition. Mathematical Theory and Computational Practice, 5635:167-177, 2009. URL: http://dx.doi.org/10.1007/978-3-642-03073-4.
http://dx.doi.org/10.1007/978-3-642-03073-4
Ross Duncan and Simon Perdrix. Rewriting measurement-based quantum computations with generalised flow. Lecture Notes in Computer Science, 6199:285-296, 2010. URL: http://dx.doi.org/10.1007/978-3-642-14162-1_24.
http://dx.doi.org/10.1007/978-3-642-14162-1_24
Ross Duncan and Simon Perdrix. Pivoting makes the zx-calculus complete for real stabilizers. Electronic Proceedings in Theoretical Computer Science, 2013. URL: http://dx.doi.org/10.4204/EPTCS.171.5.
http://dx.doi.org/10.4204/EPTCS.171.5
Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. A complete axiomatisation of the zx-calculus for clifford+ t quantum mechanics. arXiv preprint arXiv:1705.11151, 2017.
Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. Y-calculus: A language for real matrices derived from the zx-calculus. In Conference on Quantum Physics and Logics (QPL'17), 2017.
Simon Perdrix and Quanlong Wang. Supplementarity is necessary for quantum diagram reasoning. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016), volume 58 of Leibniz International Proceedings in Informatics (LIPIcs), pages 76:1-76:14, Krakow, Poland, August 2016. URL: http://dx.doi.org/10.4230/LIPIcs.MFCS.2016.76.
http://dx.doi.org/10.4230/LIPIcs.MFCS.2016.76
Christian Schröder de Witt and Vladimir Zamdzhiev. The zx-calculus is incomplete for quantum mechanics. Electronic Proceedings in Theoretical Computer Science, 2014. URL: http://dx.doi.org/10.4204/EPTCS.172.20.
http://dx.doi.org/10.4204/EPTCS.172.20
Peter Selinger. Finite dimensional hilbert spaces are complete for dagger compact closed categories. Logical Methods in Computer Science, 8(4):1-12, 2012. URL: http://dx.doi.org/10.2168/LMCS-8(3:6)2012.
http://dx.doi.org/10.2168/LMCS-8(3:6)2012
Peter Selinger. Quantum circuits of t-depth one. Phys. Rev. A, 87:042302, Apr 2013. URL: http://dx.doi.org/10.1103/PhysRevA.87.042302.
http://dx.doi.org/10.1103/PhysRevA.87.042302
Robert Spekkens. Evidence for the epistemic view of quantum states: A toy theory. Phys. Rev. A, 75:032110, Mar 2007. URL: http://dx.doi.org/10.1103/PhysRevA.75.032110.
http://dx.doi.org/10.1103/PhysRevA.75.032110
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Counting Problems for Parikh Images
Given finite-state automata (or context-free grammars) A,B over the same alphabet and a Parikh vector p, we study the complexity of deciding whether the number of words in the language of A with Parikh image p is greater than the number of such words in the language of B. Recently, this problem turned out to be tightly related to the cost problem for weighted Markov chains. We classify the complexity according to whether A and B are deterministic, the size of the alphabet, and the encoding of p (binary or unary).
Parikh images
finite automata
counting problems
12:1-12:13
Regular Paper
Christoph
Haase
Christoph Haase
Stefan
Kiefer
Stefan Kiefer
Markus
Lohrey
Markus Lohrey
10.4230/LIPIcs.MFCS.2017.12
E. Allender and K. W. Wagner. Counting hierarchies: Polynomial time and constant depth circuits. Bulletin of the EATCS, 40:182-194, 1990.
A. Bertoni, M. Goldwurm, and N. Sabadini. The complexity of computing the number of strings of given length in context-free languages. Theor. Comput. Sci., 86(2):325-342, 1991.
E. Galby, J. Ouaknine, and J. Worrell. On matrix powering in low dimensions. In Proc. STACS 2015, volume 30 of LIPIcs, pages 329-340, 2015.
F. Green, J. Köbler, K. W. Regan, T. Schwentick, and J.Torán. The power of the middle bit of a #p function. Journal of Computer and System Sciences, 50(3):456-467, 1995.
C. Haase and S. Kiefer. The odds of staying on budget. In Proc. ICALP 2015, Part II, volume 9135 of LNCS, pages 234-246. Springer, 2015.
C. Haase, S. Kiefer, and M. Lohrey. Efficient quantile computation in markov chains via counting problems for parikh images. CoRR, abs/1601.04661, 2016. URL: http://arxiv.org/abs/1601.04661.
http://arxiv.org/abs/1601.04661
C. Haase, S. Kiefer, and M. Lohrey. Computing quantiles in Markov chains with multi-dimensional costs. In Proc. LICS 2017. IEEE, 2017. To appear.
H. B. Hunt III, D. J. Rosenkrantz, and T. G. Szymanski. On the equivalence, containment, and covering problems for the regular and context-free languages. J. Comput. Syst. Sci., 12(2):222-268, 1976.
E. Kopczyński. Complexity of problems of commutative grammars. Log. Meth. Comput. Sci., 11(1), 2015.
D. Kuske and M. Lohrey. First-order and counting theories of omega-automatic structures. J. Symbolic Logic, 73:129-150, 2008.
R. E. Ladner. Polynomial space counting problems. SIAM J. Comput., 18(6):1087-1097, 1989.
M. Mahajan and V. Vinay. A combinatorial algorithm for the determinant. In Proc. SODA 1997, pages 730-738. ACM/SIAM, 1997.
C. H. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.
Z. Sawa. Efficient construction of semilinear representations of languages accepted by unary nondeterministic finite automata. Fundam. Inform., 123(1):97-106, 2013.
L. J. Stockmeyer and A. R. Meyer. Word problems requiring exponential time (preliminary report). In Proc. STOC 1973, pages 1-9. ACM, 1973.
S. Toda. PP is as hard as the polynomial-time hierarchy. SIAM J. Comput., 20(5):865-877, 1991.
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Communication Complexity of Pairs of Graph Families with Applications
Given a graph G and a pair (\mathcal{F}_1,\mathcal{F}_2) of graph families, the function {\sf GDISJ}_{G,{\cal F}_1,{\cal F}_2} takes as input, two induced subgraphs G_1 and G_2 of G, such that G_1 \in \mathcal{F}_1 and G_2 \in \mathcal{F}_2 and returns 1 if V(G_1)\cap V(G_2)=\emptyset and 0 otherwise. We study the communication complexity of this problem in the two-party model. In particular, we look at pairs of hereditary graph families. We show that the communication complexity of this function, when the two graph families are hereditary, is sublinear if and only if there are finitely many graphs in the intersection of these two families. Then, using concepts from parameterized complexity, we
obtain nuanced upper bounds on the communication complexity of GDISJ_G,\cal F_1,\cal F_2. A concept related to communication protocols is that of a (\mathcal{F}_1,\mathcal{F}_2)-separating family of a graph G. A collection \mathcal{F} of subsets of V(G) is
called a (\mathcal{F}_1,\mathcal{F}_2)-separating family} for G, if for any two vertex disjoint induced subgraphs G_1\in \mathcal{F}_1,G_2\in \mathcal{F}_2, there is a set F \in \mathcal{F} with V(G_1) \subseteq F and V(G_2) \cap F = \emptyset.
Given a graph G on n vertices, for any pair (\mathcal{F}_1,\mathcal{F}_2) of hereditary graph families with sublinear communication complexity for GDISJ_G,\cal F_1,\cal F_2, we give an enumeration algorithm that finds a subexponential sized (\mathcal{F}_1,\mathcal{F}_2)-separating
family. In fact, we give an enumeration algorithm that finds a 2^{o(k)}n^{\Oh(1)} sized (\mathcal{F}_1,\mathcal{F}_2)-separating family; where k denotes the size of a minimum sized set S of vertices such that V(G)\setminus S has a bipartition (V_1,V_2) with G[V_1] \in {\cal F}_1 and G[V_2]\in {\cal F}_2. We exhibit a wide range of applications for these separating families, to obtain combinatorial bounds, enumeration algorithms as well as exact and FPT algorithms for several problems.
Communication Complexity
Separating Family
FPT algorithms
13:1-13:13
Regular Paper
Sudeshna
Kolay
Sudeshna Kolay
Fahad
Panolan
Fahad Panolan
Saket
Saurabh
Saket Saurabh
10.4230/LIPIcs.MFCS.2017.13
Kazuyuki Amano. Some improved bounds on communication complexity via new decomposition of cliques. Discrete Applied Mathematics, 166:249-254, 2014. URL: http://dx.doi.org/10.1016/j.dam.2013.09.015.
http://dx.doi.org/10.1016/j.dam.2013.09.015
Nicolas Bousquet, Aurélie Lagoutte, and Stéphan Thomassé. Clique versus independent set. Eur. J. Comb., 40:73-92, 2014. URL: http://dx.doi.org/10.1016/j.ejc.2014.02.003.
http://dx.doi.org/10.1016/j.ejc.2014.02.003
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
http://dx.doi.org/10.1007/978-3-319-21275-3
Marek Cygan and Marcin Pilipczuk. Split vertex deletion meets vertex cover: New fixed-parameter and exact exponential-time algorithms. Inf. Process. Lett., 113(5-6):179-182, 2013. URL: http://dx.doi.org/10.1016/j.ipl.2013.01.001.
http://dx.doi.org/10.1016/j.ipl.2013.01.001
R. Diestel. Graph Theory. Springer, Berlin, second ed., electronic edition, February 2000.
Tomas Feder, Pavol Hell, Sulamita Klein, and Rajeev Motwani. Complexity of graph partition problems. In Proceedings of the Thirty-first Annual ACM Symposium on Theory of Computing, STOC'99, pages 464-472, New York, NY, USA, 1999. ACM. URL: http://dx.doi.org/10.1145/301250.301373.
http://dx.doi.org/10.1145/301250.301373
Mika Göös. Lower bounds for clique vs. independent set. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 1066-1076, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.69.
http://dx.doi.org/10.1109/FOCS.2015.69
Mika Göös, T. S. Jayram, Toniann Pitassi, and Thomas Watson. Randomized communication vs. partition number. Electronic Colloquium on Computational Complexity (ECCC), 22:169, 2015. URL: http://eccc.hpi-web.de/report/2015/169.
http://eccc.hpi-web.de/report/2015/169
Mika Göös, Toniann Pitassi, and Thomas Watson. Deterministic communication vs. partition number. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 1077-1088, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.70.
http://dx.doi.org/10.1109/FOCS.2015.70
Vince Grolmusz and Gábor Tardos. A note on non-deterministic communication complexity with few witnesses. Theory Comput. Syst., 36(4):387-391, 2003. URL: http://dx.doi.org/10.1007/s00224-003-1158-7.
http://dx.doi.org/10.1007/s00224-003-1158-7
Hao Huang and Benny Sudakov. A counterexample to the Alon-Saks-Seymour conjecture and related problems. Combinatorica, 32(2):205-219, 2012. URL: http://dx.doi.org/10.1007/s00493-012-2746-4.
http://dx.doi.org/10.1007/s00493-012-2746-4
Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, New York, NY, USA, 1997.
László Lovász. Communication complexity: a survey. Paths, flows, and VLSI-layout, pages 235-265, 1990.
László Lovász. Stable sets and polynomials. Discrete Mathematics, 124(1-3):137-153, 1994. URL: http://dx.doi.org/10.1016/0012-365X(92)00057-X.
http://dx.doi.org/10.1016/0012-365X(92)00057-X
Manami Shigeta and Kazuyuki Amano. Ordered biclique partitions and communication complexity problems. Discrete Applied Mathematics, 184:248-252, 2015. URL: http://dx.doi.org/10.1016/j.dam.2014.10.029.
http://dx.doi.org/10.1016/j.dam.2014.10.029
Mihalis Yannakakis. Expressing combinatorial optimization problems by linear programs. Journal of Computer and System Sciences, 43(3):441-466, 1991. URL: http://dx.doi.org/10.1016/0022-0000(91)90024-Y.
http://dx.doi.org/10.1016/0022-0000(91)90024-Y
Andrew Chi-Chih Yao. Some complexity questions related to distributive computing(preliminary report). In Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, STOC'79, pages 209-213, New York, NY, USA, 1979. ACM. URL: http://dx.doi.org/10.1145/800135.804414.
http://dx.doi.org/10.1145/800135.804414
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Monitor Logics for Quantitative Monitor Automata
We introduce a new logic called Monitor Logic and show that it is expressively equivalent to Quantitative Monitor Automata.
Quantitative Monitor Automata
Nested Weighted Automata
Monitor Logics
Weighted Logics
14:1-14:13
Regular Paper
Erik
Paul
Erik Paul
10.4230/LIPIcs.MFCS.2017.14
J. Richard Büchi. Weak second-order arithmetic and finite automata. Z. Math. Logik und Grundl. Math., 6:66-92, 1960.
Krishnendu Chatterjee, Laurent Doyen, and Thomas A. Henzinger. Quantitative languages. In Michael Kaminski and Simone Martini, editors, Proc. CSL, volume 5213 of LNCS, pages 385-400. Springer, 2008.
Krishnendu Chatterjee, Thomas A. Henzinger, and Jan Otop. Nested weighted automata. In Proc. LICS, pages 725-737, 2015.
Krishnendu Chatterjee, Thomas A. Henzinger, and Jan Otop. Quantitative monitor automata. In Xavier Rival, editor, Proc. SAS, volume 9837 of LNCS. Springer, 2016.
Manfred Droste and Stefan Dück. Weighted automata and logics on graphs. In Giuseppe F. Italiano, Giovanni Pighizzini, and Donald T. Sannella, editors, Proc. MFCS, volume 9234 of LNCS, pages 192-204. Springer, 2015.
Manfred Droste and Paul Gastin. Weighted automata and weighted logics. Theor. Comput. Sci., 380:69-86, 2007.
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Zoltán Ésik and Werner Kuich. A semiring-semimodule generalization of ω-regular languages II. Journal of Automata, Languages and Combinatorics, 10(2/3):243-264, 2005.
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Christian Mathissen. Weighted logics for nested words and algebraic formal power series. In Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz, editors, Proc. ICALP, volume 5126 of LNCS, pages 221-232. Springer, 2008.
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The Complexity of Quantum Disjointness
We introduce the communication problem QNDISJ, short for Quantum (Unique) Non-Disjointness, and study its complexity under different modes of communication complexity. The main motivation for the problem is that it is a candidate for the separation of the quantum communication complexity classes QMA and QCMA. The problem generalizes the Vector-in-Subspace and Non-Disjointness problems. We give tight bounds for the QMA, quantum, randomized communication complexities of the problem. We show polynomially related upper and lower bounds for the MA complexity. We also show an upper bound for QCMA protocols, and show that the bound is tight for a natural class of QCMA protocols for the problem. The latter lower bound is based on a geometric lemma, that states that every subset of the n-dimensional sphere of measure 2^-p must contain an ortho-normal set of points of size Omega(n/p).
We also study a "small-spaces" version of the problem, and give upper and lower bounds for its randomized complexity that show that the QNDISJ problem is harder than Non-disjointness for randomized protocols. Interestingly, for quantum modes the complexity depends only on the dimension of the smaller space, whereas for classical modes the dimension of the larger space matters.
Communication Complexity
Quantum Proof Systems
15:1-15:13
Regular Paper
Hartmut
Klauck
Hartmut Klauck
10.4230/LIPIcs.MFCS.2017.15
S. Aaronson. Qma/qpoly ⊆ pspace/poly: De-merlinizing quantum protocols. In Proceedings of 21st IEEE Conference on Computational Complexity, 2006.
S. Aaronson and A. Ambainis. Quantum search of spatial regions. In Proceedings of 44th IEEE FOCS, pages 200-209, 2003.
S. Aaronson and G. Kuperberg. Quantum versus classical proofs and advice. Theory of Computing, 3(1):129-157, 2007.
S. Aaronson and A. Wigderson. Algebrization: A New Barrier in Complexity Theory. ACM Transactions on Computation Theory, 1(1), 2009.
D. Aharonov and T. Naveh. Quantum np - a survey. quant-ph/0210077, 2002.
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M. Braverman, A. Garg, Young K.K., J. Mao, and D. Touchette. Near-optimal bounds on bounded-round quantum communication complexity of disjointness. In IEEE 56th Annual Symposium on Foundations of Computer Science, pages 773-791, 2015.
A. Chakrabarti, G. Cormode, A. McGregor, J. Thaler, and S. Venkatasubramanian. Verifiable stream computation and arthur-merlin communication. In 30th Conference on Computational Complexity, pages 217-243, 2015.
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M. Göös, T. Pitassi, and T. Watson. Zero-information protocols and unambiguity in arthur-merlin communication. Algorithmica, 76(3):684-719, 2016.
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H. Klauck. Rectangle size bounds and threshold covers in communication complexity. In 18th Annual IEEE Conference on Computational Complexity, pages 118-134, 2003.
H. Klauck. A strong direct product theorem for disjointness. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC, pages 77-86, 2010.
H. Klauck. On arthur merlin games in communication complexity. In Proceedings of the 26th Annual IEEE Conference on Computational Complexity, pages 189-199, 2011.
H. Klauck and S. Podder. Two Results about Quantum Messages. In Proceedings of MFCS, 2014.
I. Kremer. Quantum communication. Master’s thesis, Hebrew University, Computer Science Department, 1995.
E. Kushilevitz and N. Nisan. Communication Complexity. Cambridge University Press, 1997.
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R. Raz and A. Shpilka. On the power of quantum proofs. In 19th Annual IEEE Conference on Computational Complexity, pages 260-274, 2004.
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A. Razborov. Quantum communication complexity of symmetric predicates. Izvestiya of the Russian Academy of Sciences, mathematics, 67(1):159-176, 2003. quant-ph/0204025.
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R. de Wolf. Quantum communication and complexity. Theoretical Computer Science, 287(1):337-353, 2002.
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Smoothed and Average-Case Approximation Ratios of Mechanisms: Beyond the Worst-Case Analysis
The approximation ratio has become one of the dominant measures in mechanism design problems. In light of analysis of algorithms, we define the smoothed approximation ratio to compare the performance of the optimal mechanism and a truthful mechanism when the inputs are subject to random perturbations of the worst-case inputs, and define the average-case approximation ratio to compare the performance of these two mechanisms when the inputs follow a distribution. For the one-sided matching problem, Filos-Ratsikas et al. [2014] show that, amongst all truthful mechanisms, random priority achieves the tight approximation ratio bound of Theta(sqrt{n}). We prove that, despite of this worst-case bound, random priority has a constant smoothed approximation ratio. This is, to our limited knowledge, the first work that asymptotically differentiates the smoothed approximation ratio from the worst-case approximation ratio for mechanism design problems. For the average-case, we show that our approximation ratio can be improved to 1+e. These results partially explain why random priority has been successfully used in practice, although in the worst case the optimal social welfare is Theta(sqrt{n}) times of what random priority achieves.
These results also pave the way for further studies of smoothed and average-case analysis for approximate mechanism design problems, beyond the worst-case analysis.
mechanism design
approximation ratio
smoothed analysis
average-case analysis
16:1-16:15
Regular Paper
Xiaotie
Deng
Xiaotie Deng
Yansong
Gao
Yansong Gao
Jie
Zhang
Jie Zhang
10.4230/LIPIcs.MFCS.2017.16
Atila Abdulkadiroğlu and Tayfun Sönmez. Random serial dictatorship and the core from random endowments in house allocation problems. Econometrica, pages 689-701, 1998.
Atila Abdulkadiroğlu and Tayfun Sönmez. Matching Markets: Theory and Practice. Advances in Economics and Econometrics (Tenth World Congress), pages 3-47, 2013.
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Aaron Archer, Christos H. Papadimitriou, Kunal Talwar, and Éva Tardos. An approximate truthful mechanism for combinatorial auctions with single parameter agents. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 205-214, 2003.
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Cyril Banderier, René Beier, and Kurt Mehlhorn. Smoothed analysis of three combinatorial problems. In 28th International Symposium, Mathematical Foundations of Computer Science, MFCS, volume 2747 of Lecture Notes in Computer Science, pages 198-207. Springer, 2003.
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Luca Becchetti, Stefano Leonardi, Alberto Marchetti-Spaccamela, Guido Schäfer, and Tjark Vredeveld. Average case and smoothed competitive analysis of the multi-level feedback algorithm. In Algorithms for Optimization with Incomplete Information, pages 16-21, volume 05031 of Dagstuhl Seminar Proceedings, 2005.
Markus Bläser, Bodo Manthey, and B. V. Raghavendra Rao. Smoothed analysis of partitioning algorithms for euclidean functionals. Algorithmica, 66(2):397-418, 2013.
Avrim Blum and John Dunagan. Smoothed analysis of the perceptron algorithm for linear programming. In David Eppstein, editor, Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 905-914, 2002.
Anna Bogomolnaia and Hervé Moulin. A New Solution to the Random Assignment Problem. Journal of Economic Theory, 100:295-328, 2001.
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Endre Boros, Khaled M. Elbassioni, Mahmoud Fouz, Vladimir Gurvich, Kazuhisa Makino, and Bodo Manthey. Stochastic mean payoff games: Smoothed analysis and approximation schemes. In Automata, Languages and Programming - 38th International Colloquium, ICALP Part I, volume 6755 of Lecture Notes in Computer Science, pages 147-158. Springer, 2011.
Shuchi Chawla and Balasubramanian Sivan. Bayesian algorithmic mechanism design. SIGecom Exchanges, 13(1):5-49, 2014.
Xi Chen, Xiaotie Deng, and Shang-Hua Teng. Settling the complexity of computing two-player nash equilibria. J. ACM, 56(3), 2009. URL: http://dx.doi.org/10.1145/1516512.1516516.
http://dx.doi.org/10.1145/1516512.1516516
George Christodoulou, Elias Koutsoupias, and Angelina Vidali. A lower bound for scheduling mechanisms. Algorithmica, 55(4):729-740, 2009.
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms (3. ed.). MIT Press, 2009.
Shaddin Dughmi and Arpita Ghosh. Truthful assignment without money. In ACM Conference on Electronic Commerce, pages 325-334, 2010.
Matthias Englert, Heiko Röglin, and Berthold Vöcking. Worst case and probabilistic analysis of the 2-opt algorithm for the TSP. Algorithmica, 68(1):190-264, 2014.
Aris Filos-Ratsikas, Søren Kristoffer Stiil Frederiksen, and Jie Zhang. Social welfare in one-sided matchings: Random priority and beyond. In 7th International Symposium on Algorithmic Game Theory (SAGT), pages 1-12, 2014.
Aris Filos-Ratsikas, Minming Li, Jie Zhang, and Qiang Zhang. Facility location with double-peaked preferences. In Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, pages 893-899. AAAI Press, 2015.
Mahmoud Fouz, Manfred Kufleitner, Bodo Manthey, and Nima Zeini Jahromi. On smoothed analysis of quicksort and hoare’s find. Algorithmica, 62(3-4):879-905, 2012.
M. Guo and V. Conitzer. Strategy-proof allocation of multiple items between two agents without payments or priors. In Ninth International Joint Conference on Autonomous Agents and Multi Agent Systems (AAMAS), volume 10, pages 881-888, 2010.
Jason D. Hartline and Brendan Lucier. Bayesian algorithmic mechanism design. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, pages 301-310, 2010.
Jason D. Hartline and Tim Roughgarden. Simple versus optimal mechanisms. In Proceedings 10th ACM Conference on Electronic Commerce, ACM-EC, pages 225-234. ACM, 2009.
Aanund Hylland and Richard Zeckhauser. The Efficient Allocation of Individuals to Positions. The Journal of Political Economy, 87(2):293-314, 1979.
Elias Koutsoupias. Scheduling without payments. Theory Comput. Syst., 54(3):375-387, 2014.
Elias Koutsoupias and Angelina Vidali. A lower bound of 1+ϕ for truthful scheduling mechanisms. In the 32nd International Symposium, Mathematical Foundations of Computer Science, MFCS, volume 4708 of Lecture Notes in Computer Science, pages 454-464. Springer, 2007.
Daniel J. Lehmann, Liadan O'Callaghan, and Yoav Shoham. Truth revelation in approximately efficient combinatorial auctions. J. ACM, 49(5):577-602, 2002.
Bodo Manthey and Rüdiger Reischuk. Smoothed analysis of binary search trees. Theor. Comput. Sci., 378(3):292-315, 2007.
Bodo Manthey and Heiko Röglin. Smoothed analysis: Analysis of algorithms beyond worst case. it - Information Technology, 53(6):280-286, 2011.
Timo Mennle and Sven Seuken. An axiomatic approach to characterizing and relaxing strategyproofness of one-sided matching mechanisms. In Proceedings of the 15th ACM Conference on Economics and Computation, pages 37-38, 2014.
Ahuva Mu'alem and Noam Nisan. Truthful approximation mechanisms for restricted combinatorial auctions. Games and Economic Behavior, 64(2):612-631, 2008.
Noam Nisan and Amir Ronen. Algorithmic mechanism design. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, STOC, pages 129-140, 1999.
Noam Nisan and Amir Ronen. Algorithmic mechanism design. Games and Economic Behavior, 35(1-2):166-196, 2001.
Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay V. Vazirani, editors. Algorithmic Game Thoery. Cambridge University Press, 2007.
Jay Sethuraman Parag A. Pathak. Lotteries in student assignment: An equivalence result. Theoretical Economics, 6:1-17, 2011.
Ariel D Procaccia and Moshe Tennenholtz. Approximate mechanism design without money. In Proceedings of the 10th ACM Conference on Electronic Commerce, pages 177-186. ACM, 2009.
Arvind Sankar, Daniel A. Spielman, and Shang-Hua Teng. Smoothed analysis of the condition numbers and growth factors of matrices. SIAM J. Matrix Analysis Applications, 28(2):446-476, 2006.
Guido Schäfer and Naveen Sivadasan. Topology matters: Smoothed competitiveness of metrical task systems. Theor. Comput. Sci., 341(1-3):216-246, 2005.
Tayfun Sönmez and Utku Ünver. Matching, allocation and exchange of discrete resources. Handbook of Social Economics, 1A:781-852, 2011.
Daniel A. Spielman and Shang-Hua Teng. Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. In Proceedings on 33rd Annual ACM Symposium on Theory of Computing, STOC, pages 296-305. ACM, 2001.
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Lars-Gunnar Svensson. Strategy-proof allocation of indivisible goods. Social Choice and Welfare, 16(4):557-567, 1999.
Wojciech Szpankowsk. Average case analysis of algorithms. Chapman Hall CRC, 2010.
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Time Complexity of Constraint Satisfaction via Universal Algebra
The exponential-time hypothesis (ETH) states that 3-SAT is not solvable in subexponential time, i.e. not solvable in O(c^n) time for arbitrary c > 1, where n denotes the number of variables. Problems like k-SAT can be viewed as special cases of the constraint satisfaction problem (CSP), which is the problem of determining whether a set of constraints is satisfiable. In this paper we study the worst-case time complexity of NP-complete CSPs. Our main interest is in the CSP problem parameterized by a constraint language Gamma (CSP(Gamma)), and how the choice of Gamma affects the time complexity. It is believed that CSP(Gamma) is either tractable or NP-complete, and the algebraic CSP dichotomy conjecture gives a sharp delineation of these two classes based on algebraic properties of constraint languages. Under this conjecture and the ETH, we first rule out the existence of subexponential algorithms for finite domain NP-complete CSP(Gamma) problems. This result also extends to certain infinite-domain CSPs and structurally restricted CSP(Gamma) problems. We then begin a study of the complexity of NP-complete CSPs where one is allowed to arbitrarily restrict the values of individual variables, which is a very well-studied subclass of CSPs. For such CSPs with finite domain D, we identify a relation SD such that (1) CSP({SD}) is NP-complete and (2) if CSP(Gamma) over D is NP-complete and solvable in O(c^n) time, then CSP({SD}) is solvable in O(c^n) time, too. Hence, the time complexity of CSP({SD}) is a lower bound for all CSPs of this particular kind. We also prove that the complexity of CSP({SD}) is decreasing when |D| increases, unless the ETH is false. This implies, for instance, that for every c>1 there exists a finite-domain Gamma such that CSP(Gamma) is NP complete and solvable in O(c^n) time.
Clone Theory
Universal Algebra
Constraint Satisfaction Problems
17:1-17:15
Regular Paper
Peter
Jonsson
Peter Jonsson
Victor
Lagerkvist
Victor Lagerkvist
Biman
Roy
Biman Roy
10.4230/LIPIcs.MFCS.2017.17
L. Barto. Constraint satisfaction problem and universal algebra. ACM SIGLOG News, 1(2):14-24, October 2014.
L. Barto and M. Pinsker. The algebraic dichotomy conjecture for infinite domain constraint satisfaction problems. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2016), pages 615-622, New York, NY, USA, 2016. ACM.
M. Behrisch, M. Hermann, S. Mengel, and G. Salzer. Give me another one! In Proceedings of the 26th International Symposium on Algorithms and Computation (ISAAC-2015), pages 664-676, 2015.
M. Behrisch, M. Hermann, S. Mengel, and G. Salzer. As close as it gets. In Proceedings of the 10th International Workshop on Algorithms and Computation (WALCOM-2016), pages 222-235, 2016.
M. Bodirsky. Complexity classification in infinite-domain constraint satisfaction. Mémoire d'habilitation à diriger des recherches, Université Diderot - Paris 7. Available at arXiv:1201.0856, 2012.
M. Bodirsky, P. Jonsson, and T. V. Pham. The complexity of phylogeny constraint satisfaction. In 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016, February 17-20, 2016, Orléans, France, pages 20:1-20:13, 2016.
M. Bodirsky and J. Kára. The complexity of temporal constraint satisfaction problems. Journal of the ACM, 57(2):9:1-9:41, 2010.
M. Bodirsky and M. Pinsker. Schaefer’s theorem for graphs. J. ACM, 62(3):19:1-19:52, June 2015.
V. G. Bodnarchuk, L. A. Kaluzhnin, V. N. Kotov, and B. A. Romov. Galois theory for Post algebras. I. Cybernetics, 5:243-252, 1969.
V. G. Bodnarchuk, L. A. Kaluzhnin, V. N. Kotov, and B. A. Romov. Galois theory for Post algebras. II. Cybernetics, 5:531-539, 1969.
E. Böhler, E. Hemaspaandra, S. Reith, and H. Vollmer. Equivalence and isomorphism for boolean constraint satisfaction. In In Proceedings of the 16th International Workshop on Computer Science Logic (CSL-2002), pages 412-426, Berlin, Heidelberg, 2002. Springer Berlin Heidelberg.
A. Bulatov. Complexity of conservative constraint satisfaction problems. ACM Transactions on Computational Logic, 12(4):24:1-24:66, July 2011.
A. Bulatov. A dichotomy theorem for nonuniform csps. CoRR, abs/1703.03021, 2017. URL: http://arxiv.org/abs/1703.03021.
http://arxiv.org/abs/1703.03021
A. Bulatov and A. Hedayaty. Counting problems and clones of functions. Multiple-Valued Logic and Soft Computing, 18(2):117-138, 2012.
A. Bulatov, P. Jeavons, and A. Krokhin. Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing, 34(3):720-742, March 2005. URL: http://dx.doi.org/10.1137/S0097539700376676.
http://dx.doi.org/10.1137/S0097539700376676
N. Creignou, U. Egly, and J. Schmidt. Complexity classifications for logic-based argumentation. ACM Transactions on Computational Logic (TOCL), 15(3):19:1-19:20, 2014.
R. de Haan, I. A. Kanj, and S. Szeider. On the subexponential-time complexity of CSP. Journal of Artificial Intelligence Research (JAIR), 52:203-234, 2015. URL: http://dx.doi.org/10.1613/jair.4540.
http://dx.doi.org/10.1613/jair.4540
T. Feder and M.Y. Vardi. The computational structure of monotone monadic SNP and constraint satisfaction: A study through datalog and group theory. SIAM Journal on Computing, 28(1):57-104, 1998.
D. Geiger. Closed systems of functions and predicates. Pacific Journal of Mathematics, 27(1):95-100, 1968.
M. Grohe. The structure of tractable constraint satisfaction problems. In Proceedings of the 31st International Symposium on Mathematical Foundations of Computer Science (MFCS 2006), pages 58-72, Berlin, Heidelberg, 2006. Springer Berlin Heidelberg.
L. Ham. Gap theorems for robust satisfiability: Boolean CSPs and beyond. To appear in Theoretical Computer Science, 2017. URL: http://dx.doi.org/10.1016/j.tcs.2017.03.006.
http://dx.doi.org/10.1016/j.tcs.2017.03.006
T. Hertli. 3-SAT faster and simpler - unique-SAT bounds for PPSZ hold in general. SIAM Journal on Computing, 43(2):718-729, 2014. URL: http://dx.doi.org/10.1137/120868177.
http://dx.doi.org/10.1137/120868177
R. Impagliazzo and R. Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001.
R. Impagliazzo, R. Paturi, and F. Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63:512-530, 2001.
P. Jeavons. On the algebraic structure of combinatorial problems. Theoretical Computer Science, 200:185-204, 1998.
P. Jeavons, D. Cohen, and M. Gyssens. Closure properties of constraints. Journal of the ACM, 44(4):527-548, July 1997. URL: http://dx.doi.org/10.1145/263867.263489.
http://dx.doi.org/10.1145/263867.263489
P. Jonsson, V. Lagerkvist, G. Nordh, and B. Zanuttini. Strong partial clones and the time complexity of SAT problems. Journal of Computer and System Sciences, 84:52-78, 2017.
P. Jonsson, V. Lagerkvist, and B. Roy. Time Complexity of Constraint Satisfaction via Universal Algebra. ArXiv e-prints, June 2017. URL: http://arxiv.org/abs/1706.05902.
http://arxiv.org/abs/1706.05902
E. Post. The two-valued iterative systems of mathematical logic. Annals of Mathematical Studies, 5:1-122, 1941.
A. Rafiey, J. Kinne, and T. Feder. Dichotomy for digraph homomorphism problems. CoRR, abs/1701.02409, 2017. URL: http://arxiv.org/abs/1701.02409.
http://arxiv.org/abs/1701.02409
B.A. Romov. The algebras of partial functions and their invariants. Cybernetics, 17(2):157-167, 1981.
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S. J. Russell and P. Norvig. Artificial Intelligence - A Modern Approach (3. internat. ed.). Pearson Education, 2010.
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H. Schnoor and I. Schnoor. Partial polymorphisms and constraint satisfaction problems. In N. Creignou, P. G. Kolaitis, and H. Vollmer, editors, Complexity of Constraints, volume 5250 of Lecture Notes in Computer Science, pages 229-254. Springer Berlin Heidelberg, 2008.
M. Wahlström. Algorithms, measures and upper bounds for satisfiability and related problems. PhD thesis, Linköping University, TCSLAB - Theoretical Computer Science Laboratory, The Institute of Technology, 2007.
D. Zhuk. The proof of csp dichotomy conjecture. CoRR, abs/1704.01914, 2017. URL: https://arxiv.org/abs/1704.01914.
https://arxiv.org/abs/1704.01914
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The Hardness of Solving Simple Word Equations
We investigate the class of regular-ordered word equations. In such equations, each variable occurs at most once in each side and the order of the variables occurring in both left and right hand sides is preserved (the variables can be, however, separated by potentially distinct constant factors). Surprisingly, we obtain that solving such simple equations, even when the sides contain exactly the same variables, is NP-hard. By considerations regarding the combinatorial structure of the minimal solutions of the more general quadratic equations we obtain that the satisfiability problem for regular-ordered equations is in NP. The complexity of solving such word equations under regular constraints is also settled. Finally, we show that a related class of simple word equations, that generalises one-variable equations, is in P.
Word Equations
Regular Patterns
Regular Constraints
18:1-18:14
Regular Paper
Joel D.
Day
Joel D. Day
Florin
Manea
Florin Manea
Dirk
Nowotka
Dirk Nowotka
10.4230/LIPIcs.MFCS.2017.18
R. Da̧browski and W. Plandowski. Solving two-variable word equations. In Proc. 31th International Colloquium on Automata, Languages and Programming, ICALP 2004, volume 3142 of Lecture Notes in Computer Science, pages 408-419, 2004.
V. Diekert, A. Jez, and M. Kufleitner. Solutions of word equations over partially commutative structures. In Proc. 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, volume 55 of Leibniz International Proceedings in Informatics (LIPIcs), pages 127:1-127:14, 2016.
V. Diekert and J. M. Robson. On quadratic word equations. In Proc. 16th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1999, volume 1563 of Lecture Notes in Computer Science, pages 217-226, 1999.
A. Ehrenfeucht and G. Rozenberg. Finding a homomorphism between two words is NP-complete. Information Processing Letters, 9:86-88, 1979.
H. Fernau, F. Manea, R. Mercaş, and M.L. Schmid. Pattern matching with variables: Fast algorithms and new hardness results. In Proc. 32nd Symposium on Theoretical Aspects of Computer Science, STACS 2015, volume 30 of Leibniz International Proceedings in Informatics (LIPIcs), pages 302-315, 2015.
H. Fernau and M. L. Schmid. Pattern matching with variables: A multivariate complexity analysis. Information and Computation, 242:287-305, 2015.
H. Fernau, M. L. Schmid, and Y. Villanger. On the parameterised complexity of string morphism problems. Theory of Computing Systems, 2015. http://dx.doi.org/10.1007/s00224-015-9635-3.
D. D. Freydenberger. A logic for document spanners. In Proc. 20th International Conference on Database Theory, ICDT 2017, Leibniz International Proceedings in Informatics (LIPIcs), 2017. To appear.
D. D. Freydenberger and M. Holldack. Document spanners: From expressive power to decision problems. In Proc. 19th International Conference on Database Theory, ICDT 2016, volume 48 of Leibniz International Proceedings in Informatics (LIPIcs), pages 17:1-17:17, 2016.
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J. Jaffar. Minimal and complete word unification. Journal of the ACM, 37(1):47-85, 1990.
A. Jez. Context unification is in PSPACE. In Proc. 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014, volume 8573 of Lecture Notes in Computer Science, pages 244-255. Springer, 2014.
A. Jeż. One-variable word equations in linear time. Algorithmica, 74:1-48, 2016.
A. Jeż. Recompression: A simple and powerful technique for word equations. Journal of the ACM, 63, 2016.
J. Karhumäki, F. Mignosi, and W. Plandowski. The expressibility of languages and relations by word equations. Journal of the ACM, 47:483-505, 2000.
A. Koscielski and L. Pacholski. Complexity of Makanin’s algorithm. Journal of the ACM, 43(4):670-684, 1996.
M. Lothaire. Algebraic Combinatorics on Words. Cambridge University Press, Cambridge, New York, 2002.
R. C. Lyndon. Equations in free groups. Transactions of the American Mathematical Society, 96:445-457, 1960.
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G. S. Makanin. The problem of solvability of equations in a free semigroup. Matematicheskii Sbornik, 103:147-236, 1977.
F. Manea, D. Nowotka, and M. L. Schmid. On the solvability problem for restricted classes of word equations. In Proc. 20th International Conference on Developments in Language Theory, DLT 2016, volume 9840 of Lecture Notes in Computer Science, pages 306-318. Springer, 2016.
W. Plandowski. An efficient algorithm for solving word equations. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, STOC 2006, pages 467-476, 2006.
W. Plandowski and W. Rytter. Application of Lempel-Ziv encodings to the solution of words equations. In Proc. 25th International Colloquium on Automata, Languages and Programming, ICALP'98, volume 1443 of Lecture Notes in Computer Science, pages 731-742. Springer, 1998.
D. Reidenbach and M. L. Schmid. Patterns with bounded treewidth. Information and Computation, 239:87-99, 2014.
K. U. Schulz. Word unification and transformation of generalized equations. Journal of Automated Reasoning, 11:149-184, 1995.
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Comparison of Max-Plus Automata and Joint Spectral Radius of Tropical Matrices
Weighted automata over the tropical semiring Zmax are closely related to finitely generated semigroups of matrices over Zmax. In this paper, we use results in automata theory to study two quantities associated with sets of matrices: the joint spectral radius and the ultimate rank. We prove that these two quantities are not computable over the tropical semiring, i.e. there is no algorithm that takes as input a finite set of matrices S and provides as output the joint spectral radius (resp. the ultimate rank) of S. On the other hand, we prove that the joint spectral radius is nevertheless approximable and we exhibit restricted cases in which the joint spectral radius and the ultimate rank are computable. To reach this aim, we study the problem of comparing functions computed by weighted automata over the tropical semiring. This problem is known to be undecidable, and we prove that it remains undecidable in some specific subclasses of automata.
max-plus automata
max-plus matrices
weighted automata
tropical semiring
joint spectral radius
ultimate rank
19:1-19:14
Regular Paper
Laure
Daviaud
Laure Daviaud
Pierre
Guillon
Pierre Guillon
Glenn
Merlet
Glenn Merlet
10.4230/LIPIcs.MFCS.2017.19
Shaull Almagor, Udi Boker, and Orna Kupferman. What’s decidable about weighted automata? In ATVA 2011, pages 482-491. Springer-Verlag, oct 2011.
Yu. A. Al'pin. Bounds for joint spectral radii of a set of nonnegative matrices. Mathematical Notes, 87(1):12-14, 2010. URL: http://dx.doi.org/10.1134/S0001434610010025.
http://dx.doi.org/10.1134/S0001434610010025
François Louis Baccelli, Geert Jan Olsder, Jean-Pierre Quadrat, and Guy Cohen. Synchronization and linearity. An algebra for discrete event systems. Chichester: Wiley, 1992.
Vincent D. Blondel, Stéphane Gaubert, and John N. Tsitsiklis. Approximating the spectral radius of sets of matrices in the max-algebra is np-hard. Automatic Control, IEEE Transactions on, 45(9):1762-1765, Sep 2000. URL: http://dx.doi.org/10.1109/9.880644.
http://dx.doi.org/10.1109/9.880644
Peter Butkovič. Max-linear systems. Theory and algorithms. London: Springer, 2010. URL: http://dx.doi.org/10.1007/978-1-84996-299-5.
http://dx.doi.org/10.1007/978-1-84996-299-5
Thomas Colcombet. On distance automata and regular cost function. Presented at the Dagstuhl seminar “Advances and Applications of Automata on Words and Trees”, 2010.
Thomas Colcombet and Laure Daviaud. Approximate comparison of distance automata. In Natacha Portier and Thomas Wilke, editors, STACS, volume 20 of LIPIcs, pages 574-585. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2013. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2013.574.
http://dx.doi.org/10.4230/LIPIcs.STACS.2013.574
Thomas Colcombet, Laure Daviaud, and Florian Zuleger. Size-change abstraction and max-plus automata. In Erzsébet Csuhaj-Varjú, Martin Dietzfelbinger, and Zoltán Ésik, editors, Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Budapest, Hungary, August 25-29, 2014. Proceedings, Part I, volume 8634 of Lecture Notes in Computer Science, pages 208-219. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44522-8_18.
http://dx.doi.org/10.1007/978-3-662-44522-8_18
Ludwig Elsner and P. van den Driessche. Bounds for the perron root using max eigenvalues. Linear Algebra and its Applications, 428(8):2000-2005, 2008. URL: http://dx.doi.org/10.1016/j.laa.2007.11.014.
http://dx.doi.org/10.1016/j.laa.2007.11.014
Stéphane Gaubert and Ricardo Katz. Reachability problems for products of matrices in semirings. International Journal of Algebra and Computation, 16(3):603-627, jun 2006. URL: http://arxiv.org/abs/math/0310028, http://arxiv.org/abs/0310028, URL: http://dx.doi.org/10.1142/S021819670600313X.
http://dx.doi.org/10.1142/S021819670600313X
Stéphane Gaubert and Jean Mairesse. Task resource models and (max,+) automata. In Idempotency (Bristol, 1994), volume 11 of Publ. Newton Inst., pages 133-144. Cambridge Univ. Press, Cambridge, 1998. URL: http://dx.doi.org/10.1017/CBO9780511662508.009.
http://dx.doi.org/10.1017/CBO9780511662508.009
Stéphane Gaubert. Performance evaluation of (max,+) automata. IEEE Trans. Automat. Control, 40(12):2014-2025, 1995. URL: http://dx.doi.org/10.1109/9.478227.
http://dx.doi.org/10.1109/9.478227
Stéphane Gaubert. On the Burnside problem for semigroups of matrices in the (max,+) algebra. Semigroup Forum, 52(1):271-294, 1996. URL: http://dx.doi.org/10.1007/BF02574104.
http://dx.doi.org/10.1007/BF02574104
Stéphane Gaubert and Jean Mairesse. Modeling and analysis of timed Petri nets using heaps of pieces. IEEE Trans. Automat. Control, 44(4):683-697, 1999. URL: http://dx.doi.org/10.1109/9.754807.
http://dx.doi.org/10.1109/9.754807
Pierre Guillon, Zur Izhakian, Jean Mairesse, and Glenn Merlet. The ultimate rank of semi-groups of tropical matrices. Journal of Algebra, 437:222-248, September 2015. URL: http://dx.doi.org/10.1016/j.jalgebra.2015.02.026.
http://dx.doi.org/10.1016/j.jalgebra.2015.02.026
Bernd Heidergott, Geert Jan Oldser, and Jacob van der Woude. Max plus at work. Modeling and analysis of synchronized systems: a course on max-plus algebra and its applications. Princeton, NJ: Princeton University Press, 2006.
James P. Jones. Universal Diophantine equation. J. Symbolic Logic, 47(3):549-571, 1982. URL: http://dx.doi.org/10.2307/2273588.
http://dx.doi.org/10.2307/2273588
Raphaël Jungers. The joint spectral radius, volume 385 of Lecture Notes in Control and Information Sciences. Springer-Verlag, Berlin, 2009. Theory and applications. URL: http://dx.doi.org/10.1007/978-3-540-95980-9.
http://dx.doi.org/10.1007/978-3-540-95980-9
Jui-Yi Kao, Narad Rampersad, and Jeffrey Shallit. On NFAs where all states are final, initial, or both. Theoretical Computer Science, 410(47):5010-5021, 2009. URL: http://dx.doi.org/10.1016/j.tcs.2009.07.049.
http://dx.doi.org/10.1016/j.tcs.2009.07.049
Daniel Krob. The equality problem for rational series with multiplicities in the tropical semiring is undecidable. In Automata, languages and programming (Vienna, 1992), volume 623 of Lecture Notes in Comput. Sci., pages 101-112. Springer, Berlin, 1992. URL: http://dx.doi.org/10.1007/3-540-55719-9_67.
http://dx.doi.org/10.1007/3-540-55719-9_67
Sylvain Lombardy and Jean Mairesse. Max-plus automaton. In Handbook of Automata. European Mathematical Society, To appear.
Glenn Merlet. Semigroup of matrices acting on the max-plus projective space. Linear Algebra and its Applications, 432(8):1923-1935, 2010. URL: http://dx.doi.org/10.1016/j.laa.2009.03.029.
http://dx.doi.org/10.1016/j.laa.2009.03.029
Marvin L. Minsky. Recursive unsolvability of Post’s problem of "tag" and other topics in theory of Turing machines. Ann. of Math. (2), 74:437-455, 1961.
Marvin L. Minsky. Computation: finite and infinite machines. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. Prentice-Hall Series in Automatic Computation.
Marcel-Paul Schützenberger. On the definition of a family of automata. Information and Control, 4:245-270, 1961.
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Binary Search in Graphs Revisited
In the classical binary search in a path the aim is to detect an unknown target by asking as few queries as possible, where each query reveals the direction to the target. This binary search algorithm has been recently extended by [Emamjomeh-Zadeh et al., STOC, 2016] to the problem of detecting a target in an arbitrary graph. Similarly to the classical case in the path, the algorithm of Emamjomeh-Zadeh et al. maintains a candidates’ set for the target, while each query asks an appropriately chosen vertex– the "median"–which minimises a potential \Phi among the vertices of the candidates' set. In this paper we address three open questions posed by Emamjomeh-Zadeh et al., namely (a) detecting a target when the query response is a direction to an approximately shortest path to the target, (b) detecting a target when querying a vertex that is an approximate median of the current candidates' set (instead of an exact one), and (c) detecting multiple targets, for which to the best of our knowledge no progress has been made so far. We resolve questions (a) and (b) by providing appropriate upper and lower bounds, as well as a new potential Γ that guarantees efficient target detection even by querying an approximate median each time. With respect to (c), we initiate a systematic study for detecting two targets in graphs and we identify sufficient conditions on the queries that allow for strong (linear) lower bounds and strong (polylogarithmic) upper bounds for the number of queries. All of our positive results can be derived using our new potential \Gamma that allows querying approximate medians.
binary search
graph
approximate query
probabilistic algorithm
lower bound.
20:1-20:14
Regular Paper
Argyrios
Deligkas
Argyrios Deligkas
George B.
Mertzios
George B. Mertzios
Paul G.
Spirakis
Paul G. Spirakis
10.4230/LIPIcs.MFCS.2017.20
Yosi Ben-Asher, Eitan Farchi, and Ilan Newman. Optimal search in trees. SIAM J. Comput., 28(6):2090-2102, 1999.
Michael Ben-Or and Avinatan Hassidim. The bayesian learner is optimal for noisy binary search (and pretty good for quantum as well). In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 221-230, 2008.
Lucas Boczkowski, Amos Korman, and Yoav Rodeh. Searching on trees with noisy memory. CoRR, abs/1611.01403, 2016.
Renato Carmo, Jair Donadelli, Yoshiharu Kohayakawa, and Eduardo Sany Laber. Searching in random partially ordered sets. Theor. Comput. Sci., 321(1):41-57, 2004.
Ferdinando Cicalese, Tobias Jacobs, Eduardo Sany Laber, and Marco Molinaro. On the complexity of searching in trees and partially ordered structures. Theor. Comput. Sci., 412(50):6879-6896, 2011.
Ferdinando Cicalese, Tobias Jacobs, Eduardo Sany Laber, and Caio Dias Valentim. The binary identification problem for weighted trees. Theor. Comput. Sci., 459:100-112, 2012.
Constantinos Daskalakis, Richard M. Karp, Elchanan Mossel, Samantha Riesenfeld, and Elad Verbin. Sorting and selection in posets. SIAM J. Comput., 40(3):597-622, 2011.
Dariusz Dereniowski. Edge ranking and searching in partial orders. Discrete Applied Mathematics, 156(13):2493-2500, 2008.
Dingzhu Du and Frank K. Hwang. Combinatorial Group Testing and its Applications. World Scientific, Singapore, 1993.
Ehsan Emamjomeh-Zadeh, David Kempe, and Vikrant Singhal. Deterministic and probabilistic binary search in graphs. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 519-532, 2016.
Uriel Feige, Prabhakar Raghavan, David Peleg, and Eli Upfal. Computing with noisy information. SIAM J. Comput., 23(5):1001-1018, 1994.
Ehud Fonio, Yael Heyman, Lucas Boczkowski, Aviram Gelblum, Adrian Kosowski, Amos Korman, and Ofer Feinerman. A locally-blazed ant trail achieves efficient collective navigation despite limited information. eLife, page 23 pages, 2016.
Ananth V. Iyer, H. Donald Ratliff, and Gopalakrishnan Vijayan. Optimal node ranking of trees. Inf. Process. Lett., 28(5):225-229, 1988.
C. Jordan. Sur les assemblages de lignes. Journal f"ur die reine und angewandte Mathematik, 70:195-190, 1869.
Eduardo Sany Laber, Ruy Luiz Milidiú, and Artur Alves Pessoa. On binary searching with non-uniform costs. In Proceedings of the Twelfth Annual Symposium on Discrete Algorithms, January 7-9, 2001, Washington, DC, USA., pages 855-864, 2001.
Tak Wah Lam and Fung Ling Yue. Edge ranking of graphs is hard. Discrete Applied Mathematics, 85(1):71-86, 1998.
Tak Wah Lam and Fung Ling Yue. Optimal edge ranking of trees in linear time. Algorithmica, 30(1):12-33, 2001.
Nathan Linial and Michael E. Saks. Searching ordered structures. J. Algorithms, 6(1):86-103, 1985.
Shay Mozes, Krzysztof Onak, and Oren Weimann. Finding an optimal tree searching strategy in linear time. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, San Francisco, California, USA, January 20-22, 2008, pages 1096-1105, 2008.
Nils J. Nilsson. Problem-Solving Methods in Artificial Intelligence. McGraw-Hill Pub. Co., 1971.
Robert Nowak. Noisy generalized binary search. In Y. Bengio, D. Schuurmans, J. D. Lafferty, C. K. I. Williams, and A. Culotta, editors, Advances in Neural Information Processing Systems 22, pages 1366-1374. Curran Associates, Inc., 2009.
Krzysztof Onak and Pawel Parys. Generalization of binary search: Searching in trees and forest-like partial orders. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), 21-24 October 2006, Berkeley, California, USA, Proceedings, pages 379-388, 2006.
Judea Pearl. Heuristics - intelligent search strategies for computer problem solving. Addison-Wesley series in artificial intelligence. Addison-Wesley, 1984.
Andrzej Pelc. Searching games with errors - fifty years of coping with liars. Theor. Comput. Sci., 270(1-2):71-109, 2002.
Alfred Renyi. On a problem in information theory. Magyar Tud. Akad. Mat. Kutato Int. Kozl, 6(B):505-516, 1961.
Alejandro A. Sch"affer. Optimal node ranking of trees in linear time. Information Processing Letters, 33(2):91-96, 1989.
Stanislaw Ulam. Adventures of a Mathematician. University of California Press, 1991.
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A Formal Semantics of Influence in Bayesian Reasoning
This paper proposes a formal definition of influence in Bayesian reasoning, based on the notions of state (as probability distribution), predicate, validity and conditioning. Our approach highlights how conditioning a joint entwined/entangled state with a predicate on one of its components has 'crossover' influence on the other components. We use the total variation metric on probability
distributions to quantitatively measure such influence. These insights are applied to give a rigorous explanation of the fundamental concept of d-separation in Bayesian networks.
probability distribution
Bayesian network
influence
21:1-21:14
Regular Paper
Bart
Jacobs
Bart Jacobs
Fabio
Zanasi
Fabio Zanasi
10.4230/LIPIcs.MFCS.2017.21
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https://creativecommons.org/licenses/by/3.0/legalcode
The Complexity of SORE-definability Problems
Single occurrence regular expressions (SORE) are a special kind of deterministic regular expressions, which are extensively used in the schema languages DTD and XSD for XML documents. In this paper, with motivations from the simplification of XML schemas, we consider the SORE-definability problem: Given a regular expression, decide whether it has an equivalent SORE. We investigate extensively the complexity of the SORE-definability problem: We consider both (standard) regular expressions and regular expressions with counting, and distinguish between the alphabets of size at least two and unary alphabets. In all cases, we obtain tight complexity bounds. In addition, we consider another variant of this problem, the bounded SORE-definability problem, which is to decide, given a regular expression E and a number M (encoded in unary or binary), whether there is an SORE, which is equivalent to E on the set of words of length at most M. We show that in several cases, there is an exponential decrease in the complexity when switching from the SORE-definability problem to its bounded variant.
Single occurrence regular expressions
Definability
Complexity
22:1-22:15
Regular Paper
Ping
Lu
Ping Lu
Zhilin
Wu
Zhilin Wu
Haiming
Chen
Haiming Chen
10.4230/LIPIcs.MFCS.2017.22
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
TC^0 Circuits for Algorithmic Problems in Nilpotent Groups
Recently, Macdonald et. al. showed that many algorithmic problems for finitely generated nilpotent groups including computation of normal forms, the subgroup membership problem, the conjugacy problem, and computation of subgroup presentations can be done in LOGSPACE. Here we follow their approach and show that all these problems are complete for the uniform circuit class TC^0 - uniformly for all r-generated nilpotent groups of class at most c for fixed r and c.
Moreover, if we allow a certain binary representation of the inputs, then the word problem and computation of normal forms is still in uniform TC^0, while all the other problems we examine are shown to be TC^0-Turing reducible to the problem of computing greatest common divisors and expressing them as linear combinations.
nilpotent groups
TC^0
abelian groups
word problem
conjugacy problem
subgroup membership problem
greatest common divisors
23:1-23:14
Regular Paper
Alexei
Myasnikov
Alexei Myasnikov
Armin
Weiß
Armin Weiß
10.4230/LIPIcs.MFCS.2017.23
Norman Blackburn. Conjugacy in nilpotent groups. Proceedings of the American Mathematical Society, 16(1):143-148, 1965.
William W. Boone. The Word Problem. Ann. of Math., 70(2):207-265, 1959.
Max Dehn. Ueber unendliche diskontinuierliche Gruppen. Math. Ann., 71:116-144, 1911.
Bettina Eick and Delaram Kahrobaei. Polycyclic groups: A new platform for cryptology? ArXiv Mathematics e-prints, 2004. URL: http://arxiv.org/abs/math/0411077.
http://arxiv.org/abs/math/0411077
Michael Elberfeld, Andreas Jakoby, and Till Tantau. Algorithmic meta theorems for circuit classes of constant and logarithmic depth. Electronic Colloquium on Computational Complexity (ECCC), 18:128, 2011. URL: http://eccc.hpi-web.de/report/2011/128.
http://eccc.hpi-web.de/report/2011/128
Alberd Garreta, Alexei Miasnikov, and Denis Ovchinnikov. Properties of random nilpotent groups. ArXiv e-prints, December 2016. URL: http://arxiv.org/abs/1612.01242.
http://arxiv.org/abs/1612.01242
Philip Hall. The Edmonton notes on nilpotent groups. Queen Mary College Mathematics Notes. Mathematics Department, Queen Mary College, London, 1969.
William Hesse. Division is in uniform TC⁰. In Fernando Orejas, Paul G. Spirakis, and Jan van Leeuwen, editors, ICALP, volume 2076 of Lecture Notes in Computer Science, pages 104-114. Springer, 2001. URL: http://dx.doi.org/10.1007/3-540-48224-5_9.
http://dx.doi.org/10.1007/3-540-48224-5_9
William Hesse, Eric Allender, and David A. Mix Barrington. Uniform constant-depth threshold circuits for division and iterated multiplication. Journal of Computer and System Sciences, 65:695-716, 2002.
Mikhail I. Kargapolov and Ju. I. Merzljakov. Fundamentals of the theory of groups, volume 62 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1979. Translated from the second Russian edition by Robert G. Burns.
Mikhail I. Kargapolov, Vladimir. N. Remeslennikov, N. S. Romanovskii, Vitaly A. Roman'kov, and V. A. Čurkin. Algorithmic questions for σ-powered groups. Algebra i Logika, 8:643-659, 1969.
Daniel König and Markus Lohrey. Evaluating matrix circuits. In Computing and combinatorics, volume 9198 of Lecture Notes in Comput. Sci., pages 235-248. Springer, Cham, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21398-9_19.
http://dx.doi.org/10.1007/978-3-319-21398-9_19
Klaus-Jörn Lange and Pierre McKenzie. On the complexity of free monoid morphisms. In Kyung-Yong Chwa and Oscar H. Ibarra, editors, Algorithms and Computation, 9th International Symposium, ISAAC'98, Taejon, Korea, December 14-16, 1998, Proceedings, volume 1533 of Lecture Notes in Computer Science, pages 247-256. Springer, 1998. URL: http://dx.doi.org/10.1007/3-540-49381-6_27.
http://dx.doi.org/10.1007/3-540-49381-6_27
Charles. R. Leedham-Green and Leonard H. Soicher. Symbolic collection using Deep Thought. LMS J. Comput. Math., 1:9-24 (electronic), 1998. URL: http://dx.doi.org/10.1112/S1461157000000127.
http://dx.doi.org/10.1112/S1461157000000127
Richard J. Lipton and Yechezkel Zalcstein. Word problems solvable in logspace. J. ACM, 24(3):522-526, July 1977. URL: http://dx.doi.org/10.1145/322017.322031.
http://dx.doi.org/10.1145/322017.322031
Jeremy MacDonald, Alexei G. Myasnikov, Andrey Nikolaev, and Svetla Vassileva. Logspace and compressed-word computations in nilpotent groups. CoRR, abs/1503.03888, 2015. URL: http://arxiv.org/abs/1503.03888.
http://arxiv.org/abs/1503.03888
Anatoly I. Mal'cev. On homomorphisms onto finite groups. Transl., Ser. 2, Am. Math. Soc., 119:67-79, 1983. Translation from Uch. Zap. Ivanov. Gos. Pedagog Inst. 18, 49-60 (1958).
Andrzej Mostowski. Computational algorithms for deciding some problems for nilpotent groups. Fundamenta Mathematicae, 59(2):137-152, 1966. URL: http://eudml.org/doc/213887.
http://eudml.org/doc/213887
Alexei Myasnikov, Andrey Nikolaev, and Alexander Ushakov. The Post correspondence problem in groups. J. Group Theory, 17(6):991-1008, 2014. URL: http://dx.doi.org/10.1515/jgth-2014-0022.
http://dx.doi.org/10.1515/jgth-2014-0022
Alexei Myasnikov, Andrey Nikolaev, and Alexander Ushakov. Non-commutative lattice problems. J. Group Theory, 19(3):455-475, 2016. URL: http://dx.doi.org/10.1515/jgth-2016-0506.
http://dx.doi.org/10.1515/jgth-2016-0506
Alexei G. Myasnikov and Armin Weiß. TC^0 circuits for algorithmic problems in nilpotent groups. CoRR, abs/1702.06616, 2017. URL: http://arxiv.org/abs/1702.06616.
http://arxiv.org/abs/1702.06616
Pyotr S. Novikov. On the algorithmic unsolvability of the word problem in group theory. Trudy Mat. Inst. Steklov, pages 1-143, 1955. In Russian.
David Robinson. Parallel Algorithms for Group Word Problems. PhD thesis, University of California, San Diego, 1993.
Charles C. Sims. Computation with finitely presented groups, volume 48 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1994. URL: http://dx.doi.org/10.1017/CBO9780511574702.
http://dx.doi.org/10.1017/CBO9780511574702
Heribert Vollmer. Introduction to Circuit Complexity. Springer, Berlin, 1999.
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Better Complexity Bounds for Cost Register Automata
Cost register automata (CRAs) are one-way finite automata whose transitions have the side effect that a register is set to the result of applying a state-dependent semiring operation to a pair of registers. Here it is shown that CRAs over the tropical semiring (N U {infinity},\min,+) can simulate polynomial time computation, proving along the way that a naturally defined width-k circuit value problem over the tropical semiring is P-complete.
Then the copyless variant of the CRA, requiring that semiring operations be applied to distinct registers, is shown no more powerful than NC^1 when the semiring is (Z,+,x) or (Gamma^*,max,concat). This relates questions left open in recent work on the complexity of CRA-computable functions to long-standing class separation conjectures in complexity theory, such as NC versus P and NC^1 versus GapNC^1.
computational complexity
cost registers
24:1-24:14
Regular Paper
Eric
Allender
Eric Allender
Andreas
Krebs
Andreas Krebs
Pierre
McKenzie
Pierre McKenzie
10.4230/LIPIcs.MFCS.2017.24
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Kernelization of the Subset General Position Problem in Geometry
In this paper, we consider variants of the Geometric Subset General Position problem. In defining this problem, a geometric subsystem is specified, like a subsystem of lines, hyperplanes or spheres. The input of the problem is a set of n points in \mathbb{R}^d and a positive integer k. The objective is to find a subset of at least k input points such that this subset is in general position with respect to the specified subsystem. For example, a set of points is in
general position with respect to a subsystem of hyperplanes in \mathbb{R}^d if no d+1 points lie on the same
hyperplane. In this paper, we study the Hyperplane Subset General Position problem under two parameterizations.
When parameterized by k then we exhibit a polynomial kernelization for the problem. When parameterized by h=n-k,
or the dual parameter, then we exhibit polynomial kernels which are also tight, under standard complexity theoretic
assumptions.
We can also exhibit similar kernelization results for d-Polynomial Subset General Position, where a vector space of polynomials
of degree at most d are specified as the underlying subsystem such that the size of the basis for this vector space is b. The objective is to find a set of at least k input points, or in the dual delete at most h = n-k points, such that no b+1 points lie on the same polynomial. Notice that this is a generalization of many well-studied geometric variants of the Set Cover problem, such as Circle Subset General Position. We also study general projective variants of these problems. These problems are also related to other geometric problems like Subset Delaunay Triangulation problem.
Incidence Geometry
Kernel Lower bounds
Hyperplanes
Bounded degree polynomials
25:1-25:13
Regular Paper
Jean-Daniel
Boissonnat
Jean-Daniel Boissonnat
Kunal
Dutta
Kunal Dutta
Arijit
Ghosh
Arijit Ghosh
Sudeshna
Kolay
Sudeshna Kolay
10.4230/LIPIcs.MFCS.2017.25
Pankaj K Agarwal and Micha Sharir. Arrangements and their applications. Handbook of Computational Geometry, pages 49-119, 2000.
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Satisfiable Tseitin Formulas Are Hard for Nondeterministic Read-Once Branching Programs
We consider satisfiable Tseitin formulas TS_{G,c} based on d-regular expanders G with the absolute value of the second largest eigenvalue less than d/3. We prove that any nondeterministic read-once branching program (1-NBP) representing TS_{G,c} has size 2^{\Omega(n)}, where n is the number of vertices in G. It extends the recent result by Itsykson at el. [STACS 2017] from OBDD to 1-NBP.
On the other hand it is easy to see that TS_{G,c} can be represented as a read-2 branching program (2-BP) of size O(n), as the negation of a nondeterministic read-once branching program (1-coNBP) of size O(n) and as a CNF formula of size O(n). Thus TS_{G,c} gives the best possible separations (up to a constant in the exponent) between
1-NBP and 2-BP, 1-NBP and 1-coNBP and between 1-NBP and CNF.
Tseitin formula
read-once branching program
expander
26:1-26:12
Regular Paper
Ludmila
Glinskih
Ludmila Glinskih
Dmitry
Itsykson
Dmitry Itsykson
10.4230/LIPIcs.MFCS.2017.26
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The Complexity of Quantified Constraints Using the Algebraic Formulation
Let A be an idempotent algebra on a finite domain. We combine results of Chen, Zhuk and Carvalho et al. to argue that if A satisfies the polynomially generated powers property (PGP), then QCSP(Inv(A)) is in NP. We then use the result of Zhuk to prove a converse, that if Inv(A) satisfies the exponentially generated powers property (EGP), then QCSP(Inv(A)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying the moral correctness of what we term the Chen Conjecture.
We examine in closer detail the situation for domains of size three. Over any finite domain, the only type of PGP that can occur is switchability. Switchability was introduced by Chen as a generalisation of the already-known Collapsibility. For three-element domain algebras A that are Switchable, we prove that for every finite subset Delta of Inv(A), Pol(Delta) is Collapsible. The significance of this is that, for QCSP on finite structures (over three-element domain), all QCSP tractability explained by Switchability is already explained by Collapsibility.
Finally, we present a three-element domain complexity classification vignette, using known as well as derived results.
Quantified Constraints
Computational Complexity
Universal Algebra
Constraint Satisfaction
27:1-27:14
Regular Paper
Catarina
Carvalho
Catarina Carvalho
Barnaby
Martin
Barnaby Martin
Dmitriy
Zhuk
Dmitriy Zhuk
10.4230/LIPIcs.MFCS.2017.27
Manuel Bodirsky and Hubie Chen. Quantified equality constraints. SIAM J. Comput., 39(8):3682-3699, 2010. URL: http://dx.doi.org/10.1137/080725209.
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Manuel Bodirsky and Jan Kára. The complexity of equality constraint languages. Theory of Computing Systems, 3(2):136-158, 2008. A conference version appeared in the proceedings of CSR'06.
Ferdinand Börner, Andrei A. Bulatov, Hubie Chen, Peter Jeavons, and Andrei A. Krokhin. The complexity of constraint satisfaction games and qcsp. Inf. Comput., 207(9):923-944, 2009. URL: http://dx.doi.org/10.1016/j.ic.2009.05.003.
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A. Bulatov, A. Krokhin, and P. G. Jeavons. Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing, 34:720-742, 2005.
Catarina Carvalho, Florent R. Madelaine, and Barnaby Martin. From complexity to algebra and back: digraph classes, collapsibility and the PGP. In 30th Annual IEEE Symposium on Logic in Computer Science (LICS), 2015.
Hubie Chen. The complexity of quantified constraint satisfaction: Collapsibility, sink algebras, and the three-element case. SIAM J. Comput., 37(5):1674-1701, 2008. URL: http://dx.doi.org/10.1137/060668572.
http://dx.doi.org/10.1137/060668572
Hubie Chen. Quantified constraint satisfaction and the polynomially generated powers property. Algebra universalis, 65(3):213-241, 2011. An extended abstract appeared in ICALP B 2008. URL: http://dx.doi.org/10.1007/s00012-011-0125-4.
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Hubie Chen. Meditations on quantified constraint satisfaction. In Logic and Program Semantics - Essays Dedicated to Dexter Kozen on the Occasion of His 60th Birthday, pages 35-49, 2012. URL: http://dx.doi.org/10.1007/978-3-642-29485-3_4.
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Barnaby Martin. On the chen conjecture regarding the complexity of qcsps. CoRR, abs/1607.03819, 2016. URL: http://arxiv.org/abs/1607.03819.
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Barnaby Martin and Dmitriy Zhuk. Switchability and collapsibility of gap algebras. CoRR, abs/1510.06298, 2015. URL: http://arxiv.org/abs/1510.06298.
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Induced Embeddings into Hamming Graphs
Let d be a positive integer. Can a given graph G be realized in R^d so that vertices are mapped to distinct points, two vertices being adjacent if and only if the corresponding points lie on a common line that is parallel to some axis? Graphs admitting such realizations have been studied in the literature for decades under different names. Peterson asked in [Discrete Appl. Math., 2003] about the complexity of the recognition problem. While the two-dimensional case corresponds to the class of line graphs of bipartite graphs and is well-understood, the complexity question has remained open for all higher dimensions.
In this paper, we answer this question. We establish the NP-completeness of the recognition problem for any fixed dimension, even in the class of bipartite graphs. To do this, we strengthen a characterization of induced subgraphs of 3-dimensional Hamming graphs due to Klavžar and Peterin. We complement the hardness result by showing that for some important classes of perfect graphs –including chordal graphs and distance-hereditary graphs– the minimum dimension of the Euclidean space in which the graph can be realized, or the impossibility of doing so, can be determined in linear time.
gridline graph
Hamming graph
induced embedding
NP-completeness
chordal graph
28:1-28:15
Regular Paper
Martin
Milanic
Martin Milanic
Peter
Mursic
Peter Mursic
Marcelo
Mydlarz
Marcelo Mydlarz
10.4230/LIPIcs.MFCS.2017.28
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Structured Connectivity Augmentation
We initiate the algorithmic study of the following "structured augmentation" question: is it possible to increase the connectivity of a given graph G by superposing it with another given graph H? More precisely, graph F is the superposition of G and H with respect to injective mapping \phi:V(H)->V(G) if every edge uv of F is either an edge of G, or \phi^{-1}(u)\phi^{-1}(v) is an edge of H. Thus F contains both G and H as subgraphs, and the edge set of F is the union of the edge sets of G and \phi(H). We consider the following optimization problem. Given graphs G, H, and a weight function \omega assigning non-negative weights to pairs of vertices of V(G), the task is to find \phi of minimum weight \omega(\phi)=\sum_{xy\in E(H)}\omega(\phi(x)\phi(y)) such that the edge connectivity of the superposition F of G and H with respect to \phi is higher than the edge connectivity of G. Our main result is the following ``dichotomy'' complexity classification. We say that a class of graphs C has bounded vertex-cover number, if there is a constant t depending on C only such that the vertex-cover number of every graph from C does not exceed t. We show that for every class of graphs C with bounded vertex-cover number, the problems of superposing into a connected graph F and to 2-edge connected graph F, are solvable in polynomial time when H\in C. On the other hand, for any hereditary class C with unbounded vertex-cover number, both problems are NP-hard when H\in C. For the unweighted variants of structured augmentation problems, i.e. the problems where the task is to identify whether there is a superposition of graphs of required connectivity, we provide necessary and sufficient combinatorial conditions on the existence of such superpositions. These conditions imply polynomial time algorithms solving the unweighted variants of the problems.
connectivity augmentation
graph superposition
complexity
29:1-29:13
Regular Paper
Fedor V.
Fomin
Fedor V. Fomin
Petr A.
Golovach
Petr A. Golovach
Dimitrios M.
Thilikos
Dimitrios M. Thilikos
10.4230/LIPIcs.MFCS.2017.29
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Combinatorial Properties and Recognition of Unit Square Visibility Graphs
Unit square (grid) visibility graphs (USV and USGV, resp.) are described by axis-parallel visibility between unit squares placed (on integer grid coordinates) in the plane. We investigate combinatorial properties of these graph classes and the hardness of variants of the recognition problem, i.e., the problem of representing USGV with fixed visibilities within small area and, for USV, the general recognition problem.
Visibility graphs
visibility layout
NP-completeness
exact algorithms
30:1-30:15
Regular Paper
Katrin
Casel
Katrin Casel
Henning
Fernau
Henning Fernau
Alexander
Grigoriev
Alexander Grigoriev
Markus L.
Schmid
Markus L. Schmid
Sue
Whitesides
Sue Whitesides
10.4230/LIPIcs.MFCS.2017.30
E. N. Argyriou, M. A. Bekos, and A. Symvonis. Maximizing the total resolution of graphs. In U. Brandes and S. Cornelsen, editors, Graph Drawing, GD 2010, volume 6502 of LNCS, pages 62-67. Springer, 2011.
E. N. Argyriou, M. A. Bekos, and A. Symvonis. The straight-line RAC drawing problem is NP-hard. Journal of Graph Algorithms and Applications, 16(2):569-597, 2012.
A. Arleo, C. Binucci, E. Di Giacomo, W. S. Evans, L. Grilli, G. Liotta, H. Meijer, F. Montecchiani, S. Whitesides, and S. K. Wismath. Visibility representations of boxes in 2.5 dimensions. In Y. Hu and M. Nöllenburg, editors, Graph Drawing and Network Visualization - 24th International Symposium, GD, volume 9801 of LNCS, pages 251-265. Springer, 2016.
M. Babbitt, J. Geneson, and T. Khovanova. On k-visibility graphs. Journal of Graph Algorithms and Applications, 19(1):345-360, 2015.
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K. Casel, H. Fernau, A. Grigoriev, M L. Schmid, and S. Whitesides. Combinatorial properties and recognition of unit square visibility graphs, 2017. URL: http://arxiv.org/abs/1706.05906.
http://arxiv.org/abs/1706.05906
S. Chaplick, G. Guśpiel, G. Gutowski, T. Krawczyk, and G. Liotta. The partial visibility representation extension problem. In Y. Hu and M. Nöllenburg, editors, Graph Drawing and Network Visualization - 24th International Symposium, GD, volume 9801 of LNCS, pages 266-279. Springer, 2016.
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W. S. Evans, G. Liotta, and F. Montecchiani. Simultaneous visibility representations of plane st-graphs using L-shapes. Theoretical Computer Science, 645:100-111, 2016.
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Weighted Operator Precedence Languages
In the last years renewed investigation of operator precedence languages (OPL) led to discover important properties thereof: OPL are closed with respect to all major operations, are characterized, besides the original grammar family, in terms of an automata family (OPA) and an MSO logic; furthermore they significantly generalize the well-known visibly pushdown languages (VPL). In another area of research, quantitative models of systems are also greatly in demand. In this paper, we lay the foundation to marry these two research fields. We introduce weighted operator precedence automata and show how they are both strict extensions of OPA and weighted visibly pushdown automata. We prove a Nivat-like result which shows that quantitative OPL can be described by unweighted OPA and very particular weighted OPA. In a Büchi-like theorem, we show that weighted OPA are expressively equivalent to a weighted MSO-logic for OPL.
Quantitative automata
operator precedence languages
input-driven languages
visibly pushdown languages
quantitative logic
31:1-31:15
Regular Paper
Manfred
Droste
Manfred Droste
Stefan
Dück
Stefan Dück
Dino
Mandrioli
Dino Mandrioli
Matteo
Pradella
Matteo Pradella
10.4230/LIPIcs.MFCS.2017.31
Rajeev Alur and Dana Fisman. Colored nested words. In Adrian Horia Dediu, Jan Janousek, Carlos Martín-Vide, and Bianca Truthe, editors, Language and Automata Theory and Applications, LATA 2016, volume 9618 of LNCS, pages 143-155. Springer, 2016.
Rajeev Alur and Parthasarathy Madhusudan. Adding nesting structure to words. J. ACM, 56(3):16:1-16:43, 2009.
Alessandro Barenghi, Stefano Crespi Reghizzi, Dino Mandrioli, Federica Panella, and Matteo Pradella. Parallel parsing made practical. Sci. Comput. Program., 112(3):195-226, 2015. URL: http://dx.doi.org/10.1016/j.scico.2015.09.002.
http://dx.doi.org/10.1016/j.scico.2015.09.002
Jean Berstel and Christophe Reutenauer. Rational Series and Their Languages, volume 12 of EATCS Monographs in Theoretical Computer Science. Springer, 1988.
Benedikt Bollig and Paul Gastin. Weighted versus probabilistic logics. In Volker Diekert and Dirk Nowotka, editors, Developments in Language Theory, DLT 2009, volume 5583 of LNCS, pages 18-38. Springer, 2009. URL: http://dx.doi.org/10.1007/978-3-642-02737-6_2.
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Christian Choffrut, Andreas Malcher, Carlo Mereghetti, and Beatrice Palano. First-order logics: some characterizations and closure properties. Acta Inf., 49(4):225-248, 2012.
Stefano Crespi Reghizzi and Dino Mandrioli. Operator precedence and the visibly pushdown property. J. Comput. Syst. Sci., 78(6):1837-1867, 2012.
Manfred Droste and Stefan Dück. Weighted automata and logics for infinite nested words. Inf. Comput., 253:448-466, 2017. URL: http://dx.doi.org/10.1016/j.ic.2016.06.010.
http://dx.doi.org/10.1016/j.ic.2016.06.010
Manfred Droste, Stefan Dück, Dino Mandrioli, and Matteo Pradella. Weighted operator precedence languages. CoRR, abs/1702.04597, 2017. URL: http://arXiv.org/abs/1702.04597.
http://arXiv.org/abs/1702.04597
Manfred Droste and Paul Gastin. Weighted automata and weighted logics. Theor. Comput. Sci., 380(1-2):69-86, 2007. extended abstract in ICALP 2005. URL: http://dx.doi.org/10.1016/j.tcs.2007.02.055.
http://dx.doi.org/10.1016/j.tcs.2007.02.055
Manfred Droste, Doreen Heusel, and Heiko Vogler. Weighted unranked tree automata over tree valuation monoids and their characterization by weighted logics. In Andreas Maletti, editor, Conference Algebraic Informatics CAI 2015, volume 9270 of LNCS, pages 90-102. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-23021-4_9.
http://dx.doi.org/10.1007/978-3-319-23021-4_9
Manfred Droste, Werner Kuich, and Heiko Vogler, editors. Handbook of Weighted Automata. EATCS Monographs in Theoretical Computer Science. Springer, 2009.
Manfred Droste and Ingmar Meinecke. Weighted automata and weighted MSO logics for average and long-time behaviors. Inf. Comput., 220:44-59, 2012. URL: http://dx.doi.org/10.1016/j.ic.2012.10.001.
http://dx.doi.org/10.1016/j.ic.2012.10.001
Manfred Droste and Bundit Pibaljommee. Weighted nested word automata and logics over strong bimonoids. Int. J. Found. Comput. Sci., 25(5):641-666, 2014. URL: http://dx.doi.org/10.1142/S0129054114500269.
http://dx.doi.org/10.1142/S0129054114500269
Manfred Droste and Heiko Vogler. Weighted tree automata and weighted logics. Theor. Comput. Sci., 366(3):228-247, 2006. URL: http://dx.doi.org/10.1016/j.tcs.2006.08.025.
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Model Checking and Validity in Propositional and Modal Inclusion Logics
Propositional and modal inclusion logic are formalisms that belong to the family of logics based on team semantics. This article investigates the model checking and validity problems of these logics. We identify complexity bounds for both problems, covering both lax and strict team semantics. By doing so, we come close to finalising the programme that ultimately aims to classify the complexities of the basic reasoning problems for modal and propositional dependence, independence, and inclusion logics.
Inclusion Logic
Model Checking
Complexity
32:1-32:14
Regular Paper
Lauri
Hella
Lauri Hella
Antti
Kuusisto
Antti Kuusisto
Arne
Meier
Arne Meier
Jonni
Virtema
Jonni Virtema
10.4230/LIPIcs.MFCS.2017.32
S. R. Buss. The Boolean formula value problem is in ALOGTIME. In Proc. 19th STOC, pages 123-131, 1987.
E. Clarke, E. A. Emerson, and A. Sistla. Automatic verification of finite-state concurrent systems using temporal logic specifications. ACM ToPLS, 8(2):244-263, 1986.
S. A. Cook. The complexity of theorem proving procedures. In Proc. 3rd STOC, pages 151-158, 1971.
A. Durand, J. Kontinen, and H. Vollmer. Expressivity and complexity of dependence logic. In S. Abramsky, J. Kontinen, J. Väänänen, and H. Vollmer, editors, Dependence Logic: Theory and Applications, pages 5-32. Springer, 2016.
P. Galliani. Inclusion and exclusion dependencies in team semantics - on some logics of imperfect information. Ann. Pure Appl. Logic, 163(1):68-84, 2012. URL: http://dx.doi.org/10.1016/j.apal.2011.08.005.
http://dx.doi.org/10.1016/j.apal.2011.08.005
P. Galliani, M. Hannula, and J. Kontinen. Hierarchies in independence logic. In Proc. 22nd CSL, volume 23 of LIPIcs, pages 263-280, 2013.
P. Galliani and L. Hella. Inclusion logic and fixed point logic. In Proc. 22nd CSL, LIPIcs, pages 281-295, 2013. URL: http://dx.doi.org/10.4230/LIPIcs.CSL.2013.281.
http://dx.doi.org/10.4230/LIPIcs.CSL.2013.281
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E. Grädel and J. Väänänen. Dependence and independence. Studia Logica, 101(2):399-410, 2013.
M. Hannula and J. Kontinen. Hierarchies in independence and inclusion logic with strict semantics. J. Log. Comput., 25(3):879-897, 2015. URL: http://dx.doi.org/10.1093/logcom/exu057.
http://dx.doi.org/10.1093/logcom/exu057
M. Hannula, J. Kontinen, J. Virtema, and H. Vollmer. Complexity of propositional independence and inclusion logic. In Proc. 40th MFCS, pages 269-280, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48057-1_21.
http://dx.doi.org/10.1007/978-3-662-48057-1_21
M. Hannula, J. Kontinen, J. Virtema, and H. Vollmer. Complexity of propositional logics in team semantics. CoRR, extended version of [12], abs/1504.06135, 2015.
L. Hella, A. Kuusisto, A. Meier, and J. Virtema. Model checking and validity in propositional and modal inclusion logics. CoRR, abs/1609.06951, 2016.
L. Hella, A. Kuusisto, A. Meier, and H. Vollmer. Modal inclusion logic: Being lax is simpler than being strict. In Proc. 40th MFCS, pages 281-292, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48057-1_22.
http://dx.doi.org/10.1007/978-3-662-48057-1_22
L. Hella and J. Stumpf. The expressive power of modal logic with inclusion atoms. In Proc. 6th GandALF, pages 129-143, 2015. URL: http://dx.doi.org/10.4204/EPTCS.193.10.
http://dx.doi.org/10.4204/EPTCS.193.10
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K. Sano and J. Virtema. Characterizing frame definability in team semantics via the universal modality. In Proc. of WoLLIC 2015, pages 140-155, 2015.
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P. Schnoebelen. The complexity of temporal logic model checking. In Proc. 4th AiML, pages 393-436, 2002.
J. Väänänen. Dependence Logic. Cambridge University Press, 2007.
H. Vollmer. Introduction to Circuit Complexity - A Uniform Approach. Texts in Theoretical Computer Science. Springer Verlag, Berlin Heidelberg, 1999.
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Emptiness Problems for Integer Circuits
We study the computational complexity of emptiness problems for circuits over sets of natural numbers with the operations union, intersection, complement, addition, and multiplication. For most settings of allowed operations we precisely characterize the complexity in terms of completeness for classes like NL, NP, and PSPACE. The case where intersection, addition, and multiplication is allowed turns out to be equivalent to the complement of polynomial identity testing (PIT).
Our results imply the following improvements and insights on problems studied in earlier papers. We improve the bounds for the membership problem MC(\cup,\cap,¯,+,×) studied by McKenzie and Wagner 2007 and for the equivalence problem EQ(\cup,\cap,¯,+,×) studied by Glaßer et al. 2010. Moreover, it turns out that the following problems are equivalent to PIT, which shows that the challenge to improve their bounds is just a reformulation of a major open problem in algebraic computing complexity:
1. membership problem MC(\cap,+,×) studied by McKenzie and Wagner 2007
2. integer membership problems MC_Z(+,×), MC_Z(\cap,+,×) studied by Travers 2006
3. equivalence problem EQ(+,×) studied by Glaßer et al. 2010
computational complexity
integer expressions
integer circuits
polynomial identity testing
33:1-33:14
Regular Paper
Dominik
Barth
Dominik Barth
Moritz
Beck
Moritz Beck
Titus
Dose
Titus Dose
Christian
Glaßer
Christian Glaßer
Larissa
Michler
Larissa Michler
Marc
Technau
Marc Technau
10.4230/LIPIcs.MFCS.2017.33
M. Agrawal, N. Kayal, and N. Saxena. Primes is in P. Annals of Mathematics, 160:781-793, 2004.
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D. Barth, M. Beck, T. Dose, C. Glaßer, L. Michler, and M. Technau. Emptiness problems for integer circuits. Technical Report 17-012, Electronic Colloquium on Computational Complexity (ECCC), 2017.
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P. McKenzie and K. W. Wagner. The complexity of membership problems for circuits over sets of natural numbers. Computational Complexity, 16(3):211-244, 2007.
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S. D. Travers. The complexity of membership problems for circuits over sets of integers. Theoretical Computer Science, 369(1-3):211-229, 2006.
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Another Characterization of the Higher K-Trivials
In algorithmic randomness, the class of K-trivial sets has proved itself to be remarkable, due to its numerous different characterizations. We pursue in this paper some work already initiated on K-trivials in the context of higher randomness. In particular we give here another characterization of the non hyperarithmetic higher K-trivial sets.
Algorithmic randomness
higher computability
K-triviality
effective descriptive set theory
Kolmogorov complexity
34:1-34:13
Regular Paper
Paul-Elliot
Anglès d'Auriac
Paul-Elliot Anglès d'Auriac
Benoit
Monin
Benoit Monin
10.4230/LIPIcs.MFCS.2017.34
Laurent Bienvenu, Adam R Day, Noam Greenberg, Antonín Kučera, Joseph S Miller, André Nies, and Dan Turetsky. Computing k-trivial sets by incomplete random sets. The Bulletin of Symbolic Logic, 20(01):80-90, 2014.
Laurent Bienvenu, Noam Greenberg, and Benoit Monin. Continuous higher randomness.
Chi Tat Chong, André Nies, and Liang Yu. Lowness of higher randomness notions. Israel J. Math., 166(1):39-60, 2008.
Chi Tat Chong and Liang Yu. Randomness in the higher setting. Submitted.
Chi Tat Chong and Liang Yu. Recursion Theory: Computational Aspects of Definability, volume 8. Walter de Gruyter GmbH &Co KG, 2015.
Adam R. Day and Joseph S. Miller. Cupping with random sets. Proc. Amer. Math. Soc., 142(8):2871-2879, 2014. URL: http://dx.doi.org/10.1090/S0002-9939-2014-11997-6.
http://dx.doi.org/10.1090/S0002-9939-2014-11997-6
Rod Downey, Andre Nies, Rebecca Weber, and Liang Yu. Lowness and Π⁰₂ nullsets. J. Symbolic Logic, 71(3):1044-1052, 09 2006. URL: http://dx.doi.org/10.2178/jsl/1154698590.
http://dx.doi.org/10.2178/jsl/1154698590
Rodney G. Downey and Denis R. Hirschfeldt. Algorithmic Randomness and Complexity. Theory and Applications of Computability. Springer, 2010. URL: http://dx.doi.org/10.1007/978-0-387-68441-3.
http://dx.doi.org/10.1007/978-0-387-68441-3
N. Greenberg, J. Miller, B. Monin, and D. Turetsky. Two more characterizations of k-triviality. Notre Dame Journal of Formal Logic, To appear.
Noam Greenberg and Benoit Monin. Higher randomness and genericity.
Joel David Hamkins and Andy Lewis. Infinite time turing machines. The Journal of Symbolic Logic, 65(02):567-604, 2000.
Denis Hirschfeldt, André Nies, and Frank Stephan. Using random sets as oracles. Journal of the London Mathematical Society, 75(3):610-622, 2007.
Greg Hjorth and André Nies. Randomness via effective descriptive set theory. Journal of the London Mathematical Society, 75(2):495-508, 2007.
Alexander S. Kechris. Classical Descriptive Set Theory. Graduate Texts in Mathematics. Springer New York, 2012.
Bjørn Kjos-Hanssen, Joseph S Miller, and Reed Solomon. Lowness notions, measure and domination. Journal of the London Mathematical Society, page jdr072, 2012.
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Per Martin-Löf. On the notion of randomness. Studies in Logic and the Foundations of Mathematics, 60:73-78, 1970. URL: http://dx.doi.org/10.1016/S0049-237X(08)70741-9.
http://dx.doi.org/10.1016/S0049-237X(08)70741-9
Benoit Monin. Higher computability and randomness. PhD thesis, Universite Paris Diderot, 2014.
Yiannis Moschovakis. Descriptive Set Theory. Mathematical surveys and monographs. American Mathematical Society, 2009.
André Nies. Lowness properties and randomness. Advances in Mathematics, 197(1):274-305, 2005.
André Nies. Computability and Randomness. Oxford Logic Guides. Oxford University Press, 2009.
Gerald E. Sacks. Higher recursion theory. Perspectives in mathematical logic. Springer-Verlag, 1990.
Creative Commons Attribution 3.0 Unported license
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The Quantum Monad on Relational Structures
Homomorphisms between relational structures play a central role in finite model theory, constraint satisfaction, and database theory. A central theme in quantum computation is to show how quantum resources can be used to gain advantage in information processing tasks. In particular, non-local games have been used to exhibit quantum advantage in boolean constraint satisfaction, and to obtain quantum versions of graph invariants such as the chromatic number. We show how quantum strategies for homomorphism games between relational structures can be viewed as Kleisli morphisms for a quantum monad on the (classical) category of relational structures and homomorphisms. We use these results to exhibit a wide range of examples of contextuality-powered quantum advantage, and to unify several apparently diverse strands of previous work.
non-local games
quantum computation
monads
35:1-35:19
Regular Paper
Samson
Abramsky
Samson Abramsky
Rui Soares
Barbosa
Rui Soares Barbosa
Nadish
de Silva
Nadish de Silva
Octavio
Zapata
Octavio Zapata
10.4230/LIPIcs.MFCS.2017.35
Samson Abramsky, Rui Soares Barbosa, Giovanni Carù, Nadish de Silva, Kohei Kishida, and Shane Mansfield. Minimum quantum resources for strong non-locality, 2017. To appear in Proceedings of the 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017). Available as arXiv:1705.09312 [quant-ph].
Samson Abramsky, Rui Soares Barbosa, Kohei Kishida, Raymond Lal, and Shane Mansfield. Contextuality, cohomology and paradox. In Stephan Kreutzer, editor, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015), volume 41 of Leibniz International Proceedings in Informatics (LIPIcs), pages 211-228. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.CSL.2015.211.
http://dx.doi.org/10.4230/LIPIcs.CSL.2015.211
Samson Abramsky, Rui Soares Barbosa, and Shane Mansfield. Contextual fraction as a measure of contextuality, 2017. To appear in Physical Review Letters. Available as arXiv:1705.07918 [quant-ph].
Samson Abramsky and Adam Brandenburger. The sheaf-theoretic structure of non-locality and contextuality. New Journal of Physics, 13(11):113036, 2011. URL: http://dx.doi.org/10.1088/1367-2630/13/11/113036.
http://dx.doi.org/10.1088/1367-2630/13/11/113036
Samson Abramsky and Bob Coecke. A categorical semantics of quantum protocols. In Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS 2004), pages 415-425, 2004. URL: http://dx.doi.org/10.1109/LICS.2004.1319636.
http://dx.doi.org/10.1109/LICS.2004.1319636
Samson Abramsky and Bob Coecke. Categorical quantum mechanics. In Kurt Engesser, Dov M. Gabbay, and Daniel Lehmann, editors, Handbook of quantum logic and quantum structures: Quantum logic, pages 261-323. Elsevier, 2009. URL: http://dx.doi.org/10.1016/B978-0-444-52869-8.50010-4.
http://dx.doi.org/10.1016/B978-0-444-52869-8.50010-4
Samson Abramsky, Anuj Dawar, and Pengming Wang. The pebbling comonad in finite model theory, 2017. To appear in Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LiCS 2017). Available as arXiv:1704.05124 [cs.LO].
Albert Atserias, Laura Mančinska, David E Roberson, Robert Šámal, Simone Severini, and Antonios Varvitsiotis. Quantum and non-signalling graph isomorphisms, 2016. Available as arXiv:1611.09837 [quant-ph].
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Peter J. Cameron, Ashley Montanaro, Michael W. Newman, Simone Severini, and Andreas Winter. On the quantum chromatic number of a graph. Electronic Journal of Combinatorics, 14(1):R81, 2007.
Richard Cleve, Li Liu, and William Slofstra. Perfect commuting-operator strategies for linear system games. Journal of Mathematical Physics, 58(1):012202, 2017. URL: http://dx.doi.org/10.1063/1.4973422.
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Zhengfeng Ji. Binary constraint system games and locally commutative reductions, 2013. Available as arXiv:1310.3794 [quant-ph].
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Phokion G. Kolaitis and Moshe Y. Vardi. Conjunctive-query containment and constraint satisfaction. Journal of Computer and System Sciences, 61(2):302-332, 2000. URL: http://dx.doi.org/10.1006/jcss.2000.1713.
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Leonid Libkin. Elements of finite model theory. Texts in Theoretical Computer Science. Springer, 2004. URL: http://dx.doi.org/10.1007/978-3-662-07003-1.
http://dx.doi.org/10.1007/978-3-662-07003-1
Saunders Mac Lane. Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics. Springer, 1971. URL: http://dx.doi.org/10.1007/978-1-4757-4721-8.
http://dx.doi.org/10.1007/978-1-4757-4721-8
Laura Mančinska and David E Roberson. Quantum homomorphisms. Journal of Combinatorial Theory, Series B, 118:228-267, 2016. URL: http://dx.doi.org/10.1016/j.jctb.2015.12.009.
http://dx.doi.org/10.1016/j.jctb.2015.12.009
N. David Mermin. Quantum mysteries revisited. American Journal of Physics, 58(8):731-734, 1990. URL: http://dx.doi.org/10.1119/1.16503.
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N. David Mermin. Simple unified form for the major no-hidden-variables theorems. Physical Review Letters, 65(27):3373-3376, Dec 1990. URL: http://dx.doi.org/10.1103/PhysRevLett.65.3373.
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http://dx.doi.org/10.4230/LIPIcs.CALCO.2015.253
Ryan O'Donnell. Analysis of boolean functions. Cambridge University Press, 2014. URL: http://dx.doi.org/10.1017/CBO9781139814782.
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Sandu Popescu and Daniel Rohrlich. Quantum nonlocality as an axiom. Foundations of Physics, 24(3):379-385, 1994. URL: http://dx.doi.org/10.1007/BF02058098.
http://dx.doi.org/10.1007/BF02058098
David E. Roberson. Variations on a theme: Graph homomorphisms. PhD thesis, University of Waterloo, 2013.
Volkher B. Scholz and Reinhard F. Werner. Tsirelson’s problem, 2008. Available as arXiv:0812.4305 [math-ph].
William Slofstra. Tsirelson’s problem and an embedding theorem for groups arising from non-local games, 2016. Available as arXiv:1606.03140 [quant-ph].
Boris Tsirelson. Bell inequalities and operator algebras, 2006. Pre-print.
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Towards a Polynomial Kernel for Directed Feedback Vertex Set
In the Directed Feedback Vertex Set (DFVS) problem, the input is
a directed graph D and an integer k. The objective is to determine
whether there exists a set of at most k vertices intersecting every
directed cycle of D. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen, Liu, Lu, O'Sullivan and
Razgon [JACM 2008]; since then, the existence of a polynomial kernel for this problem has become one of the largest open problems in the area of parameterized algorithmics.
In this paper, we study DFVS parameterized by the feedback vertex
set number of the underlying undirected graph. We provide two main contributions: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus.
parameterized algorithms
kernelization
(directed) feedback vertex set
36:1-36:15
Regular Paper
Benjamin
Bergougnoux
Benjamin Bergougnoux
Eduard
Eiben
Eduard Eiben
Robert
Ganian
Robert Ganian
Sebastian
Ordyniak
Sebastian Ordyniak
M. S.
Ramanujan
M. S. Ramanujan
10.4230/LIPIcs.MFCS.2017.36
Vineet Bafna, Piotr Berman, and Toshihiro Fujito. A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Discrete Math., 12(3):289-297, 1999.
Reuven Bar-Yehuda, Dan Geiger, Joseph Naor, and Ron M. Roth. Approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and bayesian inference. SIAM J. Comput., 27(4):942-959, 1998.
Ann Becker and Dan Geiger. Optimization of pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artif. Intell., 83(1):167-188, 1996.
Hans L. Bodlaender, Fedor V. Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, and Dimitrios M. Thilikos. (meta) kernelization. J. ACM, 63(5):44:1-44:69, 2016.
Hans L. Bodlaender and Thomas C. van Dijk. A cubic kernel for feedback vertex set and loop cutset. Theory Comput. Syst., 46(3):566-597, 2010.
Yixin Cao, Jianer Chen, and Yang Liu. On feedback vertex set new measure and new structures. In Haim Kaplan, editor, Algorithm Theory - SWAT 2010, 12th Scandinavian Symposium and Workshops on Algorithm Theory, Bergen, Norway, June 21-23, 2010. Proceedings, volume 6139 of Lecture Notes in Computer Science, pages 93-104. Springer, 2010.
Chandra Chekuri and Vivek Madan. Constant factor approximation for subset feedback set problems via a new LP relaxation. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 808-820. SIAM, 2016.
Jianer Chen, Fedor V. Fomin, Yang Liu, Songjian Lu, and Yngve Villanger. Improved algorithms for feedback vertex set problems. J. Comput. Syst. Sci., 74(7):1188-1198, 2008. URL: http://dx.doi.org/10.1016/j.jcss.2008.05.002.
http://dx.doi.org/10.1016/j.jcss.2008.05.002
Jianer Chen, Yang Liu, Songjian Lu, Barry O'Sullivan, and Igor Razgon. A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM, 55(5), 2008. URL: http://dx.doi.org/10.1145/1411509.1411511.
http://dx.doi.org/10.1145/1411509.1411511
Rajesh Hemant Chitnis, Marek Cygan, Mohammad Taghi Hajiaghayi, and Dániel Marx. Directed subset feedback vertex set is fixed-parameter tractable. ACM Transactions on Algorithms, 11(4):28, 2015.
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. On the hardness of losing width. Theory Comput. Syst., 54(1):73-82, 2014.
Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In FOCS, pages 150-159, 2011.
Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk, and Jakub Onufry Wojtaszczyk. Subset feedback vertex set is fixed-parameter tractable. SIAM J. Discrete Math., 27(1):290-309, 2013. URL: http://dx.doi.org/10.1137/110843071.
http://dx.doi.org/10.1137/110843071
Reinhard Diestel. Graph Theory, volume 173 of Graduate Texts in Mathematics. Springer Verlag, New York, 2nd edition, 2000.
Rodney G. Downey and Michael R. Fellows. Fixed-parameter intractability. In Proceedings of the Seventh Annual Structure in Complexity Theory Conference, Boston, Massachusetts, USA, June 22-25, 1992, pages 36-49, 1992.
Rodney G. Downey and Michael R. Fellows. Fixed-parameter tractability and completeness I: basic results. SIAM J. Comput., 24(4):873-921, 1995.
Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013.
P Erdős and L Pósa. On independent circuits contained in a graph. Canad. J. Math, 17:347-352, 1965.
Guy Even, Joseph Naor, Baruch Schieber, and Madhu Sudan. Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica, 20(2):151-174, 1998.
Jakub Gajarský, Petr Hlinený, Jan Obdrzálek, Sebastian Ordyniak, Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, and Somnath Sikdar. Kernelization using structural parameters on sparse graph classes. J. Comput. Syst. Sci., 84:219-242, 2017.
Jonathan L. Gross and Thomas W. Tucker. Topological Graph Theory. Wiley-Interscience, New York, NY, USA, 1987.
Venkatesan Guruswami and Euiwoong Lee. Inapproximability of h-transversal/packing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2015, August 24-26, 2015, Princeton, NJ, USA, volume 40 of LIPIcs, pages 284-304. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015.
Bart M. P. Jansen and Hans L. Bodlaender. Vertex cover kernelization revisited - upper and lower bounds for a refined parameter. Theory Comput. Syst., 53(2):263-299, 2013.
Naonori Kakimura, Ken-ichi Kawarabayashi, and Yusuke Kobayashi. Erdös-pósa property and its algorithmic applications: parity constraints, subset feedback set, and subset packing. In Yuval Rabani, editor, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 1726-1736. SIAM, 2012.
Naonori Kakimura, Ken-ichi Kawarabayashi, and Dániel Marx. Packing cycles through prescribed vertices. J. Comb. Theory, Ser. B, 101(5):378-381, 2011.
Richard M. Karp. Reducibility among combinatorial problems. In Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York., pages 85-103, 1972. URL: http://www.cs.berkeley.edu/~luca/cs172/karp.pdf.
http://www.cs.berkeley.edu/~luca/cs172/karp.pdf
Ken-ichi Kawarabayashi and Yusuke Kobayashi. Fixed-parameter tractability for the subset feedback set problem and the s-cycle packing problem. J. Comb. Theory, Ser. B, 102(4):1020-1034, 2012. URL: http://dx.doi.org/10.1016/j.jctb.2011.12.001.
http://dx.doi.org/10.1016/j.jctb.2011.12.001
Ken-ichi Kawarabayashi, Daniel Král', Marek Krcál, and Stephan Kreutzer. Packing directed cycles through a specified vertex set. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 365-377, 2013.
Tomasz Kociumaka and Marcin Pilipczuk. Faster deterministic feedback vertex set. Inf. Process. Lett., 114(10):556-560, 2014. URL: http://dx.doi.org/10.1016/j.ipl.2014.05.001.
http://dx.doi.org/10.1016/j.ipl.2014.05.001
M. Pontecorvi and Paul Wollan. Disjoint cycles intersecting a set of vertices. J. Comb. Theory, Ser. B, 102(5):1134-1141, 2012.
Venkatesh Raman, Saket Saurabh, and C. R. Subramanian. Faster fixed parameter tractable algorithms for finding feedback vertex sets. ACM Transactions on Algorithms, 2(3):403-415, 2006. URL: http://dx.doi.org/10.1145/1159892.1159898.
http://dx.doi.org/10.1145/1159892.1159898
Bruce A. Reed, Neil Robertson, Paul D. Seymour, and Robin Thomas. Packing directed circuits. Combinatorica, 16(4):535-554, 1996.
Paul D. Seymour. Packing directed circuits fractionally. Combinatorica, 15(2):281-288, 1995.
Paul D. Seymour. Packing circuits in eulerian digraphs. Combinatorica, 16(2):223-231, 1996.
Stéphan Thomassé. A 4k^2 kernel for feedback vertex set. ACM Trans. Algorithms, 6(2):32:1-32:8, 2010.
Magnus Wahlström. Half-integrality, LP-branching and FPT algorithms. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1762-1781. SIAM, 2014.
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Timed Network Games
Network games are widely used as a model for selfish resource-allocation problems. In the classical model, each player selects a path connecting her source and target vertex. The cost of traversing an edge depends on the number of players that traverse it. Thus, it abstracts the fact that different users may use a resource at different times and for different durations, which plays an important role in defining the costs of the users in reality. For example, when transmitting packets in a communication network, routing traffic in a road network, or processing a task in a production system, the traversal of the network involves an inherent delay, and so sharing and congestion of resources crucially depends on time.
We study timed network games, which add a time component to network games. Each vertex v in the network is associated with a cost function, mapping the load on v to the price that a player pays for staying in v for one time unit with this load. In addition, each edge has a guard, describing time intervals in which the edge can be traversed, forcing the players to spend time on vertices. Unlike earlier work that add a time component to network games, the time in our model is continuous and cannot be discretized. In particular, players have uncountably many strategies, and a game may have uncountably many pure Nash equilibria.
We study properties of timed network games with cost-sharing or congestion cost functions: their stability, equilibrium inefficiency, and complexity. In particular, we show that the answer to the question whether we can restrict attention to boundary strategies, namely ones in which edges are traversed only at the boundaries of guards, is mixed.
Network Games
Timed Automata
Nash Equilibrium
Equilibrium Inefficiency
37:1-37:16
Regular Paper
Guy
Avni
Guy Avni
Shibashis
Guha
Shibashis Guha
Orna
Kupferman
Orna Kupferman
10.4230/LIPIcs.MFCS.2017.37
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Efficient Identity Testing and Polynomial Factorization in Nonassociative Free Rings
In this paper we study arithmetic computations in the nonassociative, and noncommutative free polynomial ring F{X}. Prior to this work, nonassociative arithmetic computation was considered by Hrubes, Wigderson, and Yehudayoff, and they showed lower bounds and proved completeness results. We consider Polynomial Identity Testing and Polynomial Factorization in F{X} and show the following results.
1. Given an arithmetic circuit C computing a polynomial f in F{X} of degree d, we give a deterministic polynomial algorithm to decide if f is identically zero. Our result is obtained by a suitable adaptation of the PIT algorithm of Raz and Shpilka for noncommutative ABPs.
2. Given an arithmetic circuit C computing a polynomial f in F{X} of degree d, we give an efficient deterministic algorithm to compute circuits for the irreducible factors of f in polynomial time when F is the field of rationals. Over finite fields of characteristic p,
our algorithm runs in time polynomial in input size and p.
Circuits
Nonassociative
Noncommutative
Polynomial Identity Testing
Factorization
38:1-38:13
Regular Paper
Vikraman
Arvind
Vikraman Arvind
Rajit
Datta
Rajit Datta
Partha
Mukhopadhyay
Partha Mukhopadhyay
S.
Raja
S. Raja
10.4230/LIPIcs.MFCS.2017.38
Avraham Shimshon Amitsur and Jacob Levitzki. Minimal identities for algebras. Proceedings of the American Mathematical Society, 1(4):449-463, 1950.
Vikraman Arvind, Pushkar S. Joglekar, and Gaurav Rattan. On the complexity of noncommutative polynomial factorization. In Mathematical Foundations of Computer Science 2015 - 40th International Symposium, MFCS 2015, Milan, Italy, August 24-28, 2015, Proceedings, Part II, pages 38-49, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48054-0_4.
http://dx.doi.org/10.1007/978-3-662-48054-0_4
Vikraman Arvind, Partha Mukhopadhyay, and Srikanth Srinivasan. New results on noncommutative and commutative polynomial identity testing. Computational Complexity, 19(4):521-558, 2010. URL: http://dx.doi.org/10.1007/s00037-010-0299-8.
http://dx.doi.org/10.1007/s00037-010-0299-8
Vikraman Arvind and S. Raja. Some lower bound results for set-multilinear arithmetic computations. Chicago J. Theor. Comput. Sci., 2016 (6), 2016.
E. R. Berlekamp. Factoring polynomials over large finite fields*. In Proceedings of the Second ACM Symposium on Symbolic and Algebraic Manipulation, SYMSAC'71, pages 223-, New York, NY, USA, 1971. ACM. URL: http://dx.doi.org/10.1145/800204.806290.
http://dx.doi.org/10.1145/800204.806290
P.M. Cohn. Noncommutative unique factorization domains. Transactions of the American Math. Society, 109(2):313-331, 1963.
Pavel Hrubes, Avi Wigderson, and Amir Yehudayoff. Relationless completeness and separations. In Proceedings of the 25th Annual IEEE Conference on Computational Complexity, CCC 2010, Cambridge, Massachusetts, June 9-12, 2010, pages 280-290, 2010. URL: http://dx.doi.org/10.1109/CCC.2010.34.
http://dx.doi.org/10.1109/CCC.2010.34
Laurent Hyafil. The power of commutativity. In 18th Annual Symposium on Foundations of Computer Science (FOCS), Providence, Rhode Island, USA, 31 October - 1 November 1977, pages 171-174, 1977. URL: http://dx.doi.org/10.1109/SFCS.1977.31.
http://dx.doi.org/10.1109/SFCS.1977.31
Erich Kaltofen. Factorization of polynomials given by straight-line programs. Randomness in Computation, vol. 5 of Advances in Computing Research:375–412, 1989.
Swastik Kopparty, Shubhangi Saraf, and Amir Shpilka. Equivalence of polynomial identity testing and polynomial factorization. Computational Complexity, 24(2):295-331, 2015. URL: http://dx.doi.org/10.1007/s00037-015-0102-y.
http://dx.doi.org/10.1007/s00037-015-0102-y
Guillaume Lagarde, Guillaume Malod, and Sylvain Perifel. Non-commutative computations: lower bounds and polynomial identity testing. Electronic Colloquium on Computational Complexity (ECCC), 23:94, 2016. URL: http://eccc.hpi-web.de/report/2016/094.
http://eccc.hpi-web.de/report/2016/094
Noam Nisan. Lower bounds for non-commutative computation (extended abstract). In STOC, pages 410-418, 1991. URL: http://dx.doi.org/10.1145/103418.103462.
http://dx.doi.org/10.1145/103418.103462
Ran Raz and Amir Shpilka. Deterministic polynomial identity testing in non-commutative models. Computational Complexity, 14(1):1-19, 2005. URL: http://dx.doi.org/10.1007/s00037-005-0188-8.
http://dx.doi.org/10.1007/s00037-005-0188-8
Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science, 5(3-4):207-388, 2010. URL: http://dx.doi.org/10.1561/0400000039.
http://dx.doi.org/10.1561/0400000039
Joachim von zur Gathen and Victor Shoup. Computing frobenius maps and factoring polynomials. Computational Complexity, 2:187-224, 1992.
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Faster Algorithms for Mean-Payoff Parity Games
Graph games provide the foundation for modeling and synthesis of reactive processes. Such games are played over graphs where the vertices are controlled by two adversarial players. We consider graph games where the objective of the first player is the
conjunction of a qualitative objective (specified as a parity condition) and a quantitative objective (specified as a mean-payoff condition). There are two variants of the problem, namely, the threshold problem where the quantitative goal is to ensure that the mean-payoff value is above a threshold, and the value problem where the quantitative goal is to ensure the optimal mean-payoff value; in both cases ensuring the qualitative parity objective. The previous best-known algorithms for game graphs with n vertices, m edges,
parity objectives with d priorities, and maximal absolute reward value W for mean-payoff objectives, are as follows: O(n^(d+1)·m·W) for the threshold problem, and O(n^(d+2)·m·W) for the value problem.
Our main contributions are faster algorithms, and the running times of our algorithms are as follows: O(n^(d-1)·m·W) for the threshold problem, and O(n^d·m·W·log(n·W)) for the value problem. For mean-payoff parity objectives with two priorities, our algorithms match the best-known bounds of the algorithms for mean-payoff games (without conjunction with parity objectives). Our results are relevant in synthesis of reactive systems with both functional
requirement (given as a qualitative objective) and performance requirement (given as a quantitative objective).
graph games
mean-payoff parity games
39:1-39:14
Regular Paper
Krishnendu
Chatterjee
Krishnendu Chatterjee
Monika
Henzinger
Monika Henzinger
Alexander
Svozil
Alexander Svozil
10.4230/LIPIcs.MFCS.2017.39
R. Bloem, K. Chatterjee, T. A. Henzinger, and B. Jobstmann. Better quality in synthesis through quantitative objectives. In Proc. of CAV, LNCS 5643, pages 140-156. Springer, 2009.
Roderick Bloem, Krishnendu Chatterjee, Karin Greimel, Thomas A. Henzinger, Georg Hofferek, Barbara Jobstmann, Bettina Könighofer, and Robert Könighofer. Synthesizing robust systems. Acta Inf., 51(3-4):193-220, 2014.
P. Bouyer, U. Fahrenberg, K. G. Larsen, N. Markey, and J. Srba. Infinite runs in weighted timed automata with energy constraints. In Proc. of FORMATS, LNCS 5215, pages 33-47. Springer, 2008.
P. Bouyer, N. Markey, J. Olschewski, and M. Ummels. Measuring permissiveness in parity games: Mean-payoff parity games revisited. In Proc. of ATVA, LNCS 6996, pages 135-149. Springer, 2011.
L. Brim, J. Chaloupka, L. Doyen, R. Gentilini, and J. F. Raskin. Faster algorithms for mean-payoff games. Form. Methods Syst. Des., 38(2):97-118, April 2011. URL: http://dx.doi.org/10.1007/s10703-010-0105-x.
http://dx.doi.org/10.1007/s10703-010-0105-x
J. R. Büchi and L. H. Landweber. Solving sequential conditions by finite-state strategies. Transactions of the AMS, 138:295-311, 1969.
P. Cerný, K. Chatterjee, T. A. Henzinger, A. Radhakrishna, and R. Singh. Quantitative synthesis for concurrent programs. In Proc. of CAV, LNCS 6806, pages 243-259. Springer, 2011.
A. Chakrabarti, L. de Alfaro, T. A. Henzinger, and M. Stoelinga. Resource interfaces. In Proc. of EMSOFT, LNCS 2855, pages 117-133. Springer, 2003.
K. Chatterjee and L. Doyen. Energy parity games. In Proc. of ICALP: Automata, Languages and Programming (B), LNCS 6199, pages 599-610. Springer, 2010.
K. Chatterjee and L. Doyen. Energy and mean-payoff parity Markov decision processes. In Proc. of MFCS, LNCS 6907, pages 206-218. Springer, 2011.
K. Chatterjee, T. A. Henzinger, and M. Jurdziński. Mean-payoff parity games. In Proc. of LICS, pages 178-187. IEEE Computer Society, 2005.
Krishnendu Chatterjee and Laurent Doyen. Games and markov decision processes with mean-payoff parity and energy parity objectives. In MEMICS, pages 37-46, 2011.
Krishnendu Chatterjee, Laurent Doyen, Hugo Gimbert, and Youssouf Oualhadj. Perfect-information stochastic mean-payoff parity games. In FOSSACS, pages 210-225, 2014.
Carlo Comin and Romeo Rizzi. Improved pseudo-polynomial bound for the value problem and optimal strategy synthesis in mean payoff games. Algorithmica, 77(4):995-1021, 2017. URL: http://dx.doi.org/10.1007/s00453-016-0123-1.
http://dx.doi.org/10.1007/s00453-016-0123-1
A. Ehrenfeucht and J. Mycielski. Positional strategies for mean payoff games. International Journal of Game Theory, 8(2):109-113, 1979. URL: http://dx.doi.org/10.1007/BF01768705.
http://dx.doi.org/10.1007/BF01768705
Y. M. Lifshits and D. S. Pavlov. Potential theory for mean payoff games. Journal of Mathematical Sciences, 145(3):4967-4974, 2007. URL: http://dx.doi.org/10.1007/s10958-007-0331-y.
http://dx.doi.org/10.1007/s10958-007-0331-y
A. Pnueli and R. Rosner. On the synthesis of a reactive module. In Proc. of POPL, pages 179-190. ACM Press, 1989.
P. J. Ramadge and W. M. Wonham. Supervisory control of a class of discrete-event processes. SIAM Journal of Control and Optimization, 25(1):206-230, 1987.
W. Thomas. Languages, automata, and logic. In Handbook of Formal Languages, volume 3, Beyond Words, chapter 7, pages 389-455. Springer, 1997.
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Attainable Values of Reset Thresholds
An automaton is synchronizing if there exists a word that sends all states of the automaton to a single state. The reset threshold is the length of the shortest such word. We study the set RT_n of attainable reset thresholds by automata with n states. Relying on constructions of digraphs with known local exponents we show that the intervals [1, (n^2-3n+4)/2] and
[(p-1)(q-1), p(q-2)+n-q+1], where 2 <= p < q <= n, p+q > n, gcd(p,q)=1, belong to RT_n, even if restrict our attention to strongly connected automata. Moreover, we prove that in this case the smallest value that does not belong to RT_n is at least n^2 - O(n^{1.7625} log n / log log n).
This value is increased further assuming certain conjectures about the gaps between consecutive prime numbers.
We also show that any value smaller than n(n-1)/2 is attainable by an automaton with a sink state and any value smaller than n^2-O(n^{1.5}) is attainable in general case.
Furthermore, we solve the problem of existence of slowly synchronizing automata over an arbitrarily large alphabet, by presenting for every fixed size of the alphabet an infinite series of irreducibly synchronizing automata with the reset threshold n^2-O(n).
Cerny conjecture
exponent
primitive digraph
reset word
synchronizing automaton
40:1-40:14
Regular Paper
Michalina
Dzyga
Michalina Dzyga
Robert
Ferens
Robert Ferens
Vladimir V.
Gusev
Vladimir V. Gusev
Marek
Szykula
Marek Szykula
10.4230/LIPIcs.MFCS.2017.40
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Lower Bounds and PIT for Non-Commutative Arithmetic Circuits with Restricted Parse Trees
We investigate the power of Non-commutative Arithmetic Circuits, which compute polynomials over the free non-commutative polynomial ring F<x_1,...,x_N>, where variables do not commute. We consider circuits that are restricted in the ways in which they can compute monomials: this can be seen as restricting the families of parse trees that appear in the circuit. Such restrictions capture essentially all non-commutative circuit models for which lower bounds are known. We prove several results about such circuits.
- We show explicit exponential lower bounds for circuits with up to an exponential number of parse trees, strengthening the work of Lagarde, Malod, and Perifel (ECCC 2016), who prove such a result for Unique Parse Tree (UPT) circuits which have a single parse tree.
- We show explicit exponential lower bounds for circuits whose parse trees are rotations of a single tree. This simultaneously generalizes recent lower bounds of Limaye, Malod, and Srinivasan (Theory of Computing 2016) and the above lower bounds of Lagarde et al., which are known to be incomparable.
- We make progress on a question of Nisan (STOC 1991) regarding separating the power of Algebraic Branching Programs (ABPs) and Formulas in the non-commutative setting by showing a tight lower bound of n^{Omega(log d)} for any UPT formula computing the product of d n*n matrices.
When d <= log n, we can also prove superpolynomial lower bounds for formulas with up to 2^{o(d)} many parse trees (for computing the same polynomial). Improving this bound to allow for 2^{O(d)} trees would yield an unconditional separation between ABPs and Formulas.
- We give deterministic white-box PIT algorithms for UPT circuits over any field (strengthening a result of Lagarde et al. (2016)) and also for sums of a constant number of UPT circuits with different parse trees.
Non-commutative Arithemetic circuits
Partial derivatives
restrictions
41:1-41:14
Regular Paper
Guillaume
Lagarde
Guillaume Lagarde
Nutan
Limaye
Nutan Limaye
Srikanth
Srinivasan
Srikanth Srinivasan
10.4230/LIPIcs.MFCS.2017.41
Eric Allender, Jia Jiao, Meena Mahajan, and V. Vinay. Non-commutative arithmetic circuits: Depth reduction and size lower bounds. Theor. Comput. Sci., 209(1-2):47-86, 1998. URL: http://dx.doi.org/10.1016/S0304-3975(97)00227-2.
http://dx.doi.org/10.1016/S0304-3975(97)00227-2
Vikraman Arvind, Pushkar S. Joglekar, Partha Mukhopadhyay, and S Raja. Identity testing for +-regular noncommutative arithmetic circuits. Electronic Colloquium on Computational Complexity (ECCC), 23:193, 2016. URL: http://eccc.hpi-web.de/report/2016/193.
http://eccc.hpi-web.de/report/2016/193
Vikraman Arvind, Pushkar S. Joglekar, and Srikanth Srinivasan. Arithmetic circuits and the hadamard product of polynomials. In Ravi Kannan and K. Narayan Kumar, editors, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2009, December 15-17, 2009, IIT Kanpur, India, volume 4 of LIPIcs, pages 25-36. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2009. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2009.2304.
http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2009.2304
Vikraman Arvind, Partha Mukhopadhyay, and S Raja. Randomized polynomial time identity testing for noncommutative circuits. Electronic Colloquium on Computational Complexity (ECCC), 23:89, 2016. URL: http://eccc.hpi-web.de/report/2016/089.
http://eccc.hpi-web.de/report/2016/089
Vikraman Arvind and S. Raja. The complexity of two register and skew arithmetic computation. Electronic Colloquium on Computational Complexity (ECCC), 21:28, 2014. URL: http://eccc.hpi-web.de/report/2014/028.
http://eccc.hpi-web.de/report/2014/028
Steve Chien, Lars Eilstrup Rasmussen, and Alistair Sinclair. Clifford algebras and approximating the permanent. J. Comput. Syst. Sci., 67(2):263-290, 2003. URL: http://dx.doi.org/10.1016/S0022-0000(03)00010-2.
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Steve Chien and Alistair Sinclair. Algebras with polynomial identities and computing the determinant. In 45th Symposium on Foundations of Computer Science (FOCS 2004), 17-19 October 2004, Rome, Italy, Proceedings, pages 352-361, 2004. URL: http://dx.doi.org/10.1109/FOCS.2004.9.
http://dx.doi.org/10.1109/FOCS.2004.9
Hervé Fournier, Nutan Limaye, Guillaume Malod, and Srikanth Srinivasan. Lower bounds for depth-4 formulas computing iterated matrix multiplication. SIAM J. Comput., 44(5):1173-1201, 2015. URL: http://dx.doi.org/10.1137/140990280.
http://dx.doi.org/10.1137/140990280
Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. Approaching the chasm at depth four. In Proceedings of the Conference on Computational Complexity (CCC), 2013.
Rohit Gurjar, Arpita Korwar, Nitin Saxena, and Thomas Thierauf. Deterministic identity testing for sum of read-once oblivious arithmetic branching programs. In 30th Conference on Computational Complexity, CCC 2015, June 17-19, 2015, Portland, Oregon, USA, pages 323-346, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2015.323.
http://dx.doi.org/10.4230/LIPIcs.CCC.2015.323
Pavel Hrubeš, Avi Wigderson, and Amir Yehudayoff. Non-commutative circuits and the sum-of-squares problem. Journal of the American Mathematical Society, 24(3):871-898, 2011.
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http://dx.doi.org/10.1109/FOCS.2014.15
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http://dx.doi.org/10.1145/2591796.2591847
Mrinal Kumar and Shubhangi Saraf. On the power of homogeneous depth 4 arithmetic circuits. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 364-373, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.46.
http://dx.doi.org/10.1109/FOCS.2014.46
Guillaume Lagarde, Nutan Limaye, and Srikanth Srinivasan. Lower bounds and PIT for non-commutative arithmetic circuits with restricted parse trees. Electronic Colloquium on Computational Complexity (ECCC), 24:77, 2017. URL: https://eccc.weizmann.ac.il/report/2017/077.
https://eccc.weizmann.ac.il/report/2017/077
Guillaume Lagarde, Guillaume Malod, and Sylvain Perifel. Non-commutative computations: lower bounds and polynomial identity testing. Electronic Colloquium on Computational Complexity (ECCC), 23:94, 2016. URL: http://eccc.hpi-web.de/report/2016/094.
http://eccc.hpi-web.de/report/2016/094
Nutan Limaye, Guillaume Malod, and Srikanth Srinivasan. Lower bounds for non-commutative skew circuits. Theory of Computing, 12(1):1-38, 2016. URL: http://dx.doi.org/10.4086/toc.2016.v012a012.
http://dx.doi.org/10.4086/toc.2016.v012a012
Guillaume Malod and Natacha Portier. Characterizing valiant’s algebraic complexity classes. J. Complexity, 24(1):16-38, 2008. URL: http://dx.doi.org/10.1016/j.jco.2006.09.006.
http://dx.doi.org/10.1016/j.jco.2006.09.006
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http://dx.doi.org/10.1145/103418.103462
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http://dx.doi.org/10.1007/BF01294256
Ran Raz. Multi-linear formulas for permanent and determinant are of super-polynomial size. J. ACM, 56(2):8:1-8:17, 2009. URL: http://dx.doi.org/10.1145/1502793.1502797.
http://dx.doi.org/10.1145/1502793.1502797
Ran Raz and Amir Shpilka. Deterministic polynomial identity testing in non-commutative models. Computational Complexity, 14(1):1-19, 2005. URL: http://dx.doi.org/10.1007/s00037-005-0188-8.
http://dx.doi.org/10.1007/s00037-005-0188-8
Amir Shpilka and Avi Wigderson. Depth-3 arithmetic circuits over fields of characteristic zero. Computational Complexity, 10(1):1-27, 2001. URL: http://dx.doi.org/10.1007/PL00001609.
http://dx.doi.org/10.1007/PL00001609
Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science, 5(3-4):207-388, 2010. URL: http://dx.doi.org/10.1561/0400000039.
http://dx.doi.org/10.1561/0400000039
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https://creativecommons.org/licenses/by/3.0/legalcode
Approximation and Parameterized Algorithms for Geometric Independent Set with Shrinking
Consider the Maximum Weight Independent Set problem for rectangles: given a family of weighted axis-parallel rectangles in the plane, find a maximum-weight subset of non-overlapping rectangles. The problem is notoriously hard both in the approximation and in the parameterized setting. The best known polynomial-time approximation algorithms achieve super-constant approximation ratios [Chalermsook & Chuzhoy, Proc. SODA 2009; Chan & Har-Peled, Discrete & Comp. Geometry, 2012], even though there is a (1+epsilon)-approximation running in quasi-polynomial time [Adamaszek & Wiese, Proc. FOCS 2013; Chuzhoy & Ene, Proc. FOCS 2016]. When parameterized by the target size of the solution, the problem is W[1]-hard even in the unweighted setting [Marx, ESA 2005].
To achieve tractability, we study the following shrinking model: one is allowed to shrink each input rectangle by a multiplicative factor 1-delta for some fixed delta > 0, but the performance is still compared against the optimal solution for the original, non-shrunk instance. We prove that in this regime, the problem admits an EPTAS with running time f(epsilon,delta) n^{O(1)}, and an FPT algorithm with running time f(k,delta) n^{O(1)}, in the setting where a maximum-weight solution of size at most k is to be computed. This improves and significantly simplifies a PTAS given earlier for this problem [Adamaszek, Chalermsook & Wiese, Proc. APPROX/RANDOM 2015], and provides the first parameterized results for the shrinking model. Furthermore, we explore kernelization in the shrinking model, by giving efficient kernelization procedures for several variants of the problem when the input rectangles are squares.
Combinatorial optimization
Approximation algorithms
Fixed-parameter algorithms
42:1-42:13
Regular Paper
Michal
Pilipczuk
Michal Pilipczuk
Erik Jan
van Leeuwen
Erik Jan van Leeuwen
Andreas
Wiese
Andreas Wiese
10.4230/LIPIcs.MFCS.2017.42
Anna Adamaszek, Parinya Chalermsook, and Andreas Wiese. How to Tame Rectangles: Solving Independent Set and Coloring of Rectangles via Shrinking. In Proc. APPROX/RANDOM 2015, volume 40 of LIPIcs, pages 43-60. Schloss Dagstuhl, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.43.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.43
Anna Adamaszek and Andreas Wiese. Approximation schemes for maximum weight independent set of rectangles. In Proc. FOCS 2013, pages 400-409. IEEE, 2013. URL: http://dx.doi.org/10.1109/FOCS.2013.50.
http://dx.doi.org/10.1109/FOCS.2013.50
Anna Adamaszek and Andreas Wiese. A QPTAS for maximum weight independent set of polygons with polylogarithmically many vertices. In Proc. SODA 2014, pages 645-656. SIAM, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.49.
http://dx.doi.org/10.1137/1.9781611973402.49
Jochen Alber and Jiří Fiala. Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. J. Algorithms, 52(2):134-151, 2004. URL: http://dx.doi.org/10.1016/j.jalgor.2003.10.001.
http://dx.doi.org/10.1016/j.jalgor.2003.10.001
Parinya Chalermsook and Julia Chuzhoy. Maximum independent set of rectangles. In Proc. SODA 2009, pages 892-901. SIAM, 2009.
Timothy M. Chan. Polynomial-time approximation schemes for packing and piercing fat objects. Journal of Algorithms, 46(2):178-189, 2003. URL: http://dx.doi.org/10.1016/S0196-6774(02)00294-8.
http://dx.doi.org/10.1016/S0196-6774(02)00294-8
Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete &Comp. Geometry, 48(2):373-392, 2012. URL: http://dx.doi.org/10.1007/s00454-012-9417-5.
http://dx.doi.org/10.1007/s00454-012-9417-5
Julia Chuzhoy and Alina Ene. On approximating maximum independent set of rectangles. In Proc. FOCS 2016, pages 820-829. IEEE, 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.92.
http://dx.doi.org/10.1109/FOCS.2016.92
Rod G. Downey and Michael R. Fellows. Fixed-parameter tractability and completeness II: On completeness for W[1]. Theoretical Computer Science, 141(1):109-131, 1995. URL: http://dx.doi.org/10.1016/0304-3975(94)00097-3.
http://dx.doi.org/10.1016/0304-3975(94)00097-3
Thomas Erlebach, Klaus Jansen, and Eike Seidel. Polynomial-time approximation schemes for geometric intersection graphs. SIAM J. on Computing, 34(6):1302-1323, 2005. URL: http://dx.doi.org/10.1137/S0097539702402676.
http://dx.doi.org/10.1137/S0097539702402676
Sariel Har-Peled. Quasi-polynomial time approximation scheme for sparse subsets of polygons. In Proc. SOCG 2014, pages 120-129. ACM, 2014. URL: http://dx.doi.org/10.1145/2582112.2582157.
http://dx.doi.org/10.1145/2582112.2582157
Dániel Marx. Efficient approximation schemes for geometric problems? In Proc. ESA 2005, volume 3669 of LNCS, pages 448-459. Springer, 2005. URL: http://dx.doi.org/10.1007/11561071_41.
http://dx.doi.org/10.1007/11561071_41
Dániel Marx and Michał Pilipczuk. Optimal parameterized algorithms for planar facility location problems using Voronoi diagrams. In Proc. ESA 2015, volume 9294 of LNCS, pages 865-877. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_72.
http://dx.doi.org/10.1007/978-3-662-48350-3_72
Michał Pilipczuk, Erik Jan van Leeuwen, and Andreas Wiese. Approximation and parameterized algorithms for geometric independent set with shrinking. CoRR, abs/1611.06501, 2016. URL: http://arxiv.org/abs/1611.06501.
http://arxiv.org/abs/1611.06501
Andreas Wiese. Independent set of convex polygons: From n^ε to 1+ε via shrinking. In Proc. LATIN 2016, volume 9644 of LNCS, pages 700-711. Springer, 2016. URL: http://dx.doi.org/10.1007/978-3-662-49529-2_52.
http://dx.doi.org/10.1007/978-3-662-49529-2_52
David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. Theory of Computing, 3:103-128, 2007. URL: http://dx.doi.org/10.4086/toc.2007.v003a006.
http://dx.doi.org/10.4086/toc.2007.v003a006
Creative Commons Attribution 3.0 Unported license
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Eilenberg Theorems for Free
Eilenberg-type correspondences, relating varieties of languages (e.g., of finite words, infinite words, or trees) to pseudovarieties of finite algebras, form the backbone of algebraic language theory. We show that they all arise from the same recipe: one models languages and the algebras recognizing them by monads on an algebraic category, and applies a Stone-type duality. Our main
contribution is a variety theorem that covers e.g. Wilke's and Pin's
work on infinity-languages, the variety theorem for cost functions of Daviaud, Kuperberg, and Pin, and unifies the two categorical
approaches of Bojanczyk and of Adamek et al. In addition we derive new results, such as an extension of the local variety theorem of Gehrke, Grigorieff, and Pin from finite to infinite words.
Eilenberg's theorem
variety of languages
pseudovariety
monad
duality
43:1-43:15
Regular Paper
Henning
Urbat
Henning Urbat
Jiri
Adámek
Jiri Adámek
Liang-Ting
Chen
Liang-Ting Chen
Stefan
Milius
Stefan Milius
10.4230/LIPIcs.MFCS.2017.43
J. Adámek, S. Milius, R. Myers, and H. Urbat. Generalized Eilenberg Theorem I: Local Varieties of Languages. In A. Muscholl, editor, Proc. FoSSaCS'14, volume 8412 of LNCS, pages 366-380. Springer, 2014. Full version: URL: http://arxiv.org/pdf/1501.02834v1.pdf.
http://arxiv.org/pdf/1501.02834v1.pdf
J. Adámek, S. Milius, and H. Urbat. Syntactic monoids in a category. In Proc. CALCO'15, LIPIcs. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2015. Full version: URL: http://arxiv.org/abs/1504.02694.
http://arxiv.org/abs/1504.02694
J. Adámek, R. Myers, S. Milius, and H. Urbat. Varieties of languages in a category. In 30th Annual ACM/IEEE Symposium on Logic in Computer Science. IEEE, 2015.
J. Almeida. On pseudovarieties, varieties of languages, filters of congruences, pseudoidentities and related topics. Algebra Universalis, 27(3):333-350, 1990.
N. Bedon and O. Carton. An Eilenberg theorem for words on countable ordinals. In Proc. LATIN'98, volume 1380 of LNCS, pages 53-64. Springer, 1998.
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M. Bojańczyk. Recognisable languages over monads. In I. Potapov, editor, Proc. DLT'15, volume 9168 of LNCS, pages 1-13. Springer, 2015. URL: http://arxiv.org/abs/1502.04898.
http://arxiv.org/abs/1502.04898
L.-T. Chen, J. Adámek, S. Milius, and H. Urbat. Profinite monads, profinite equations and Reiterman’s theorem. In B. Jacobs and C. Löding, editors, Proc. FoSSaCS'16, volume 9634 of LNCS. Springer, 2016. URL: http://arxiv.org/abs/1511.02147.
http://arxiv.org/abs/1511.02147
L.-T. Chen and H. Urbat. A fibrational approach to automata theory. In L. S. Moss and P. Sobocinski, editors, Proc. CALCO'15, LIPIcs. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2015.
L. Daviaud, D. Kuperberg, and J.-É. Pin. Varieties of cost functions. In N. Ollinger and H. Vollmer, editors, Proc. STACS 2016, volume 47 of LIPIcs, pages 30:1-30:14. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2016.
S. Eilenberg. Automata, Languages, and Machines Vol. B. Academic Press, 1976.
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M. Gehrke, S. Grigorieff, and J.-É. Pin. Duality and equational theory of regular languages. In L. Aceto and al., editors, Proc. ICALP'08, Part II, volume 5126 of LNCS, pages 246-257. Springer, 2008.
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D. Perrin and J.-É. Pin. Infinite Words. Elsevier, 2004.
J.-É. Pin. A variety theorem without complementation. Russ. Math., 39:80-90, 1995.
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J.-É. Pin. Mathematical foundations of automata theory. Available at http://www.liafa.jussieu.fr/~jep/PDF/MPRI/MPRI.pdf, November 2016.
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http://dx.doi.org/10.1007/BF02679444
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J. Salamánca. Unveiling Eilenberg-type Correspondences: Birkhoff’s Theorem for (finite) Algebras + Duality. https://arxiv.org/abs/1702.02822, February 2017.
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S. Salehi and M. Steinby. Tree algebras and varieties of tree languages. Theor. Comput. Sci., 377(1-3):1-24, 2007.
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https://www.tu-braunschweig.de/Medien-DB/iti/hspnote.pdf
H. Urbat, J. Adámek, L.-T. Chen, and S. Milius. Eilenberg Theorems for Free. https://arxiv.org/abs/1602.05831. February 2017.
https://arxiv.org/abs/1602.05831
T. Wilke. An Eilenberg theorem for ∞-languages. In Proc. ICALP'91, volume 510 of LNCS, pages 588-599. Springer, 1991.
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Membership Problem in GL(2, Z) Extended by Singular Matrices
We consider the membership problem for matrix semigroups, which is the problem to decide whether a matrix belongs to a given finitely generated matrix semigroup.
In general, the decidability and complexity of this problem for two-dimensional matrix semigroups remains open. Recently there was a significant progress with this open problem by showing that the membership is decidable for 2x2 nonsingular integer matrices. In this paper we focus on the membership for singular integer matrices and prove that this problem is decidable for 2x2 integer matrices whose determinants are equal to 0, 1, -1 (i.e. for matrices from GL(2,Z) and any singular matrices). Our algorithm relies on a translation of numerical problems on matrices into combinatorial problems on words and conversion of the membership problem into decision problem on regular languages.
Matrix Semigroups
Membership Problem
General Linear Group
Singular Matrices
Automata and Formal Languages
44:1-44:13
Regular Paper
Igor
Potapov
Igor Potapov
Pavel
Semukhin
Pavel Semukhin
10.4230/LIPIcs.MFCS.2017.44
László Babai, Robert Beals, Jin-yi Cai, Gábor Ivanyos, and Eugene M. Luks. Multiplicative equations over commuting matrices. In Proceedings of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'96, pages 498-507, Philadelphia, PA, USA, 1996. Society for Industrial and Applied Mathematics.
Paul Bell and Igor Potapov. On undecidability bounds for matrix decision problems. Theoretical Computer Science, 391(1-2):3-13, 2008.
Paul C. Bell, Mika Hirvensalo, and Igor Potapov. Mortality for 2x2 matrices is NP-hard. In Branislav Rovan, Vladimiro Sassone, and Peter Widmayer, editors, Mathematical Foundations of Computer Science 2012, volume 7464 of Lecture Notes in Computer Science, pages 148-159. Springer Berlin Heidelberg, 2012.
Paul C. Bell, Mika Hirvensalo, and Igor Potapov. The identity problem for matrix semigroups in SL₂(ℤ) is NP-complete. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 187-206, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.13.
http://dx.doi.org/10.1137/1.9781611974782.13
Paul C. Bell and Igor Potapov. On the undecidability of the identity correspondence problem and its applications for word and matrix semigroups. Int. J. Found. Comput. Sci., 21(6):963-978, 2010.
Paul C. Bell and Igor Potapov. On the computational complexity of matrix semigroup problems. Fundam. Inf., 116(1-4):1-13, January 2012.
Vincent D. Blondel, Emmanuel Jeandel, Pascal Koiran, and Natacha Portier. Decidable and undecidable problems about quantum automata. SIAM J. Comput., 34(6):1464-1473, June 2005.
Julien Cassaigne, Vesa Halava, Tero Harju, and François Nicolas. Tighter undecidability bounds for matrix mortality, zero-in-the-corner problems, and more. CoRR, abs/1404.0644, 2014.
Christian Choffrut and Juhani Karhumaki. Some decision problems on integer matrices. RAIRO-Theor. Inf. Appl., 39(1):125-131, 2005.
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Joël Ouaknine and James Worrell. On the positivity problem for simple linear recurrence sequences,. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part II, pages 318-329, 2014.
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http://dx.doi.org/10.1137/1.9781611974782.12
Robert A. Rankin. Modular forms and functions. Cambridge University Press, Cambridge-New York-Melbourne, 1977.
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Grammars for Indentation-Sensitive Parsing
Adams' extension of parsing expression grammars enables specifying
indentation sensitivity using two non-standard grammar constructs - indentation by a binary relation and alignment. This paper is a theoretical study of Adams' grammars. It proposes a
step-by-step transformation of well-formed Adams' grammars for
elimination of the alignment construct from the grammar. The idea that alignment could be avoided was suggested by Adams but no
process for achieving this aim has been described before.
This paper also establishes general conditions that binary
relations used in indentation constructs must satisfy in order to enable efficient parsing.
Parsing expression grammars
indentation
grammar transformation
45:1-45:13
Regular Paper
Härmel
Nestra
Härmel Nestra
10.4230/LIPIcs.MFCS.2017.45
Michael D. Adams. The indentation package. URL: http://hackage.haskell.org/package/indentation.
http://hackage.haskell.org/package/indentation
Michael D. Adams. Principled parsing for indentation-sensitive languages: Revisiting Landin’s offside rule. In Roberto Giacobazzi and Radhia Cousot, editors, The 40th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL'13, Rome, Italy - January 23 - 25, 2013, pages 511-522. ACM, 2013. URL: http://dx.doi.org/10.1145/2429069.2429129.
http://dx.doi.org/10.1145/2429069.2429129
Michael D. Adams and Ömer S. Aŭgacan. Indentation-sensitive parsing for Parsec. In Wouter Swierstra, editor, Proceedings of the 2014 ACM SIGPLAN symposium on Haskell, Gothenburg, Sweden, September 4-5, 2014, pages 121-132. ACM, 2014. URL: http://dx.doi.org/10.1145/2633357.2633369.
http://dx.doi.org/10.1145/2633357.2633369
Alfred V. Aho and Jeffrey D. Ullman. The Theory of Parsing, Translation, and Compiling. Prentice-Hall, Inc., Upper Saddle River, NJ, USA, 1972.
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Bryan Ford. Parsing expression grammars: A recognition-based syntactic foundation. In Neil D. Jones and Xavier Leroy, editors, Proceedings of the 31st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2004, Venice, Italy, January 14-16, 2004, pages 111-122. ACM, 2004. URL: http://dx.doi.org/10.1145/964001.964011.
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Tetsuro Matsumura and Kimio Kuramitsu. A declarative extension of parsing expression grammars for recognizing most programming languages. JIP, 24(2):256-264, 2016. URL: http://dx.doi.org/10.2197/ipsjjip.24.256.
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Sérgio Medeiros, Fabio Mascarenhas, and Roberto Ierusalimschy. Left recursion in parsing expression grammars. In Francisco Heron de Carvalho Junior and Luís Soares Barbosa, editors, Programming Languages - 16th Brazilian Symposium, SBLP 2012, Natal, Brazil, September 23-28, 2012. Proceedings, volume 7554 of Lecture Notes in Computer Science, pages 27-41. Springer, 2012. URL: http://dx.doi.org/10.1007/978-3-642-33182-4_4.
http://dx.doi.org/10.1007/978-3-642-33182-4_4
Härmel Nestra. Alignment elimination from Adams' grammars, 2017. URL: http://arxiv.org/abs/1706.06497.
http://arxiv.org/abs/1706.06497
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The Power of Linear-Time Data Reduction for Maximum Matching
Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph primitives. For m-edge and n-vertex graphs, it is well-known to be solvable in O(m\sqrt{n}) time; however, for several applications this running time is still too slow. We investigate how linear-time (and almost linear-time) data reduction (used as preprocessing) can alleviate the situation. More specifically, we focus on linear-time kernelization. We start a deeper and systematic study both for general graphs and for bipartite graphs. Our data reduction algorithms easily comply (in form of preprocessing) with every solution strategy (exact, approximate, heuristic), thus making them attractive in various settings.
Maximum-cardinality matching
bipartite graphs
linear-time algorithms
kernelization
parameterized complexity analysis
FPT in P
46:1-46:14
Regular Paper
George B.
Mertzios
George B. Mertzios
André
Nichterlein
André Nichterlein
Rolf
Niedermeier
Rolf Niedermeier
10.4230/LIPIcs.MFCS.2017.46
Reuven Bar-Yehuda, Dan Geiger, Joseph Naor, and Ron M. Roth. Approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and Bayesian inference. SIAM Journal on Computing, 27(4):942-959, 1998. URL: http://dx.doi.org/10.1137/S0097539796305109.
http://dx.doi.org/10.1137/S0097539796305109
Norbert Blum. A new approach to maximum matching in general graphs. In Proceedings of the 17th International Colloquium on Automata, Languages, and Programming (ICALP '90), volume 443 of LNCS, pages 586-597. Springer, 1990. URL: http://dx.doi.org/10.1007/BFb0032060.
http://dx.doi.org/10.1007/BFb0032060
Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Preprocessing for treewidth: A combinatorial analysis through kernelization. SIAM Journal on Discrete Mathematics, 27(4):2108-2142, 2013. URL: http://dx.doi.org/10.1137/120903518.
http://dx.doi.org/10.1137/120903518
Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM Journal on Discrete Mathematics, 28(1):277-305, 2014. URL: http://dx.doi.org/10.1137/120880240.
http://dx.doi.org/10.1137/120880240
Andreas Brandstädt, Van Bang Le, and Jeremy P. Spinrad. Graph Classes: a Survey, volume 3 of SIAM Monographs on Discrete Mathematics and Applications. SIAM, 1999.
Leizhen Cai. Parameterized complexity of Vertex Colouring. Discrete Applied Mathematics, 127(1):415-429, 2003. URL: http://dx.doi.org/10.1016/S0166-218X(02)00242-1.
http://dx.doi.org/10.1016/S0166-218X(02)00242-1
Maw-Shang Chang. Algorithms for maximum matching and minimum fill-in on chordal bipartite graphs. In Proceedings of the 7th International Symposium on Algorithms and Computation (ISAAC '96), volume 1178 of LNCS, pages 146-155. Springer, 1996. URL: http://dx.doi.org/10.1007/BFb0009490.
http://dx.doi.org/10.1007/BFb0009490
Elias Dahlhaus and Marek Karpinski. Matching and multidimensional matching in chordal and strongly chordal graphs. Discrete Applied Mathematics, 84(1–3):79-91, 1998. URL: http://dx.doi.org/10.1016/S0166-218X(98)00006-7.
http://dx.doi.org/10.1016/S0166-218X(98)00006-7
Ran Duan and Seth Pettie. Linear-time approximation for maximum weight matching. Journal of the ACM, 61(1):1:1-1:23, 2014. URL: http://dx.doi.org/10.1145/2529989.
http://dx.doi.org/10.1145/2529989
Fedor V. Fomin, Daniel Lokshtanov, Michal Pilipczuk, Saket Saurabh, and Marcin Wrochna. Fully polynomial-time parameterized computations for graphs and matrices of low treewidth. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '17), pages 1419-1432. SIAM, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.92.
http://dx.doi.org/10.1137/1.9781611974782.92
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Harold N. Gabow and Robert Endre Tarjan. Faster scaling algorithms for general graph-matching problems. Journal of the ACM, 38(4):815-853, 1991. URL: http://dx.doi.org/10.1145/115234.115366.
http://dx.doi.org/10.1145/115234.115366
Archontia C. Giannopoulou, George B. Mertzios, and Rolf Niedermeier. Polynomial fixed-parameter algorithms: A case study for longest path on interval graphs. Theoretical Computer Science, 2017. Available online. URL: http://dx.doi.org/10.1016/j.tcs.2017.05.017.
http://dx.doi.org/10.1016/j.tcs.2017.05.017
Jiong Guo, Falk Hüffner, and Rolf Niedermeier. A structural view on parameterizing problems: Distance from triviality. In Proceedings of the 1st International Workshop on Parameterized and Exact Computation (IWPEC '04), volume 3162 of LNCS, pages 162-173. Springer, 2004. URL: http://dx.doi.org/10.1007/978-3-540-28639-4_15.
http://dx.doi.org/10.1007/978-3-540-28639-4_15
Manoj Gupta and Richard Peng. Fully dynamic (1+ e)-approximate matchings. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS '13), pages 548-557. IEEE, 2013. URL: http://dx.doi.org/10.1109/FOCS.2013.65.
http://dx.doi.org/10.1109/FOCS.2013.65
John E. Hopcroft and Richard M. Karp. An n^5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing, 2(4):225-231, 1973. URL: http://dx.doi.org/10.1137/0202019.
http://dx.doi.org/10.1137/0202019
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http://dx.doi.org/10.1109/SFCS.1981.21
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http://dx.doi.org/10.1007/978-3-642-32589-2_2
George B. Mertzios, André Nichterlein, and Rolf Niedermeier. Linear-time algorithm for maximum-cardinality matching on cocomparability graphs. CoRR, abs/1703.05598, 2017.
Silvio Micali and Vijay V. Vazirani. An O(√|V| |E|) algorithm for finding maximum matching in general graphs. In Proceedings of the 21st Annual IEEE Symposium on Foundations of Computer Science (FOCS '80), pages 17-27. IEEE, 1980. URL: http://dx.doi.org/10.1109/SFCS.1980.12.
http://dx.doi.org/10.1109/SFCS.1980.12
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http://dx.doi.org/10.1109/FOCS.2004.40
Steven S. Skiena. The Algorithm Design Manual. Springer, 2010.
G. Steiner and J. S. Yeomans. A linear time algorithm for maximum matchings in convex bipartite graphs. Comput. Math. Appl., 31:91-96, 1996. URL: http://dx.doi.org/10.1016/0898-1221(96)00079-X.
http://dx.doi.org/10.1016/0898-1221(96)00079-X
Ryan Williams, Carla P. Gomes, and Bart Selman. Backdoors to typical case complexity. In Proceedings of the 18th International Joint Conference on Artificial Intelligence (IJCAI '03), pages 1173-1178. Morgan Kaufmann, 2003.
Raphael Yuster. Maximum matching in regular and almost regular graphs. Algorithmica, 66(1):87-92, 2013. URL: http://dx.doi.org/10.1007/s00453-012-9625-7.
http://dx.doi.org/10.1007/s00453-012-9625-7
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Two-Planar Graphs Are Quasiplanar
It is shown that every 2-planar graph is quasiplanar, that is, if a simple graph admits a drawing in the plane such that every edge is crossed at most twice, then it also admits a drawing in which no three edges pairwise cross. We further show that quasiplanarity is witnessed by a simple topological drawing, that is, any two edges cross at most once and adjacent edges do not cross.
graph drawing
near-planar graph
simple topological plane graph
47:1-47:14
Regular Paper
Michael
Hoffmann
Michael Hoffmann
Csaba D.
Tóth
Csaba D. Tóth
10.4230/LIPIcs.MFCS.2017.47
Eyal Ackerman. On the maximum number of edges in topological graphs with no four pairwise crossing edges. Discrete Comput. Geom., 41(3):365-375, 2009. URL: http://dx.doi.org/10.1007/s00454-009-9143-9.
http://dx.doi.org/10.1007/s00454-009-9143-9
Eyal Ackerman. On topological graphs with at most four crossings per edge. CoRR, abs/1509.01932:1-41, 2015. URL: https://arxiv.org/abs/1509.01932.
https://arxiv.org/abs/1509.01932
Eyal Ackerman and Gábor Tardos. On the maximum number of edges in quasi-planar graphs. J. Combin. Theory Ser. A, 114(3):563-571, 2007. URL: http://dx.doi.org/10.1016/j.jcta.2006.08.002.
http://dx.doi.org/10.1016/j.jcta.2006.08.002
Pankaj K. Agarwal, Boris Aronov, János Pach, Richard Pollack, and Micha Sharir. Quasi-planar graphs have a linear number of edges. Combinatorica, 17:1-9, 1997. URL: http://dx.doi.org/10.1007/BF01196127.
http://dx.doi.org/10.1007/BF01196127
Miklós Ajtai, Václav Chvátal, Monroe Newborn, and Endre Szemerédi. Crossing-free subgraphs. Ann. Discrete Math., 12:9-12, 1982. URL: http://dx.doi.org/10.1016/S0304-0208(08)73484-4.
http://dx.doi.org/10.1016/S0304-0208(08)73484-4
Patrizio Angelini, Michael A. Bekos, Franz J. Brandenburg, Giordano Da Lozzo, Giuseppe Di Battista, Walter Didimo, Giuseppe Liotta, Fabrizio Montecchiani, and Ignaz Rutter. On the relationship between k-planar and k-quasi planar graphs. In Proc. 43rd Internat. Workshop Graph-Theoret. Concepts Comput. Sci., Lecture Notes Comput. Sci. Springer, 2017. to appear; preliminary version available at arXiv:1702.08716. URL: http://arxiv.org/abs/1702.08716.
http://arxiv.org/abs/1702.08716
Michael Bekos, Michael Kaufmann, and Chrysanthi Raftopoulou. On optimal 2- and 3-planar graphs. In Proc. 33rd Internat. Sympos. Comput. Geom., LIPIcs. SchloßDagstuhl, 2017. to appear; preliminary version available at arXiv:1703.06526. URL: http://arxiv.org/abs/1702.08716.
http://arxiv.org/abs/1702.08716
Franz J. Brandenburg, Walter Didimo, William S. Evans, Philipp Kindermann, Giuseppe Liotta, and Fabrizio Montecchiani. Recognizing and drawing IC-planar graphs. Theoret. Comput. Sci., 636:1-16, 2016. URL: http://dx.doi.org/10.1016/j.tcs.2016.04.026.
http://dx.doi.org/10.1016/j.tcs.2016.04.026
Christian A. Duncan. On graph thickness, geometric thickness, and separator theorems. Comput. Geom. Theory Appl., 44(2):95-99, 2011. URL: http://dx.doi.org/10.1016/j.comgeo.2010.09.005.
http://dx.doi.org/10.1016/j.comgeo.2010.09.005
David Eppstein, Philipp Kindermann, Stephen G. Kobourov, Giuseppe Liotta, Anna Lubiw, Aude Maignan, Debajyoti Mondal, Hamideh Vosoughpour, Sue Whitesides, and Stephen K. Wismath. On the planar split thickness of graphs. In Proc. 12th Latin Amer. Sympos. Theoretical Informatics, volume 9644 of Lecture Notes Comput. Sci., pages 403-415. Springer, 2016. URL: http://dx.doi.org/10.1007/978-3-662-49529-2_30.
http://dx.doi.org/10.1007/978-3-662-49529-2_30
Jacob Fox and János Pach. Applications of a new separator theorem for string graphs. Combinatorics, Probability and Computing, 23(1):66-74, 2012. URL: http://dx.doi.org/10.1017/S0963548313000412.
http://dx.doi.org/10.1017/S0963548313000412
Jacob Fox and János Pach. Coloring K_k-free intersection graphs of geometric objects in the plane. European J. Combin., 33(5):853-866, 2012. URL: http://dx.doi.org/10.1016/j.ejc.2011.09.021.
http://dx.doi.org/10.1016/j.ejc.2011.09.021
Jacob Fox, János Pach, and Andrew Suk. The number of edges in k-quasi-planar graphs. SIAM J. Discrete Math., 27(1):550-561, 2013. URL: http://dx.doi.org/10.1137/110858586.
http://dx.doi.org/10.1137/110858586
Radoslav Fulek, Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič. Adjacent crossings do matter. J. Graph Algorithms Appl., 16(3):759-782, 2012. URL: http://dx.doi.org/10.7155/jgaa.00266.
http://dx.doi.org/10.7155/jgaa.00266
Michael Hoffmann and Csaba D. Tóth. Two-planar graphs are quasiplanar. CoRR, abs/1705.05569:1-22, 2017. URL: https://arxiv.org/abs/1705.05569.
https://arxiv.org/abs/1705.05569
Stephen G. Kobourov, Giuseppe Liotta, and Fabrizio Montecchiani. An annotated bibliography on 1-planarity. CoRR, abs/1703.02261:1-38, 2017. URL: https://arxiv.org/abs/1703.02261.
https://arxiv.org/abs/1703.02261
László Lovász. Graph minor theory. Bulletin of the American Mathematical Society, 43(1):75-86, 2006. URL: http://dx.doi.org/10.1090/S0273-0979-05-01088-8.
http://dx.doi.org/10.1090/S0273-0979-05-01088-8
János Pach, Radoš Radoičić, Gábor Tardos, and Géza Tóth. Improving the crossing lemma by finding more crossings in sparse graphs. Discrete Comput. Geom., 36(4):527-552, 2006. URL: http://dx.doi.org/10.1007/s00454-006-1264-9.
http://dx.doi.org/10.1007/s00454-006-1264-9
János Pach, Radoš Radoičić, and Géza Tóth. Relaxing planarity for topological graphs. In More Graphs, Sets and Numbers, volume 15 of Bolyai Soc. Math. Studies, pages 285-300. Springer, 2006. URL: http://dx.doi.org/10.1007/978-3-540-32439-3_12.
http://dx.doi.org/10.1007/978-3-540-32439-3_12
János Pach, Farhad Shahrokhi, and Mario Szegedy. Applications of the crossing number. Algorithmica, 16(1):111-117, 1996. URL: http://dx.doi.org/10.1007/BF02086610.
http://dx.doi.org/10.1007/BF02086610
János Pach and Géza Tóth. Graphs drawn with few crossings per edge. Combinatorica, 17(3):427-439, 1997. URL: http://dx.doi.org/10.1007/BF01215922.
http://dx.doi.org/10.1007/BF01215922
Marcus Schaefer. The graph crossing number and its variants: a survey. Electronic J. Combinatorics, #DS21:1-100, 2014. URL: http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS21.
http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS21
Andrew Suk and Bartosz Walczak. New bounds on the maximum number of edges in k-quasi-planar graphs. Comput. Geom. Theory Appl., 50:24-33, 2015. URL: http://dx.doi.org/10.1016/j.comgeo.2015.06.001.
http://dx.doi.org/10.1016/j.comgeo.2015.06.001
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
The Shortest Identities for Max-Plus Automata with Two States
Max-plus automata are quantitative extensions of automata designed to associate an integer with every non-empty word. A pair of distinct words is said to be an identity for a class of max-plus automata if each of the automata in the class computes the same value on the two words. We give the shortest identities holding for the class of max-plus automata with two states. For this, we exhibit an interesting list of necessary conditions for an identity to hold. Moreover, this result provides a counter-example of a conjecture of Izhakian, concerning the minimality of certain identities.
Max-plus automata
Weighted automata
Identities
Tropical matrices
48:1-48:13
Regular Paper
Laure
Daviaud
Laure Daviaud
Marianne
Johnson
Marianne Johnson
10.4230/LIPIcs.MFCS.2017.48
Y. Chen, X. Hu, Y. Luo, and O. Sapir. The finite basis problem for the monoid of two-by-two upper triangular tropical matrices. Bull. Aust. Math. Soc., 94(1):54-64, 2016. URL: http://dx.doi.org/10.1017/S0004972715001483.
http://dx.doi.org/10.1017/S0004972715001483
T. Colcombet, L. Daviaud, and F. Zuleger. Size-change abstraction and max-plus automata. In Erzsébet Csuhaj-Varjú, Martin Dietzfelbinger, and Zoltán Ésik, editors, Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Budapest, Hungary, August 25-29, 2014. Proceedings, Part I, volume 8634 of Lecture Notes in Computer Science, pages 208-219. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44522-8_18.
http://dx.doi.org/10.1007/978-3-662-44522-8_18
L. Daviaud, M. Johnson, and M. Kambites. Identities in upper triangular tropical matrix semigroups and the bicyclic monoid, 2017. preprint, http://arxiv.org/abs/1612.04219.
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Z. Izhakian. Semigroup identities of tropical matrix semigroups of maximal rank. Semigroup Forum, 92(3):712-732, 2016. URL: http://dx.doi.org/10.1007/s00233-015-9765-6.
http://dx.doi.org/10.1007/s00233-015-9765-6
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http://dx.doi.org/10.1007/s00233-009-9203-8
J. Okniński. Identities of the semigroup of upper triangular tropical matrices. Comm. Algebra, 43(10):4422-4426, 2015. URL: http://dx.doi.org/10.1080/00927872.2014.946141.
http://dx.doi.org/10.1080/00927872.2014.946141
J. Sakarovitch. Elements of Automata Theory. Cambridge University Press, 2009.
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Y. Shitov. A semigroup identity for tropical 3×3 matrices, 2014. To appear in Ars Mathematica Contemporanea 14 (2018), 15-23.
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On the Upward/Downward Closures of Petri Nets
We study the size and the complexity of computing finite state automata (FSA) representing and approximating the downward and the upward closure of Petri net languages with coverability as the acceptance condition.
We show how to construct an FSA recognizing the upward closure of a Petri net language in doubly-exponential time, and therefore the size is at most doubly exponential.
For downward closures, we prove that the size of the minimal automata can be non-primitive recursive.
In the case of BPP nets, a well-known subclass of Petri nets, we show that an FSA accepting the downward/upward closure can be constructed in exponential time.
Furthermore, we consider the problem of checking whether a simple regular language is included in the downward/upward closure of a Petri net/BPP net language.
We show that this problem is EXPSPACE-complete (resp. NP-complete) in the case of Petri nets (resp. BPP nets).
Finally, we show that it is decidable whether a Petri net language is upward/downward closed.
Petri nets
BPP nets
downward closure
upward closure
49:1-49:14
Regular Paper
Mohamed Faouzi
Atig
Mohamed Faouzi Atig
Roland
Meyer
Roland Meyer
Sebastian
Muskalla
Sebastian Muskalla
Prakash
Saivasan
Prakash Saivasan
10.4230/LIPIcs.MFCS.2017.49
P. A. Abdulla, A. Collomb-Annichini, A. Bouajjani, and B. Jonsson. Using forward reachability analysis for verification of lossy channel systems. FMSD, 25(1), 2004.
P. A. Abdulla, G. Delzanno, and L. V. Begin. Comparing the expressive power of well-structured transition systems. In CSL, LNCS. Springer, 2007.
M. F. Atig, A. Bouajjani, K. Narayan Kumar, and P. Saivasan. On bounded reachability analysis of shared memory systems. In FSTTCS, LIPIcs. Dagstuhl, 2014.
M. F. Atig, A. Bouajjani, and S. Qadeer. Context-bounded analysis for concurrent programs with dynamic creation of threads. LMCS, 7(4), 2011.
M. F. Atig, A. Bouajjani, and T. Touili. On the reachability analysis of acyclic networks of pushdown systems. In CONCUR, LNCS. Springer, 2008.
M. F. Atig, D. Chistikov, P. Hofman, K. N. Kumar, P. Saivasan, and G. Zetzsche. Complexity of regular abstractions of one-counter languages. In LICS, pages 207-216. ACM, 2016.
M. F. Atig, R. Meyer, S. Muskalla, and P. Saivasan. On the upward/downward closures of petri nets. CoRR, abs/1701.02927, 2017. URL: http://arxiv.org/abs/1701.02927.
http://arxiv.org/abs/1701.02927
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M. Hague, J. Kochems, and C.-H. Luke Ong. Unboundedness and downward closures of higher-order pushdown automata. In POPL. ACM, 2016.
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S. La Torre, A. Muscholl, and I. Walukiewicz. Safety of parametrized asynchronous shared-memory systems is almost always decidable. In CONCUR, LIPIcs. Dagstuhl, 2015.
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J. Leroux. Vector addition system reachability problem: a short self-contained proof. In POPL. ACM, 2011.
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J. Leroux, M. Praveen, and G. Sutre. A relational trace logic for vector addition systems with application to context-freeness. In CONCUR, pages 137-151. Springer, 2013.
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On Multidimensional and Monotone k-SUM
The well-known k-SUM conjecture is that integer k-SUM requires time Omega(n^{\ceil{k/2}-o(1)}). Recent work has studied multidimensional k-SUM in F_p^d, where the best known algorithm takes time \tilde O(n^{\ceil{k/2}}). Bhattacharyya et al. [ICS 2011] proved a min(2^{\Omega(d)},n^{\Omega(k)}) lower bound for k-SUM in F_p^d under the Exponential Time Hypothesis. We give a more refined lower bound under the standard k-SUM conjecture: for sufficiently large p, k-SUM in F_p^d requires time Omega(n^{k/2-o(1)}) if k is even, and Omega(n^{\ceil{k/2}-2k(log k)/(log p)-o(1)}) if k is odd.
For a special case of the multidimensional problem, bounded monotone d-dimensional 3SUM, Chan and Lewenstein [STOC 2015] gave a surprising \tilde O(n^{2-2/(d+13)}) algorithm using additive combinatorics. We show this algorithm is essentially optimal. To be more precise, bounded monotone d-dimensional 3SUM requires time Omega(n^{2-\frac{4}{d}-o(1)}) under the standard 3SUM conjecture, and time Omega(n^{2-\frac{2}{d}-o(1)}) under the so-called strong 3SUM conjecture. Thus, even though one might hope to further exploit the structural advantage of monotonicity, no substantial improvements beyond those obtained by Chan and Lewenstein are possible for bounded monotone d-dimensional 3SUM.
3SUM
kSUM
monotone 3SUM
strong 3SUM conjecture
50:1-50:13
Regular Paper
Chloe Ching-Yun
Hsu
Chloe Ching-Yun Hsu
Chris
Umans
Chris Umans
10.4230/LIPIcs.MFCS.2017.50
Amir Abboud and Kevin Lewi. Exact weight subgraphs and the k-SUM conjecture. In International Colloquium on Automata, Languages, and Programming, pages 1-12. Springer, 2013.
Amir Abboud, Kevin Lewi, and Ryan Williams. Losing weight by gaining edges. In European Symposium on Algorithms, pages 1-12. Springer, 2014.
Nir Ailon and Bernard Chazelle. Lower bounds for linear degeneracy testing. Journal of the ACM (JACM), 52(2):157-171, 2005.
Amihood Amir, Timothy M Chan, Moshe Lewenstein, and Noa Lewenstein. On hardness of jumbled indexing. In Automata, Languages, and Programming, pages 114-125. Springer, 2014.
Ilya Baran, Erik D Demaine, and Mihai Pătraşcu. Subquadratic algorithms for 3SUM. Algorithmica, 50(4):584-596, 2008.
Arnab Bhattacharyya, Piotr Indyk, David P Woodruff, and Ning Xie. The complexity of linear dependence problems in vector spaces. In ICS, pages 496-508, 2011.
Timothy M Chan and Moshe Lewenstein. Clustered integer 3SUM via additive combinatorics. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, pages 31-40. ACM, 2015.
Rod G Downey, Michael R Fellows, Alexander Vardy, and Geoff Whittle. The parametrized complexity of some fundamental problems in coding theory. SIAM Journal on Computing, 29(2):545-570, 1999.
Jeff Erickson. Lower bounds for linear satisfiability problems. In Chicago Journal of Theoretical Computer Science, volume 8, 1999.
Ari Freund. Improved subquadratic 3sum. Algorithmica, pages 1-19, 2015.
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Omer Gold and Micha Sharir. Improved bounds for 3sum, k-sum, and linear degeneracy. arXiv preprint arXiv:1512.05279, 2015.
Isaac Goldstein, Tsvi Kopelowitz, Moshe Lewenstein, and Ely Porat. How hard is it to find (honest) witnesses? In LIPIcs-Leibniz International Proceedings in Informatics, volume 57. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016.
Allan Grønlund and Seth Pettie. Threesomes, degenerates, and love triangles. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 621-630. IEEE, 2014.
Zahra Jafargholi and Emanuele Viola. 3SUM, 3XOR, triangles. Algorithmica, 74(1):326-343, 2016.
Tsvi Kopelowitz, Seth Pettie, and Ely Porat. Higher lower bounds from the 3sum conjecture. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1272-1287. Society for Industrial and Applied Mathematics, 2016.
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Mihai Pătraşcu and Ryan Williams. On the possibility of faster SAT algorithms. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1065-1075. SIAM, 2010.
Virginia Vassilevska and Ryan Williams. Finding, minimizing, and counting weighted subgraphs. In Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, pages 455-464. ACM, 2009.
Emanuele Viola. Reducing 3XOR to listing triangles, an exposition. In Electronic Colloquium on Computational Complexity (ECCC), volume 18, page 113, 2011.
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Parameterized Complexity of the List Coloring Reconfiguration Problem with Graph Parameters
Let G be a graph such that each vertex has its list of available colors, and assume that each list is a subset of the common set consisting of k colors. For two given list colorings of G, we study the problem of transforming one into the other by changing only one vertex color assignment at a time, while at all times maintaining a list coloring. This problem is known to be PSPACE-complete even for bounded bandwidth graphs and a fixed constant k. In this paper, we study the fixed-parameter tractability of the problem when parameterized by several graph parameters. We first give a fixed-parameter algorithm for the problem when parameterized by k and the modular-width of an input graph. We next give a fixed-parameter algorithm for the shortest variant which computes the length of a shortest transformation when parameterized by k and the size of a minimum vertex cover of an input graph. As corollaries, we show that the problem for cographs and the shortest variant for split graphs are fixed-parameter tractable even when only k is taken as a parameter. On the other hand, we prove that the problem is W[1]-hard when parameterized only by the size of a minimum vertex cover of an input graph.
combinatorial reconfiguration
fixed-parameter tractability
graph algorithm
list coloring
W[1]-hardness
51:1-51:13
Regular Paper
Tatsuhiko
Hatanaka
Tatsuhiko Hatanaka
Takehiro
Ito
Takehiro Ito
Xiao
Zhou
Xiao Zhou
10.4230/LIPIcs.MFCS.2017.51
Marthe Bonamy and Nicolas Bousquet. Recoloring bounded treewidth graphs. Electronic Notes in Discrete Mathematics, 44:257-262, 2013. URL: http://dx.doi.org/10.1016/j.endm.2013.10.040.
http://dx.doi.org/10.1016/j.endm.2013.10.040
Marthe Bonamy, Matthew Johnson, Ioannis Lignos, Viresh Patel, and Daniël Paulusma. Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs. Journal of Combinatorial Optimization, 27(1):132-143, 2014. URL: http://dx.doi.org/10.1007/s10878-012-9490-y.
http://dx.doi.org/10.1007/s10878-012-9490-y
Paul Bonsma and Luis Cereceda. Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theoretical Computer Science, 410(50):5215-5226, 2009. URL: http://dx.doi.org/10.1016/j.tcs.2009.08.023.
http://dx.doi.org/10.1016/j.tcs.2009.08.023
Paul Bonsma, Amer E. Mouawad, Naomi Nishimura, and Venkatesh Raman. The complexity of bounded length graph recoloring and CSP reconfiguration. In Parameterized and Exact Computation - 9th International Symposium, IPEC 2014, Wroclaw, Poland, September 10-12, 2014. Revised Selected Papers, pages 110-121, 2014. URL: http://dx.doi.org/10.1007/978-3-319-13524-3_10.
http://dx.doi.org/10.1007/978-3-319-13524-3_10
Richard C. Brewster, Sean McGuinness, Benjamin Moore, and Jonathan A. Noel. A dichotomy theorem for circular colouring reconfiguration. Theoretical Computer Science, 639:1-13, 2016. URL: http://dx.doi.org/10.1016/j.tcs.2016.05.015.
http://dx.doi.org/10.1016/j.tcs.2016.05.015
Luis Cereceda. Mixing Graph Colourings. PhD thesis, The London School of Economics and Political Science, 2007.
Luis Cereceda, Jan van den Heuvel, and Matthew Johnson. Finding paths between 3-colorings. Journal of Graph Theory, 67(1):69-82, 2011. URL: http://dx.doi.org/10.1002/jgt.20514.
http://dx.doi.org/10.1002/jgt.20514
Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Springer-Verlag, 1999.
Martin Dyer, Abraham D. Flaxman, Alan M. Frieze, and Eric Vigoda. Randomly coloring sparse random graphs with fewer colors than the maximum degree. Random Structures &Algorithms, 29(4):450-465, 2006. URL: http://dx.doi.org/10.1002/rsa.20129.
http://dx.doi.org/10.1002/rsa.20129
Jakub Gajarský, Michael Lampis, and Sebastian Ordyniak. Parameterized algorithms for modular-width. In Parameterized and Exact Computation - 8th International Symposium, IPEC 2013, Sophia Antipolis, France, September 4-6, 2013, Revised Selected Papers, pages 163-176, 2013. URL: http://dx.doi.org/10.1007/978-3-319-03898-8_15.
http://dx.doi.org/10.1007/978-3-319-03898-8_15
Tibor Gallai. Transitiv orientierbare graphen. Acta Mathematica Academiae Scientiarum Hungarica, 18(1):25-66, 1967. URL: http://dx.doi.org/10.1007/BF02020961.
http://dx.doi.org/10.1007/BF02020961
Michel Habib and Christophe Paul. A survey of the algorithmic aspects of modular decomposition. Computer Science Review, 4(1):41-59, 2010. URL: http://dx.doi.org/10.1016/j.cosrev.2010.01.001.
http://dx.doi.org/10.1016/j.cosrev.2010.01.001
Tatsuhiko Hatanaka, Takehiro Ito, and Xiao Zhou. The list coloring reconfiguration problem for bounded pathwidth graphs. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E98.A(6):1168-1178, 2015. URL: http://dx.doi.org/10.1587/transfun.E98.A.1168.
http://dx.doi.org/10.1587/transfun.E98.A.1168
Takehiro Ito, Erik D. Demaine, Nicholas J.A. Harvey, Christos H. Papadimitriou, Martha Sideri, Ryuhei Uehara, and Yushi Uno. On the complexity of reconfiguration problems. Theoretical Computer Science, 412(12):1054-1065, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2010.12.005.
http://dx.doi.org/10.1016/j.tcs.2010.12.005
Matthew Johnson, Dieter Kratsch, Stefan Kratsch, Viresh Patel, and Daniël Paulusma. Finding shortest paths between graph colourings. Algorithmica, 75(2):295-321, 2016. URL: http://dx.doi.org/10.1007/s00453-015-0009-7.
http://dx.doi.org/10.1007/s00453-015-0009-7
Sampath Kannan, Moni Naor, and Steven Rudich. Implicat representation of graphs. SIAM Journal on Discrete Mathematics, 5(4):596-603, 1992. URL: http://dx.doi.org/10.1137/0405049.
http://dx.doi.org/10.1137/0405049
Ross M. McConnell and Fabien de Montgolfier. Linear-time modular decomposition of directed graphs. Discrete Applied Mathematics, 145(2):198-209, 2005. URL: http://dx.doi.org/10.1016/j.dam.2004.02.017.
http://dx.doi.org/10.1016/j.dam.2004.02.017
Jan van den Heuvel. The complexity of change. In Surveys in Combinatorics 2013, pages 127-160. 2013. URL: http://dx.doi.org/10.1017/CBO9781139506748.005.
http://dx.doi.org/10.1017/CBO9781139506748.005
Marcin Wrochna. Reconfiguration in bounded bandwidth and treedepth. CoRR, abs/1405.0847, 2014.
Marcin Wrochna. Homomorphism reconfiguration via homotopy. In 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, pages 730-742, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.730.
http://dx.doi.org/10.4230/LIPIcs.STACS.2015.730
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Automata in the Category of Glued Vector Spaces
In this paper we adopt a category-theoretic approach to the conception of automata classes enjoying minimization by design. The main instantiation of our construction is a new class of automata that are hybrid between deterministic automata and automata weighted over a field.
hybrid set-vector automata
automata minimization in a category
52:1-52:14
Regular Paper
Thomas
Colcombet
Thomas Colcombet
Daniela
Petrisan
Daniela Petrisan
10.4230/LIPIcs.MFCS.2017.52
Jirí Adámek, Stefan Milius, Lawrence S. Moss, and Lurdes Sousa. Well-pointed coalgebras. Logical Methods in Computer Science, 9(3), 2013.
Jiří Adámek, Filippo Bonchi, Mathias Hülsbusch, Barbara König, Stefan Milius, and Alexandra Silva. A coalgebraic perspective on minimization and determinization. In Proceedings of the 15th International Conference on Foundations of Software Science and Computational Structures, FOSSACS'12, pages 58-73, Berlin, Heidelberg, 2012. Springer-Verlag. URL: http://dx.doi.org/10.1007/978-3-642-28729-9_4.
http://dx.doi.org/10.1007/978-3-642-28729-9_4
Michael A. Arbib and Ernest G. Manes. Adjoint machines, state-behavior machines, and duality. Journal of Pure and Applied Algebra, 6(3):313 - 344, 1975. URL: http://dx.doi.org/10.1016/0022-4049(75)90028-6.
http://dx.doi.org/10.1016/0022-4049(75)90028-6
Nick Bezhanishvili, Clemens Kupke, and Prakash Panangaden. Minimization via Duality, pages 191-205. Springer Berlin Heidelberg, Berlin, Heidelberg, 2012. URL: http://dx.doi.org/10.1007/978-3-642-32621-9_14.
http://dx.doi.org/10.1007/978-3-642-32621-9_14
Filippo Bonchi, Marcello Bonsangue, Michele Boreale, Jan Rutten, and Alexandra Silva. A coalgebraic perspective on linear weighted automata. Inf. Comput., 211:77-105, February 2012. URL: http://dx.doi.org/10.1016/j.ic.2011.12.002.
http://dx.doi.org/10.1016/j.ic.2011.12.002
Filippo Bonchi, Marcello M. Bonsangue, Helle Hvid Hansen, Prakash Panangaden, Jan J. M. M. Rutten, and Alexandra Silva. Algebra-coalgebra duality in Brzozowski’s minimization algorithm. ACM Trans. Comput. Log., 15(1):3:1-3:29, 2014.
Christian Choffrut. A generalization of Ginsburg and Rose’s characterization of G-S-M mappings. In ICALP, volume 71 of Lecture Notes in Computer Science, pages 88-103. Springer, 1979.
Thomas Colcombet and Daniela Petrişan. Automata minimization: a functorial approach. CALCO, 72:8:1-8:15, 2017.
Brian J. Day and Stephen Lack. Limits of small functors. Journal of Pure and Applied Algebra, 210(3):651 - 663, 2007. URL: http://dx.doi.org/10.1016/j.jpaa.2006.10.019.
http://dx.doi.org/10.1016/j.jpaa.2006.10.019
J. A. Goguen. Minimal realization of machines in closed categories. Bull. Amer. Math. Soc., 78(5):777-783, 09 1972. URL: http://projecteuclid.org/euclid.bams/1183533991.
http://projecteuclid.org/euclid.bams/1183533991
Helle Hvid Hansen. Subsequential transducers: a coalgebraic perspective. Inf. Comput., 208(12):1368-1397, 2010.
Sylvain Lombardy and Jacques Sakarovitch. Sequential? Theor. Comput. Sci., 356(1-2):224-244, 2006. URL: http://dx.doi.org/10.1016/j.tcs.2006.01.028.
http://dx.doi.org/10.1016/j.tcs.2006.01.028
M.P. Schützenberger. On the definition of a family of automata. Information and Control, 4(2):245 - 270, 1961. URL: http://dx.doi.org/10.1016/S0019-9958(61)80020-X.
http://dx.doi.org/10.1016/S0019-9958(61)80020-X
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The Equivalence, Unambiguity and Sequentiality Problems of Finitely Ambiguous Max-Plus Tree Automata are Decidable
We show that the equivalence, unambiguity and sequentiality problems are decidable for finitely ambiguous max-plus tree automata.
Tree Automata
Max-Plus Automata
Equivalence
Unambiguity
Sequentiality
Decidability
53:1-53:13
Regular Paper
Erik
Paul
Erik Paul
10.4230/LIPIcs.MFCS.2017.53
Sebastian Bala and Artur Koniński. Unambiguous automata denoting finitely sequential functions. In Adrian-Horia Dediu, Carlos Martín-Vide, and Bianca Truthe, editors, Proc. LATA, volume 7810 of LNCS, pages 104-115. Springer, 2013.
Jean Berstel and Christophe Reutenauer. Rational Series and Their Languages. Springer, 1988.
Matthias Büchse, Jonathan May, and Heiko Vogler. Determinization of weighted tree automata using factorizations. Journal of Automata, Languages and Combinatorics, 15(3/4):229-254, 2010.
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Andreas Weber. Finite-valued distance automata. Theor. Comput. Sci., 134(1):225-251, 1994.
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New Insights on the (Non-)Hardness of Circuit Minimization and Related Problems
The Minimum Circuit Size Problem (MCSP) and a related problem (MKTP) that deals with time-bounded Kolmogorov complexity are prominent candidates for NP-intermediate status. We show that, under very modest cryptographic assumptions (such as the existence of one-way functions), the problem of approximating the minimum circuit size (or time-bounded Kolmogorov complexity) within a factor of n^{1 - o(1)} is indeed NP-intermediate. To the best of our knowledge, these problems are the first natural NP-intermediate problems under the existence of an arbitrary one-way function.
We also prove that MKTP is hard for the complexity class DET under
non-uniform NC^0 reductions. This is surprising, since prior work on MCSP and MKTP had highlighted weaknesses of "local" reductions such as NC^0 reductions. We exploit this local reduction to obtain several new consequences:
* MKTP is not in AC^0[p].
* Circuit size lower bounds are equivalent to hardness of a relativized version MKTP^A of MKTP under a class of uniform AC^0 reductions, for a large class of sets A.
* Hardness of MCSP^A implies hardness of MKTP^A for a wide class of
sets A. This is the first result directly relating the complexity of
MCSP^A and MKTP^A, for any A.
computational complexity
Kolmogorov complexity
circuit size
54:1-54:14
Regular Paper
Eric
Allender
Eric Allender
Shuichi
Hirahara
Shuichi Hirahara
10.4230/LIPIcs.MFCS.2017.54
Manindra Agrawal. The isomorphism conjecture for constant depth reductions. Journal of Computer and System Sciences, 77(1):3-13, 2011. URL: http://dx.doi.org/10.1145/28395.28404.
http://dx.doi.org/10.1145/28395.28404
Manindra Agrawal, Eric Allender, and Steven Rudich. Reductions in circuit complexity: An isomorphism theorem and a gap theorem. Journal of Computer and System Sciences, 57(2):127-143, 1998. URL: http://dx.doi.org/10.1006/jcss.1998.1583.
http://dx.doi.org/10.1006/jcss.1998.1583
Eric Allender, Harry Buhrman, Michal Koucký, Dieter van Melkebeek, and Detlef Ronneburger. Power from random strings. SIAM Journal on Computing, 35:1467-1493, 2006. URL: http://dx.doi.org/10.1137/050628994.
http://dx.doi.org/10.1137/050628994
Eric Allender and Bireswar Das. Zero knowledge and circuit minimization. Information and Computation, 2017. to appear. URL: http://dx.doi.org/10.1016/j.ic.2017.04.004.
http://dx.doi.org/10.1016/j.ic.2017.04.004
Eric Allender, Joshua Grochow, and Cristopher Moore. Graph isomorphism and circuit size. Technical Report TR15-162, Electronic Colloquium on Computational Complexity, 2015. URL: https://eccc.weizmann.ac.il/report/2015/162/.
https://eccc.weizmann.ac.il/report/2015/162/
Eric Allender and Shuichi Hirahara. New insights on the (non)-hardness of circuit minimization and related problems. Technical Report TR17-073, Electronic Colloquium on Computational Complexity, 2017. URL: https://eccc.weizmann.ac.il/report/2017/073/.
https://eccc.weizmann.ac.il/report/2017/073/
Eric Allender, Dhiraj Holden, and Valentine Kabanets. The minimum oracle circuit size problem. Computational Complexity, 26(2):469-496, 2017. URL: http://dx.doi.org/10.1007/s00037-016-0124-0.
http://dx.doi.org/10.1007/s00037-016-0124-0
Eric Allender and Michal Koucký. Amplifying lower bounds by means of self-reducibility. Journal of the ACM, 57:14:1-14:36, 2010. URL: http://dx.doi.org/10.1145/1706591.1706594.
http://dx.doi.org/10.1145/1706591.1706594
Eric Allender, Michal Koucký, Detlef Ronneburger, and Sambuddha Roy. The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory. Journal of Computer and System Sciences, 77:14-40, 2010. URL: http://dx.doi.org/10.1016/j.jcss.2010.06.004.
http://dx.doi.org/10.1016/j.jcss.2010.06.004
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Shuichi Hirahara and Rahul Santhanam. On the average-case complexity of mcsp and its variants. In 32nd Conference on Computational Complexity, CCC, LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. to appear.
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http://dx.doi.org/10.4230/LIPIcs.CCC.2016.18
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Igor Oliveira and Rahul Santhanam. Conspiracies between learning algorithms, circuit lower bounds and pseudorandomness. In 32nd Conference on Computational Complexity, CCC, LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. to appear.
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Michael Rudow. Discrete logarithm and minimum circuit size. Technical Report TR16-23, Electronic Colloquium on Computational Complexity, 2016. URL: https://eccc.weizmann.ac.il/report/2016/108/.
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http://dx.doi.org/10.1007/978-3-662-03927-4
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Strategy Complexity of Concurrent Safety Games
We consider two player, zero-sum, finite-state concurrent reachability games, played for an infinite number of rounds, where in every round, each player simultaneously and independently of the other players chooses an action, whereafter the successor state is determined by a probability distribution given by the current state and the chosen actions. Player 1 wins iff a designated goal state is eventually visited. We are interested in the complexity of stationary strategies measured by their patience, which is defined as the inverse of the smallest non-zero probability employed. Our main results are as follows: We show that: (i) the optimal bound on the patience of optimal and epsilon-optimal strategies, for both players is doubly exponential; and (ii) even in games with a single non-absorbing state exponential (in the number of actions) patience is necessary.
Concurrent games
Reachability and safety
Patience of strategies
55:1-55:13
Regular Paper
Krishnendu
Chatterjee
Krishnendu Chatterjee
Kristoffer Arnsfelt
Hansen
Kristoffer Arnsfelt Hansen
Rasmus
Ibsen-Jensen
Rasmus Ibsen-Jensen
10.4230/LIPIcs.MFCS.2017.55
R. Alur, T.A. Henzinger, and O. Kupferman. Alternating-time temporal logic. Journal of the ACM, 49:672-713, 2002.
K. Chatterjee. Concurrent games with tail objectives. Theoretical Computer Science, 388:181-198, 2007.
K. Chatterjee, L. de Alfaro, and T.A. Henzinger. Qualitative concurrent parity games. ACM ToCL, 2011.
Krishnendu Chatterjee, Kristoffer Arnsfelt Hansen, and Rasmus Ibsen-Jensen. Strategy complexity of concurrent stochastic games with safety and reachability objectives. CoRR, abs/1506.02434, 2015. URL: http://arxiv.org/abs/1506.02434.
http://arxiv.org/abs/1506.02434
Krishnendu Chatterjee and Rasmus Ibsen-Jensen. The Complexity of Ergodic Mean-payoff Games. In ICALP 2014, pages 122-133, 2014.
L. de Alfaro, T.A. Henzinger, and F.Y.C. Mang. The control of synchronous systems. In CONCUR'00, LNCS 1877, pages 458-473. Springer, 2000.
L. de Alfaro, T.A. Henzinger, and F.Y.C. Mang. The control of synchronous systems, Part II. In CONCUR'01, LNCS 2154, pages 566-580. Springer, 2001.
Luca de Alfaro, Thomas A. Henzinger, and Orna Kupferman. Concurrent reachability games. Theor. Comput. Sci, 386(3):188-217, 2007.
K. Etessami and M. Yannakakis. Recursive concurrent stochastic games. In ICALP'06 (2), LNCS 4052, Springer, pages 324-335, 2006.
H. Everett. Recursive games. In CTG, volume 39 of AMS, pages 47-78, 1957.
J. Filar and K. Vrieze. Competitive Markov Decision Processes. Springer-Verlag, 1997.
K. A. Hansen, R. Ibsen-Jensen, and P. B. Miltersen. The complexity of solving reachability games using value and strategy iteration. In CSR, pages 77-90, 2011.
K. A. Hansen, M. Koucký, and P. B. Miltersen. Winning concurrent reachability games requires doubly-exponential patience. In LICS, pages 332-341, 2009.
C. J. Himmelberg, T. Parthasarathy, T. E. S. Raghavan, and F. S. Van Vleck. Existence of p-equilibrium and optimal stationary strategies in stochastic games. Proc. Amer. Math. Soc., 60:245-251, 1976.
R. Ibsen-Jensen. Strategy complexity of two-player, zero-sum games. PhD thesis, Aarhus University, 2013.
Rasmus Ibsen-Jensen and Peter Bro Miltersen. Solving simple stochastic games with few coin toss positions. In ESA, pages 636-647, 2012.
R.J. Lipton, E. Markakis, and A. Mehta. Playing large games using simple strategies. In EC 03: Electronic Commerce, pages 36-41. ACM Press, 2003.
P. B. Miltersen and T. B. Sørensen. A near-optimal strategy for a heads-up no-limit texas hold'em poker tournament. In AAMAS'07, pages 191-197, 2007.
G. Owen. Game Theory. Academic Press, 1995.
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A. Pnueli and R. Rosner. On the synthesis of a reactive module. In Proc. of POPL, pages 179-190. ACM Press, 1989.
P. J. Ramadge and W. M. Wonham. Supervisory control of a class of discrete-event processes. SIAM Journal of Control and Optimization, 25(1):206-230, 1987.
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Eilon Solan and Nicolas Vieille. Computing uniformly optimal strategies in two-player stochastic games. Economic Theory, 42(1):237-253, 2010.
M.Y. Vardi. Automatic verification of probabilistic concurrent finite-state systems. In FOCS'85, pages 327-338. IEEE, 1985.
J. von Neumann and O. Morgenstern. Theory of games and economic behavior. Princeton University Press, 1947.
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O.J. Vrieze and S.H. Tijs. Fictitious play applied to sequences of games and discounted stochastic games. International Journal of Game Theory, 11(2):71-85, 1982.
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A Characterisation of Pi^0_2 Regular Tree Languages
We show an algorithm that for a given regular tree language L decides if L is in Pi^0_2, that is if L belongs to the second level of Borel Hierarchy. Moreover, if L is in Pi^0_2, then we construct a weak alternating automaton of index (0, 2) which recognises L. We also prove that for a given language L, L is recognisable by a weak alternating (1, 3)-automaton if and only if it is recognisable by a weak non-deterministic (1, 3)-automaton.
infinite trees
Rabin-Mostowski hierarchy
regular languages
56:1-56:14
Regular Paper
Filippo
Cavallari
Filippo Cavallari
Henryk
Michalewski
Henryk Michalewski
Michal
Skrzypczak
Michal Skrzypczak
10.4230/LIPIcs.MFCS.2017.56
Mikołaj Bojańczyk and Thomas Place. Regular languages of infinite trees that are Boolean combinations of open sets. In ICALP, pages 104-115, 2012.
Julius Richard Büchi and Lawrence H. Landweber. Solving sequential conditions by finite-state strategies. Transactions of the American Mathematical Society, 138:295-311, 1969.
Thomas Colcombet. Fonctions régulières de coût. Habilitation thesis, Université Paris Diderot - Paris 7, 2013.
Thomas Colcombet, Denis Kuperberg, Christof Löding, and Michael Vanden Boom. Deciding the weak definability of Büchi definable tree languages. In CSL, pages 215-230, 2013.
Thomas Colcombet and Christof Löding. The non-deterministic Mostowski hierarchy and distance-parity automata. In ICALP (2), pages 398-409, 2008.
Jacques Duparc and Filip Murlak. On the topological complexity of weakly recognizable tree languages. Fundamentals of computation theory, 2007.
Alessandro Facchini and Henryk Michalewski. Deciding the Borel complexity of regular tree languages. In CiE 2014, pages 163-172, 2014.
Alessandro Facchini, Filip Murlak, and Michał Skrzypczak. Index problems for game automata. ACM Trans. Comput. Log., 17(4):24:1-24:38, 2016.
Tomasz Gogacz, Henryk Michalewski, Matteo Mio, and Michał Skrzypczak. Measure properties of game tree languages. In MFCS, pages 303-314, 2014.
Erich Grädel, Wolfgang Thomas, and Thomas Wilke, editors. Automata, Logics, and Infinite Games: A Guide to Current Research, volume 2500 of Lecture Notes in Computer Science. Springer, 2002.
Alexander Kechris. Classical descriptive set theory. Springer-Verlag, New York, 1995.
Denis Kuperberg and Michael Vanden Boom. Quasi-weak cost automata: A new variant of weakness. In FSTTCS, volume 13 of LIPIcs, pages 66-77, 2011.
Orna Kupferman and Moshe Y. Vardi. The weakness of self-complementation. In STACS, pages 455-466, 1999.
Christof Löding. Logic and automata over infinite trees. Habilitation thesis, RWTH Aachen, Germany, 2009.
Satoru Miyano and Takeshi Hayashi. Alternating finite automata on omega-words. Theor. Comput. Sci., 32:321-330, 1984.
Filip Murlak. The Wadge hierarchy of deterministic tree languages. Logical Methods in Computer Science, 4(4), 2008.
Damian Niwiński and Igor Walukiewicz. A gap property of deterministic tree languages. Theor. Comput. Sci., 1(303):215-231, 2003.
Damian Niwiński and Igor Walukiewicz. Deciding nondeterministic hierarchy of deterministic tree automata. Electr. Notes Theor. Comput. Sci., 123:195-208, 2005.
Dominique Perrin and Jean-Éric Pin. Infinite Words: Automata, Semigroups, Logic and Games. Elsevier, 2004.
Michael Oser Rabin. Decidability of second-order theories and automata on infinite trees. Trans. of the American Math. Soc., 141:1-35, 1969.
Michael Oser Rabin and Dana Scott. Finite automata and their decision problems. IBM Journal of Research and Development, 3(2):114-125, April 1959.
Michał Skrzypczak. Descriptive Set Theoretic Methods in Automata Theory - Decidability and Topological Complexity, volume 9802 of Lecture Notes in Computer Science. Springer, 2016.
Michał Skrzypczak and Igor Walukiewicz. Deciding the topological complexity of Büchi languages. In ICALP (2), pages 99:1-99:13, 2016.
Jerzy Skurczyński. The Borel hierarchy is infinite in the class of regular sets of trees. Theoretical Computer Science, 112(2):413-418, 1993.
Wolfgang Thomas. Languages, automata, and logic. In Handbook of Formal Languages, pages 389-455. Springer, 1996.
Wolfgang Thomas and Helmut Lescow. Logical specifications of infinite computations. In REX School/Symposium, pages 583-621, 1993.
Boris A. Trakhtenbrot. Finite automata and the monadic predicate calculus. Siberian Mathematical Journal, 3(1):103-131, 1962.
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On the Exact Amount of Missing Information that Makes Finding Possible Winners Hard
We consider election scenarios with incomplete information, a situation that arises often in practice. There are several models of incomplete information and accordingly, different notions of outcomes of such elections. In one well-studied model of incompleteness, the votes are given by partial orders over the candidates. In this context we can frame the problem of finding a possible winner, which involves determining whether a given candidate wins in at least one completion of a given set of partial votes for a specific voting rule.
The Possible Winner problem is well-known to be NP-Complete in general, and it is in fact known to be NP-Complete for several voting rules where the number of undetermined pairs in every vote is bounded only by some constant. In this paper, we address the question of determining precisely the smallest number of undetermined pairs for which the Possible Winner problem remains NP-Complete. In particular, we find the exact values of t for which the Possible Winner problem transitions to being NP-Complete from being in P, where t is the maximum number of undetermined pairs in every vote. We demonstrate tight results for a broad subclass of scoring rules which includes all the commonly used scoring rules (such as plurality, veto, Borda, and k-approval), Copeland^\alpha for every \alpha in [0,1], maximin, and Bucklin voting rules. A somewhat surprising aspect of our results is that for many of these rules, the Possible Winner problem turns out to be hard even if every vote has at most one undetermined pair of candidates.
Computational Social Choice
Dichotomy
NP-completeness
Maxflow
Voting
Possible winner
57:1-57:14
Regular Paper
Palash
Dey
Palash Dey
Neeldhara
Misra
Neeldhara Misra
10.4230/LIPIcs.MFCS.2017.57
Dorothea Baumeister, Magnus Roos, and Jörg Rothe. Computational complexity of two variants of the possible winner problem. In Proc. International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pages 853-860, 2011.
Dorothea Baumeister and Jörg Rothe. Taking the final step to a full dichotomy of the possible winner problem in pure scoring rules. Inf. Process. Lett., 112(5):186-190, 2012. URL: http://dx.doi.org/10.1016/j.ipl.2011.11.016.
http://dx.doi.org/10.1016/j.ipl.2011.11.016
Piotr Berman, Marek Karpinski, and Alex D. Scott. Approximation hardness and satisfiability of bounded occurrence instances of SAT. Electronic Colloquium on Computational Complexity (ECCC), 10(022), 2003. URL: http://eccc.hpi-web.de/eccc-reports/2003/TR03-022/index.html.
http://eccc.hpi-web.de/eccc-reports/2003/TR03-022/index.html
Nadja Betzler, Robert Bredereck, and Rolf Niedermeier. Partial kernelization for rank aggregation: theory and experiments. In Proc. 5th International Symposium on Parameterized and Exact Computation (IPEC), pages 26-37. Springer, 2010.
Nadja Betzler, Robert Bredereck, and Rolf Niedermeier. Theoretical and empirical evaluation of data reduction for exact kemeny rank aggregation. Autonomous Agents and Multi-Agent Systems, 28(5):721-748, 2014. URL: http://dx.doi.org/10.1007/s10458-013-9236-y.
http://dx.doi.org/10.1007/s10458-013-9236-y
Nadja Betzler and Britta Dorn. Towards a dichotomy of finding possible winners in elections based on scoring rules. In Proc. 34th Mathematical Foundations of Computer Science (MFCS), pages 124-136. Springer, 2009.
Nadja Betzler, Susanne Hemmann, and Rolf Niedermeier. A Multivariate Complexity Analysis of Determining Possible Winners given Incomplete Votes. In Proc. 21st International Joint Conference on Artificial Intelligence (IJCAI), volume 9, pages 53-58, 2009.
Yann Chevaleyre, Jérôme Lang, Nicolas Maudet, and Jérôme Monnot. Possible winners when new candidates are added: The case of scoring rules. In Proc. 24th International Conference on Artificial Intelligence (AAAI), 2010.
William W. Cohen, Robert E. Schapire, and Yoram Singer. Learning to order things. J. Artif. Int. Res., 10(1):243-270, May 1999. URL: http://dl.acm.org/citation.cfm?id=1622859.1622867.
http://dl.acm.org/citation.cfm?id=1622859.1622867
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
http://dx.doi.org/10.1007/978-3-319-21275-3
Palash Dey. Resolving the complexity of some fundamental problems in computational social choice. CoRR, abs/1703.08041, 2017. URL: http://arxiv.org/abs/1703.08041.
http://arxiv.org/abs/1703.08041
Palash Dey and Neeldhara Misra. On the exact amount of missing information that makes finding possible winners hard. CoRR, abs/1610.08407, 2016. URL: http://arxiv.org/abs/1610.08407.
http://arxiv.org/abs/1610.08407
Palash Dey, Neeldhara Misra, and Y. Narahari. Kernelization complexity of possible winner and coalitional manipulation problems in voting. In Proc. 14th International Conference on Autonomous Agents and Multiagent Systems, AAMAS 2015, Istanbul, Turkey, May 4-8, 2015, pages 87-96, 2015. URL: http://dl.acm.org/citation.cfm?id=2772894.
http://dl.acm.org/citation.cfm?id=2772894
Palash Dey, Neeldhara Misra, and Y. Narahari. Complexity of manipulation with partial information in voting. In Proc. 25th International Joint Conference on Artificial Intelligence, IJCAI 2016, New York, USA, pages 229-235, 2016. URL: http://www.ijcai.org/Abstract/16/040.
http://www.ijcai.org/Abstract/16/040
Palash Dey, Neeldhara Misra, and Y. Narahari. Frugal bribery in voting. In Proc. 30th AAAI Conference on Artificial Intelligence, February 12-17, 2016, Phoenix, Arizona, USA., pages 2466-2472, 2016. URL: http://www.aaai.org/ocs/index.php/AAAI/AAAI16/paper/view/12133.
http://www.aaai.org/ocs/index.php/AAAI/AAAI16/paper/view/12133
Palash Dey, Neeldhara Misra, and Y. Narahari. Kernelization complexity of possible winner and coalitional manipulation problems in voting. Theor. Comput. Sci., 616:111-125, 2016. URL: http://dx.doi.org/10.1016/j.tcs.2015.12.023.
http://dx.doi.org/10.1016/j.tcs.2015.12.023
Palash Dey, Neeldhara Misra, and Y. Narahari. Frugal bribery in voting. Theor. Comput. Sci., 676:15-32, 2017. URL: http://dx.doi.org/10.1016/j.tcs.2017.02.031.
http://dx.doi.org/10.1016/j.tcs.2017.02.031
Piotr Faliszewski, Yannick Reisch, Jörg Rothe, and Lena Schend. Complexity of manipulation, bribery, and campaign management in bucklin and fallback voting. In Proc. 13th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pages 1357-1358. International Foundation for Autonomous Agents and Multiagent Systems, 2014.
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Fractal Intersections and Products via Algorithmic Dimension
Algorithmic dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions of intersections and Cartesian products of fractals in Euclidean spaces. This approach shows that a known intersection formula for Borel sets holds for arbitrary sets, and it significantly simplifies the proof of a known product formula. Both of these formulas are prominent, fundamental results in fractal geometry that are taught in typical undergraduate courses on the subject.
algorithmic randomness
geometric measure theory
Hausdorff dimension
Kolmogorov complexity
58:1-58:12
Regular Paper
Neil
Lutz
Neil Lutz
10.4230/LIPIcs.MFCS.2017.58
Krishna B. Athreya, John M. Hitchcock, Jack H. Lutz, and Elvira Mayordomo. Effective strong dimension in algorithmic information and computational complexity. SIAM Journal of Computing, 37(3):671-705, 2007. URL: http://dx.doi.org/10.1137/s0097539703446912.
http://dx.doi.org/10.1137/s0097539703446912
Verónica Becher, Jan Reimann, and Theodore A. Slaman. Irrationality exponent, Hausdorff dimension and effectivization. arXiv:1601.00153 [math.NT], 2016.
Christopher J. Bishop. Personal communication, April 27, 2017.
Christopher J. Bishop and Yuval Peres. Fractals in Probability and Analysis. Cambridge University Press, 2017. URL: http://dx.doi.org/10.1017/9781316460238.
http://dx.doi.org/10.1017/9781316460238
Jin-Yi Cai and Juris Hartmanis. On Hausdorff and topological dimensions of the Kolmogorov complexity of the real line. Journal of Computer and System Sciences, 49(3):605-619, 1994. URL: http://dx.doi.org/10.1016/S0022-0000(05)80073-X.
http://dx.doi.org/10.1016/S0022-0000(05)80073-X
Adam Case and Jack H. Lutz. Mutual dimension. ACM Transactions on Computation Theory, 7(3):12, 2015. URL: http://dx.doi.org/10.1145/2786566.
http://dx.doi.org/10.1145/2786566
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http://dx.doi.org/10.1007/978-0-387-68441-3
Kenneth J. Falconer. The Geometry of Fractal Sets. Cambridge University Press, 1985. URL: http://dx.doi.org/10.1017/cbo9780511623738.
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Kenneth J. Falconer. Fractal Geometry: Mathematical Foundations and Applications. Wiley, third edition, 2014. URL: http://dx.doi.org/10.1002/0470013850.
http://dx.doi.org/10.1002/0470013850
Felix Hausdorff. Dimension und äusseres Mass. Mathematische Annalen, 79:157-179, 1919. URL: http://dx.doi.org/10.1007/978-3-642-59483-0_2.
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http://dx.doi.org/10.1016/s0924-6509(09)70272-7
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Jack H. Lutz. The dimensions of individual strings and sequences. Information and Computation, 187(1):49-79, 2003. URL: http://dx.doi.org/10.1016/s0890-5401(03)00187-1.
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Jack H. Lutz and Neil Lutz. Algorithmic information, plane Kakeya sets, and conditional dimension. In Proceedings of the 34th Symposium on Theoretical Aspects of Computer Science, STACS 2017, March 8-11, 2017, Hannover, Germany, pages 53:1-53:13, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2017.53.
http://dx.doi.org/10.4230/LIPIcs.STACS.2017.53
Jack H. Lutz and Elvira Mayordomo. Dimensions of points in self-similar fractals. SIAM Journal of Computing, 38(3):1080-1112, 2008. URL: http://dx.doi.org/10.1007/978-3-540-69733-6_22.
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Neil Lutz and D. M. Stull. Bounding the dimension of points on a line. In TV Gopal, Gerhard Jaeger, and Silvia Steila, editors, Theory and Applications of Models of Computation: 14th Annual Conference, TAMC 2017, Bern, Switzerland, April 20-22, 2017, Proceedings, pages 425-439, 2017. URL: http://dx.doi.org/10.1007/978-3-319-55911-7_31.
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http://dx.doi.org/10.1016/s0020-0190(02)00343-5
Elvira Mayordomo. Effective fractal dimension in algorithmic information theory. In S. Barry Cooper, Benedikt Löwe, and Andrea Sorbi, editors, New Computational Paradigms: Changing Conceptions of What is Computable, pages 259-285. Springer New York, 2008. URL: http://dx.doi.org/10.1007/978-0-387-68546-5_12.
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Jan Reimann. Computability and fractal dimension. PhD thesis, Heidelberg University, 2004.
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Ludwig Staiger. Kolmogorov complexity and Hausdorff dimension. Information and Computation, 103:159-194, 1989. URL: http://dx.doi.org/10.1007/3-540-51498-8_42.
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Elias M. Stein and Rami Shakarchi. Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton Lectures in Analysis. Princeton University Press, 2005.
Dennis Sullivan. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Mathematica, 153(1):259-277, 1984. URL: http://dx.doi.org/10.1007/bf02392379.
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Claude Tricot. Two definitions of fractional dimension. Mathematical Proceedings of the Cambridge Philosophical Society, 91(1):57-74, 1982. URL: http://dx.doi.org/10.1017/s0305004100059119.
http://dx.doi.org/10.1017/s0305004100059119
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Domains for Higher-Order Games
We study two-player inclusion games played over word-generating higher-order recursion schemes.
While inclusion checks are known to capture verification problems, two-player games generalize this relationship to program synthesis.
In such games, non-terminals of the grammar are controlled by opposing players.
The goal of the existential player is to avoid producing a word that lies outside of a regular language of safe words.
We contribute a new domain that provides a representation of the winning region of such games. Our domain is based on (functions over) potentially infinite Boolean formulas with words as atomic propositions. We develop an abstract interpretation framework that we instantiate to abstract this domain into a domain where the propositions are replaced by states of a finite automaton.
This second domain is therefore finite and we obtain, via standard fixed-point techniques, a direct algorithm for the analysis of two-player inclusion games. We show, via a second instantiation of the framework, that our finite domain can be optimized, leading to a (k+1)EXP algorithm for order-k recursion schemes. We give a matching lower bound, showing that our approach is optimal. Since our approach is based on standard Kleene iteration, existing techniques and tools for fixed-point computations can be applied.
higher-order recursion schemes
games
semantics
abstract interpretation
fixed points
59:1-59:15
Regular Paper
Matthew
Hague
Matthew Hague
Roland
Meyer
Roland Meyer
Sebastian
Muskalla
Sebastian Muskalla
10.4230/LIPIcs.MFCS.2017.59
P. A. Abdulla, Y. Chen, L. Clemente, L. Holík, C.-D. Hong, R. Mayr, and T. Vojnar. Simulation subsumption in Ramsey-based Büchi automata universality and inclusion testing. In CAV, volume 6174 of LNCS, pages 132-147. Springer, 2010.
P. A. Abdulla, Y. Chen, L. Clemente, L. Holík, C.-D. Hong, R. Mayr, and T. Vojnar. Advanced Ramsey-based Büchi automata inclusion testing. In CONCUR, volume 6901 of LNCS, pages 187-202. Springer, 2011.
S. Abramsky. Abstract interpretation, logical relations and Kan extensions. J. Log. Comp., 1(1):5-40, 1990.
S. Abramsky and C. Hankin. An introduction to abstract interpretation. In Abstract Interpretation of declarative languages, volume 1, pages 63-102. Ellis Horwood, 1987.
K. Aehlig. A finite semantics of simply-typed lambda terms for infinite runs of automata. LMCS, 3(3):1-23, 2007.
K. Backhouse and R. C. Backhouse. Safety of abstract interpretations for free, via logical relations and Galois connections. Sci. Comp. Prog., 51(1-2):153-196, 2004.
A. Bouajjani, J. Esparza, and O. Maler. Reachability analysis of pushdown automata: Application to model-checking. In CONCUR, volume 1243 of LNCS, pages 135-150. Springer, 1997.
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Azadeh Farzan, Zachary Kincaid, and Andreas Podelski. Proofs that count. In POPL, pages 151-164. ACM, 2014.
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M. Hague, R. Meyer, and S. Muskalla. Domains for higher-order games. CoRR, abs/1705.00355, 2017. URL: http://arxiv.org/abs/1705.00355.
http://arxiv.org/abs/1705.00355
M. Hague, A. S. Murawski, C.-H. L. Ong, and O. Serre. Collapsible pushdown automata and recursion schemes. In LICS, pages 452-461. IEEE, 2008.
M. Hague and C.-H. L. Ong. Symbolic backwards-reachability analysis for higher-order pushdown systems. In FoSSaCS, volume 4423 of LNCS, pages 213-227. Springer, 2007.
M. Hague and C.-H. L. Ong. Winning regions of pushdown parity games: A saturation method. In CONCUR, volume 5710 of LNCS, pages 384-398. Springer, 2009.
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M. Hofmann and J. Ledent. A cartesian-closed category for higher-order model checking. In LICS. IEEE, 2017. To appear.
L. Holík, R. Meyer, and S. Muskalla. Summaries for context-free games. In FSTTCS, volume 65 of LIPIcs, pages 41:1-41:16. Dagstuhl, 2016.
Lukás Holík and Roland Meyer. Antichains for the verification of recursive programs. In NETYS, volume 9466 of LNCS, pages 322-336. Springer, 2015.
T. Knapik, D. Niwinski, and P. Urzyczyn. Higher-order pushdown trees are easy. In FoSSaCS, volume 2303 of LNCS, pages 205-222. Springer, 2002.
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N. Kobayashi. Types and higher-order recursion schemes for verification of higher-order programs. In POPL, pages 416-428. ACM, 2009.
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R. Meyer, S. Muskalla, and E. Neumann. Liveness verification and synthesis: New algorithms for recursive programs. URL: https://arxiv.org/abs/1701.02947.
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C.-H. L. Ong. On model-checking trees generated by higher-order recursion schemes. In LICS, pages 81-90. IEEE, 2006.
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S. Salvati and I. Walukiewicz. Using models to model-check recursive schemes. LMCS, 11(2):1-23, 2015.
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Fine-Grained Complexity of Rainbow Coloring and its Variants
Consider a graph G and an edge-coloring c_R:E(G) \rightarrow [k]. A rainbow path between u,v \in V(G) is a path P from u to v such that for all e,e' \in E(P), where e \neq e' we have c_R(e) \neq c_R(e'). In the Rainbow k-Coloring problem we are given a graph G, and the objective is to decide if there exists c_R: E(G) \rightarrow [k] such that for all u,v \in V(G) there is a rainbow path between u and v in G. Several variants of Rainbow k-Coloring have been studied, two of which are defined as follows. The Subset Rainbow k-Coloring takes as an input a graph G and a set S \subseteq V(G) \times V(G), and the objective is to decide if there exists c_R: E(G) \rightarrow [k] such that for all (u,v) \in S there is a rainbow path between u and v in G. The problem Steiner Rainbow k-Coloring takes as an input a graph G and a set S \subseteq V(G), and the objective is to decide if there exists c_R: E(G) \rightarrow [k] such that for all u,v \in S there is a rainbow path between u and v in G. In an attempt to resolve open problems posed by Kowalik et al. (ESA 2016), we obtain the following results.
- For every k \geq 3, Rainbow k-Coloring does not admit an algorithm running in time 2^{o(|E(G)|)}n^{O(1)}, unless ETH fails.
- For every k \geq 3, Steiner Rainbow k-Coloring does not admit an algorithm running in time 2^{o(|S|^2)}n^{O(1)}, unless ETH fails.
- Subset Rainbow k-Coloring admits an algorithm running in time 2^{\OO(|S|)}n^{O(1)}. This also implies an algorithm running in time 2^{o(|S|^2)}n^{O(1)} for Steiner Rainbow k-Coloring, which matches the lower bound we obtain.
Rainbow Coloring
Lower bound
ETH
Fine-grained Complexity
60:1-60:14
Regular Paper
Akanksha
Agrawal
Akanksha Agrawal
10.4230/LIPIcs.MFCS.2017.60
Akanksha Agrawal, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Split contraction: The untold story. In 34th Symposium on Theoretical Aspects of Computer Science, (STACS), pages 5:1-5:14, 2017.
Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. Journal of the ACM (JACM), 42(4):844-856, 1995.
Prabhanjan Ananth, Meghana Nasre, and Kanthi K. Sarpatwar. Rainbow Connectivity: Hardness and Tractability. In Foundations of Software Technology and Theoretical Computer Science (FSTTCS), volume 13, pages 241-251, 2011.
Yair Caro, Arie Lev, Yehuda Roditty, Zsolt Tuza, and Raphael Yuster. On rainbow connection. Electronic Journal of Combinatorics, 15(1):R57, 2008.
Sourav Chakraborty, Eldar Fischer, Arie Matsliah, and Raphael Yuster. Hardness and algorithms for rainbow connection. Journal of Combinatorial Optimization, 21(3):330-347, 2011.
L. Sunil Chandran and Deepak Rajendraprasad. Rainbow colouring of split and threshold graphs. In 18th Annual International Conference: Computing and Combinatorics, (COCOON), pages 181-192, 2012.
L. Sunil Chandran and Deepak Rajendraprasad. Inapproximability of rainbow colouring. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pages 153-162, 2013.
Gary Chartrand, Garry L Johns, Kathleen A McKeon, and Ping Zhang. Rainbow connection in graphs. Mathematica Bohemica, 133(1):85-98, 2008.
Gary Chartrand and Ping Zhang. Chromatic graph theory. CRC press, 2008.
Marek Cygan, Fedor V. Fomin, Alexander Golovnev, Alexander S. Kulikov, Ivan Mihajlin, Jakub Pachocki, and Arkadiusz Socala. Tight bounds for graph homomorphism and subgraph isomorphism. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms, (SODA), pages 1643-1649, 2016.
Marek Cygan, Fedor V. Fomin, Alexander Golovnev, Alexander S. Kulikov, Ivan Mihajlin, Jakub Pachocki, and Arkadiusz Socala. Tight bounds for graph homomorphism and subgraph isomorphism. In ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1643-1649, 2016.
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012.
Keith Edwards. The harmonious chromatic number and the achromatic number. Surveys in Combinatorics, pages 13-48, 1997.
Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman &Co., New York, NY, USA, 1979.
Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. Journal of Computer and System Sciences, 62(2):367-375, 2001.
Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001.
Christian Komusiewicz. Tight running time lower bounds for vertex deletion problems. arXiv preprint arXiv:1511.05449, 2015.
Lukasz Kowalik and Juho Lauri. On finding rainbow and colorful paths. Theoretical Computer Science, 628(C):110-114, 2016.
Lukasz Kowalik, Juho Lauri, and Arkadiusz Socala. On the fine-grained complexity of rainbow coloring. In 24th Annual European Symposium on Algorithms, (ESA), pages 58:1-58:16, 2016.
V.B. Le and Z. Tuza. Finding Optimal Rainbow Connection is Hard. Preprints aus dem Institut für Informatik / CS. Inst. für Informatik, 2009. URL: https://books.google.no/books?id=0ErVPgAACAAJ.
https://books.google.no/books?id=0ErVPgAACAAJ
Sin-Min Lee and John Mitchem. An upper bound for the harmonious chromatic number. Journal of Graph Theory, 11(4):565-567, 1987.
Xueliang Li, Yongtang Shi, and Yuefang Sun. Rainbow connections of graphs: A survey. Graphs and Combinatorics, 29(1):1-38, 2013.
Xueliang Li and Yuefang Sun. Rainbow connections of graphs. Springer Science &Business Media, 2012.
Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Lower bounds based on the exponential time hypothesis. Bulletin of the EATCS, pages 41-71, 2011.
Colin McDiarmid and Luo Xinhua. Upper bounds for harmonious coloring. Journal of Graph Theory, 15(6):629-636, 1991.
M. Naor, L. J. Schulman, and A. Srinivasan. Splitters and near-optimal derandomization. In Proceedings of the 36th Annual Symposium on Foundations of Computer Science (FOCS), pages 182-191, 1995.
Kei Uchizawa, Takanori Aoki, Takehiro Ito, Akira Suzuki, and Xiao Zhou. On the rainbow connectivity of graphs: Complexity and fpt algorithms. Algorithmica, 67(2):161-179, 2013.
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Faster Monte-Carlo Algorithms for Fixation Probability of the Moran Process on Undirected Graphs
Evolutionary graph theory studies the evolutionary dynamics in a population structure given as a connected graph. Each node of the graph represents an individual of the population, and edges determine how offspring are placed. We consider the classical birth-death Moran process where there are two types of individuals, namely, the residents with fitness 1 and mutants with fitness r. The fitness indicates the reproductive strength. The evolutionary dynamics happens as follows: in the initial step, in a population of all resident individuals a mutant is introduced, and then at each step, an individual is chosen proportional to the fitness of its type to reproduce, and the offspring replaces a neighbor uniformly at random. The process stops when all individuals are either residents or mutants. The probability that all individuals in the end are mutants is called the fixation probability, which is a key factor in the rate of evolution. We consider the problem of approximating the fixation probability. The class of algorithms that is extremely relevant for approximation of the fixation probabilities is the Monte-Carlo simulation of the process. Previous results present a polynomial-time Monte-Carlo algorithm for undirected graphs when $r$ is given in unary. First, we present a simple modification: instead of simulating each step, we discard ineffective steps, where no node changes type (i.e., either residents replace residents, or mutants replace mutants). Using the above simple modification and our result that the number of effective steps is concentrated around the expected number of effective steps, we present faster polynomial-time Monte-Carlo algorithms for undirected graphs. Our algorithms are always at least a factor O(n^2/log n) faster as compared to the previous algorithms, where n is the number of nodes, and is polynomial even if r is given in binary. We also present lower bounds showing that the upper bound on the expected number of effective steps we present is asymptotically tight for undirected graphs.
Graph algorithms
Evolutionary biology
Monte-Carlo algorithms
61:1-61:13
Regular Paper
Krishnendu
Chatterjee
Krishnendu Chatterjee
Rasmus
Ibsen-Jensen
Rasmus Ibsen-Jensen
Martin A.
Nowak
Martin A. Nowak
10.4230/LIPIcs.MFCS.2017.61
B. Adlam, K. Chatterjee, and M. A. Nowak. Amplifiers of selection. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 471(2181), 2015. URL: http://dx.doi.org/10.1098/rspa.2015.0114.
http://dx.doi.org/10.1098/rspa.2015.0114
Krishnendu Chatterjee, Rasmus Ibsen-Jensen, and Martin Nowak. Faster monte-carlo algorithms for fixation probability of the moran process on undirected graphs. CoRR, abs/1706.06931, 2017. URL: http://arxiv.org/abs/1706.06931.
http://arxiv.org/abs/1706.06931
F. Débarre, C. Hauert, and M. Doebeli. Social evolution in structured populations. Nature Communications, 2014.
Josep Díaz, Leslie Ann Goldberg, George B. Mertzios, David Richerby, Maria Serna, and Paul G. Spirakis. On the fixation probability of superstars. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 469(2156), 2013.
Josep Díaz, Leslie Ann Goldberg, George B. Mertzios, David Richerby, Maria Serna, and Paul G. Spirakis. Approximating Fixation Probabilities in the Generalized Moran Process. Algorithmica, 69(1):78-91, 2014 (Conference version SODA 2012).
Josep Díaz, Leslie Ann Goldberg, David Richerby, and Maria Serna. Absorption time of the Moran process. Random Structures &Algorithms, 48(1):137-159, 2016.
W.J. Ewens. Mathematical Population Genetics 1: I. Theoretical Introduction. Interdisciplinary Applied Mathematics. Springer, 2004.
Marcus Frean, Paul B. Rainey, and Arne Traulsen. The effect of population structure on the rate of evolution. Proceedings of the Royal Society B: Biological Sciences, 280(1762), 2013.
Andreas Galanis, Andreas Göbel, Leslie Ann Goldberg, John Lapinskas, and David Richerby. Amplifiers for the Moran Process. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016), volume 55, pages 62:1-62:13, 2016.
Rasmus Ibsen-Jensen, Krishnendu Chatterjee, and Martin A Nowak. Computational complexity of ecological and evolutionary spatial dynamics. Proceedings of the National Academy of Sciences, 112(51):15636-15641, 2015.
Samuel Karlin and Howard M. Taylor. A First Course in Stochastic Processes, Second Edition. Academic Press, 2 edition, April 1975.
Erez Lieberman, Christoph Hauert, and Martin A. Nowak. Evolutionary dynamics on graphs. Nature, 433(7023):312-316, January 2005. URL: http://dx.doi.org/10.1038/nature03204.
http://dx.doi.org/10.1038/nature03204
P. A. P. Moran. The Statistical Processes of Evolutionary Theory. Oxford University Press, Oxford, 1962.
Martin A. Nowak. Evolutionary Dynamics: Exploring the Equations of Life. Harvard University Press, 2006.
Paulo Shakarian, Patrick Roos, and Anthony Johnson. A review of evolutionary graph theory with applications to game theory. Biosystems, 107(2):66-80, 2012.
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The 2CNF Boolean Formula Satisfiability Problem and the Linear Space Hypothesis
We aim at investigating the solvability/insolvability of nondeterministic logarithmic-space (NL) decision, search, and optimization problems parameterized by size parameters using simultaneously polynomial time and sub-linear space on multi-tape deterministic Turing machines. We are particularly focused on a special NL-complete problem, 2SAT - the 2CNF Boolean formula satisfiability problem-parameterized by the number of Boolean variables. It is shown that 2SAT with n variables and m clauses can be solved simultaneously polynomial time and (n/2^{c sqrt{log(n)}}) polylog(m+n) space for an absolute constant c>0. This fact inspires us to propose a new, practical working hypothesis, called the linear space hypothesis (LSH), which states that 2SAT_3-a restricted variant of 2SAT in which each variable of a given 2CNF formula appears as literals in at most 3 clauses-cannot be solved simultaneously in polynomial time using strictly "sub-linear" (i.e., n^{epsilon} polylog(n) for a certain constant epsilon in (0,1)) space. An immediate consequence of this working hypothesis is L neq NL. Moreover, we use our hypothesis as a plausible basis to lead to the insolvability of various NL search problems as well as the nonapproximability of NL optimization problems.
For our investigation, since standard logarithmic-space reductions may no longer preserve polynomial-time sub-linear-space complexity, we need to introduce a new, practical notion of "short reduction." It turns out that overline{2SAT}_3 is complete for a restricted version of NL, called Syntactic NL or simply SNL, under such short reductions. This fact supports the legitimacy of our working hypothesis.
sub-linear space
linear space hypothesis
short reduction
Boolean formula satisfiability problem
NL search
NL optimization
Syntactic NL
62:1-62:14
Regular Paper
Tomoyuki
Yamakami
Tomoyuki Yamakami
10.4230/LIPIcs.MFCS.2017.62
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Variations on Inductive-Recursive Definitions
Dybjer and Setzer introduced the definitional principle of inductive-recursively defined families - i.e. of families (U : Set, T : U -> D) such that the inductive definition of U may depend on the recursively defined T --- by defining a type DS D E of codes. Each c : DS D E defines a functor [c] : Fam D -> Fam E, and
(U, T) = \mu [c] : Fam D is exhibited as the initial algebra of [c].
This paper considers the composition of DS-definable functors: Given F : Fam C -> Fam D and G : Fam D -> Fam E, is G \circ F : Fam C -> Fam E DS-definable, if F and G are? We show that this is the case if and only if powers of families are DS-definable, which seems unlikely. To construct composition, we present two new systems UF and PN of codes for inductive-recursive definitions, with UF a subsytem of DS a subsystem of PN. Both UF and PN are closed under composition. Since PN defines a potentially larger class of functors, we show that there is a model where initial algebras of PN-functors exist by adapting Dybjer-Setzer's proof for DS.
Type Theory
induction-recursion
initial-algebra semantics
63:1-63:13
Regular Paper
Neil
Ghani
Neil Ghani
Conor
McBride
Conor McBride
Fredrik
Nordvall Forsberg
Fredrik Nordvall Forsberg
Stephan
Spahn
Stephan Spahn
10.4230/LIPIcs.MFCS.2017.63
Michael Abbott, Thorsten Altenkirch, and Neil Ghani. Containers: Constructing strictly positive types. TCS, 342(1):3 - 27, 2005.
Thorsten Altenkirch, Neil Ghani, Peter Hancock, Conor McBride, and Peter Morris. Indexed containers. Journal Functional Programming, 25, 2015.
James Chapman, Pierre-Évariste Dagand, Conor McBride, and Peter Morris. The gentle art of levitation. In ICFP 2010, pages 3-14, 2010.
Peter Dybjer. Inductive families. Formal aspects of computing, 6(4):440-465, 1994.
Peter Dybjer. A general formulation of simultaneous inductive-recursive definitions in type theory. Journal of Symbolic Logic, 65(2), 2000.
Peter Dybjer and Anton Setzer. A finite axiomatization of inductive-recursive definitions. In TLCA, pages 129-146. Springer Verlag, 1999.
Peter Dybjer and Anton Setzer. Induction-recursion and initial algebras. Annals of Pure and Applied Logic, 124(1-3):1-47, 2003.
Peter Dybjer and Anton Setzer. Indexed induction-recursion. Journal of logic and algebraic programming, 66(1):1-49, 2006.
Nicola Gambino and Martin Hyland. Wellfounded trees and dependent polynomial functors. In Types for Proofs and Programs, pages 210-225, 2004.
Nicola Gambino and Joachim Kock. Polynomial functors and polynomial monads. Mathematical Proceedings of the Cambridge Philosophical Society, 154:153-192, 2013.
Neil Ghani and Peter Hancock. Containers, monads and induction recursion. Mathematical Structures in Computer Science, 26(1):89-113, 2016.
Peter Hancock. Private communication.
Per Martin-Löf. An intuitionistic theory of types: predicative part. In H.E. Rose and J.C. Shepherdson, editors, Logic Colloquium '73, Proceedings of the Logic Colloquium, volume 80 of Studies in Logic and the Foundations of Mathematics, pages 73-118. North-Holland, 1975.
Per Martin-Löf. Intuitionistic type theory, volume 1 of Studies in Proof Theory. Bibliopolis, 1984.
Robert Pollack. Dependently typed records in type theory. Formal Aspects of Computing, 13(3):386-402, 2002.
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One-Dimensional Logic over Trees
A one-dimensional fragment of first-order logic is obtained by restricting quantification to blocks of existential quantifiers that leave at most one variable free. This fragment contains two-variable logic, and it is known that over words both formalisms have the same complexity and expressive power. Here we investigate the one-dimensional fragment over trees. We consider unranked unordered trees accessible by one or both of the descendant and child relations, as well as ordered trees equipped additionally with sibling relations. We show that over unordered trees the satisfiability problem is ExpSpace-complete when only the descendant relation is available and 2ExpTime-complete with both the descendant and child or with only the child relation. Over ordered trees the problem remains 2ExpTime-complete. Regarding expressivity, we show that over ordered trees and over unordered trees accessible by both the descendant and child the one-dimensional fragment is equivalent to the two-variable fragment with counting quantifiers.
satisfiability
expressivity
trees
fragments of first-order logic
64:1-64:13
Regular Paper
Emanuel
Kieronski
Emanuel Kieronski
Antti
Kuusisto
Antti Kuusisto
10.4230/LIPIcs.MFCS.2017.64
Bartosz Bednarczyk, Witold Charatonik, and Emanuel Kieronski. Extending two-variable logic on trees. In Computer Science Logic, pages 11:1-11:20, 2017.
Saguy Benaim, Michael Benedikt, Witold Charatonik, Emanuel Kieronski, Rastislav Lenhardt, Filip Mazowiecki, and James Worrell. Complexity of two-variable logic on finite trees. ACM Trans. Comput. Log., 17(4):32:1-32:38, 2016.
Ashok K. Chandra, Dexter Kozen, and Larry J. Stockmeyer. Alternation. J. ACM, 28(1):114-133, 1981. URL: http://dx.doi.org/10.1145/322234.322243.
http://dx.doi.org/10.1145/322234.322243
Witold Charatonik, Emanuel Kieronski, and Filip Mazowiecki. Satisfiability of the two-variable fragment of first-order logic over trees. CoRR, abs/1304.7204, 2013.
Witold Charatonik and Piotr Witkowski. Two-variable logic with counting and a linear order. In Computer Science Logic, pages 631-647, 2015.
Heinz-Dieter Ebbinghaus and Jörg Flum. Finite model theory. Perspectives in Mathematical Logic. Springer, 1995.
Kousha Etessami, Moshe Y. Vardi, and Thomas Wilke. First-order logic with two variables and unary temporal logic. Inf. Comput., 179(2):279-295, 2002. URL: http://dx.doi.org/10.1006/inco.2001.2953.
http://dx.doi.org/10.1006/inco.2001.2953
Lauri Hella and Antti Kuusisto. One-dimensional fragment of first-order logic. In Advances in Modal Logic 10, pages 274-293, 2014.
Emanuel Kieronski. On the complexity of the two-variable guarded fragment with transitive guards. Inf. Comput., 204(11):1663-1703, 2006.
Emanuel Kieronski. One-dimensional logic over words. In Computer Science Logic, pages 38:1-38:15, 2016.
Emanuel Kieronski and Antti Kuusisto. Complexity and expressivity of uniform one-dimensional fragment with equality. In Mathematical Foundations of Computer Science, Part I, pages 365-376, 2014.
Emanuel Kieronski and Antti Kuusisto. Uniform one-dimensional fragments with one equivalence relation. In Computer Science Logic, pages 597-615, 2015.
Andreas Krebs, Kamal Lodaya, Paritosh Pandya, and Howard Straubing. Two-variable logic with a between predicate. In Logic in Computer Science, 2016.
Antti Kuusisto. On the uniform one-dimensional fragment. In Proceedings of Description Logic Workshop, 2016.
Maarten Marx. First order paths in ordered trees. In International Conference on Database Theory, pages 114-128, 2005.
Dana Scott. A decision method for validity of sentences in two variables. Journal Symbolic Logic, 27:477, 1962.
Luc Segoufin and Balder ten Cate. Unary negation. Logical Methods in Computer Science, 9(3), 2013.
Larry J. Stockmeyer. The Complexity of Decision Problems in Automata Theory and Logic. PhD thesis, MIT, Cambridge, Massachusetts, USA, 1974.
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An Improved FPT Algorithm for the Flip Distance Problem
Given a set \cal P of points in the Euclidean plane and two triangulations of \cal P, the flip distance between these two triangulations is the minimum number of flips required to transform one triangulation into the other. The Parameterized Flip Distance problem is to decide if the flip distance between two given triangulations is equal to a given integer k. The previous best FPT algorithm runs in time O^*(k\cdot c^k) (c\leq 2\times 14^11), where each step has fourteen possible choices, and the length of the action sequence is bounded by 11k. By applying the backtracking strategy and analyzing the underlying property of the flip sequence, each step of our algorithm has only five possible choices. Based on an auxiliary graph G, we prove that the length of the action sequence for our algorithm is bounded by 2|G|. As a result, we present an FPT algorithm running in time O^*(k\cdot 32^k).
triangulation
flip distance
FPT algorithm
65:1-65:13
Regular Paper
Shaohua
Li
Shaohua Li
Qilong
Feng
Qilong Feng
Xiangzhong
Meng
Xiangzhong Meng
Jianxin
Wang
Jianxin Wang
10.4230/LIPIcs.MFCS.2017.65
Oswin Aichholzer, Ferran Hurtado, and Marc Noy. A lower bound on the number of triangulations of planar point sets. Computational Geometry, 29(2):135-145, 2004. URL: http://dx.doi.org/10.1016/j.comgeo.2004.02.003.
http://dx.doi.org/10.1016/j.comgeo.2004.02.003
Oswin Aichholzer, Wolfgang Mulzer, and Alexander Pilz. Flip distance between triangulations of a simple polygon is NP-complete. Discrete & Computational Geometry, 54(2):368-389, 2015. URL: http://dx.doi.org/10.1007/s00454-015-9709-7.
http://dx.doi.org/10.1007/s00454-015-9709-7
Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, Michael T. Hallett, and Harold T. Wareham. Parameterized complexity analysis in computational biology. Computer Applications in the Biosciences, 11(1):49-57, 1995. URL: http://dx.doi.org/10.1093/bioinformatics/11.1.49.
http://dx.doi.org/10.1093/bioinformatics/11.1.49
Jianer Chen. Parameterized computation and complexity: A new approach dealing with np-hardness. J. Comput. Sci. Technol., 20(1):18-37, 2005. URL: http://dx.doi.org/10.1007/s11390-005-0003-7.
http://dx.doi.org/10.1007/s11390-005-0003-7
Jianer Chen, Donald K. Friesen, Weijia Jia, and Iyad A. Kanj. Using nondeterminism to design efficient deterministic algorithms. Algorithmica, 40(2):83-97, 2004. URL: http://dx.doi.org/10.1007/s00453-004-1096-z.
http://dx.doi.org/10.1007/s00453-004-1096-z
Jianer Chen, Chao Xu, and Jianxin Wang. Dealing with 4-variables by resolution: An improved maxsat algorithm. Theor. Comput. Sci., 670:33-44, 2017. URL: http://dx.doi.org/10.1016/j.tcs.2017.01.020.
http://dx.doi.org/10.1016/j.tcs.2017.01.020
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
http://dx.doi.org/10.1007/978-3-319-21275-3
Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, FOCS, Palm Springs, USA, pages 150-159, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.23.
http://dx.doi.org/10.1109/FOCS.2011.23
Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk, and Jakub Onufry Wojtaszczyk. Subset feedback vertex set is fixed-parameter tractable. SIAM J. Discrete Math., 27(1):290-309, 2013. URL: http://dx.doi.org/10.1137/110843071.
http://dx.doi.org/10.1137/110843071
Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational geometry: algorithms and applications, 3rd Edition. Springer, 2008.
Gerald E. Farin. Curves and surfaces for computer-aided geometric design - a practical guide (4. ed.). Computer science and scientific computing. Academic Press, 1997.
Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Efficient computation of representative sets with applications in parameterized and exact algorithms. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, Portland, USA, pages 142-151, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.10.
http://dx.doi.org/10.1137/1.9781611973402.10
Bernd Hamann. Modeling contours of trivariatc data. Mathematical Modeling and Numerical Analysis, 26:51-75, 1992.
Ferran Hurtado, Marc Noy, and Jorge Urrutia. Flipping edges in triangulations. Discrete & Computational Geometry, 22(3):333-346, 1999. URL: http://dx.doi.org/10.1007/PL00009464.
http://dx.doi.org/10.1007/PL00009464
Iyad A. Kanj and Ge Xia. Flip Distance is in FPT time O(n+k⋅ c^k). In proceedings of the 32nd International Symposium on Theoretical Aspects of Computer Science, STACS, Garching, Germany, pages 500-512, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.500.
http://dx.doi.org/10.4230/LIPIcs.STACS.2015.500
Iyad A. Kanj and Ge Xia. Computing the flip distance between triangulations. to appear in Discrete & Computational Geometry, 2017.
Charles L. Lawson. Transforming triangulations. Discrete Mathematics, 3(4):365-372, 1972. URL: http://dx.doi.org/10.1016/0012-365X(72)90093-3.
http://dx.doi.org/10.1016/0012-365X(72)90093-3
Anna Lubiw and Vinayak Pathak. Flip distance between two triangulations of a point set is NP-complete. Computational Geometry, 49:17-23, 2015. URL: http://dx.doi.org/10.1016/j.comgeo.2014.11.001.
http://dx.doi.org/10.1016/j.comgeo.2014.11.001
Joan M. Lucas. An improved kernel size for rotation distance in binary trees. Information Processing Letters, 110(12-13):481-484, 2010. URL: http://dx.doi.org/10.1016/j.ipl.2010.04.022.
http://dx.doi.org/10.1016/j.ipl.2010.04.022
Alexander Pilz. Flip distance between triangulations of a planar point set is APX-hard. Computational Geometry, 47(5):589-604, 2014. URL: http://dx.doi.org/10.1016/j.comgeo.2014.01.001.
http://dx.doi.org/10.1016/j.comgeo.2014.01.001
Larry L. Schumaker. Triangulations in CAGD. IEEE Computer Graphics and Applications, 13(1):47-52, 1993. URL: http://dx.doi.org/10.1109/38.180117.
http://dx.doi.org/10.1109/38.180117
Feng Shi, Jianxin Wang, Yufei Yang, Qilong Feng, Weilong Li, and Jianer Chen. A fixed-parameter algorithm for the maximum agreement forest problem on multifurcating trees. SCIENCE CHINA Information Sciences, 59(1):1-14, 2016. URL: http://dx.doi.org/10.1007/s11432-015-5355-1.
http://dx.doi.org/10.1007/s11432-015-5355-1
Daniel D. Sleator, Robert E. Tarjan, and William P. Thurston. Rotation distance, triangulations, and hyperbolic geometry. In Proceedings of the 18th Annual ACM Symposium on Theory of Computing, STOC, Berkeley, USA, pages 122-135, 1986. URL: http://dx.doi.org/10.1145/12130.12143.
http://dx.doi.org/10.1145/12130.12143
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Reversible Kleene lattices
We investigate the equational theory of reversible Kleene lattices, that is algebras of languages with the regular operations (union, composition and Kleene star), together with intersection and mirror image. Building on results by Andréka, Mikulás and Németi from 2011, we construct the free representation of this algebra. We then provide an automaton model to compare representations. These automata are adapted from Petri automata, which we introduced with Pous in 2015 to tackle a similar problem for algebras of binary relations. This allows us to show that testing the validity of
equations in this algebra is decidable, and in fact ExpSpace-complete.
Kleene algebra
Automata
Petri nets
Decidability
Complexity
Formal languages
Lattice
66:1-66:14
Regular Paper
Paul
Brunet
Paul Brunet
10.4230/LIPIcs.MFCS.2017.66
Hajnal Andréka, Szabolcs Mikulás, and István Németi. The equational theory of Kleene lattices. TCS, 412(52):7099-7108, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2011.09.024.
http://dx.doi.org/10.1016/j.tcs.2011.09.024
Hajnal Andréka and Dmitry A. Bredikhin. The equational theory of union-free algebras of relations. Alg. Univ., 33(4):516-532, 1995. URL: http://dx.doi.org/10.1007/BF01225472.
http://dx.doi.org/10.1007/BF01225472
Stephen L. Bloom, Zoltán Ésik, and Gheorghe Stefanescu. Notes on equational theories of relations. Alg. Univ., 33(1):98-126, 1995. URL: http://dx.doi.org/10.1007/BF01190768.
http://dx.doi.org/10.1007/BF01190768
Paul Brunet. Algebras of Relations: From algorithms to formal proofs. PhD thesis, Université de Lyon, 2016. URL: https://tel.archives-ouvertes.fr/tel-01455083v1.
https://tel.archives-ouvertes.fr/tel-01455083v1
Paul Brunet. Reversible Kleene lattices. extended abstract, 2017. URL: https://hal.archives-ouvertes.fr/hal-01474911.
https://hal.archives-ouvertes.fr/hal-01474911
Paul Brunet and Damien Pous. Petri Automata for Kleene Allegories. In Proc. LICS, pages 68-79, July 2015. URL: http://dx.doi.org/10.1109/LICS.2015.17.
http://dx.doi.org/10.1109/LICS.2015.17
Paul Brunet and Damien Pous. A formal exploration of Nominal Kleene Algebra. In Proc. MFCS, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.MFCS.2016.22.
http://dx.doi.org/10.4230/LIPIcs.MFCS.2016.22
Paul Brunet and Damien Pous. Petri automata. Logical Methods in Computer Science, 2017. submitted. URL: http://arxiv.org/abs/1702.01804.
http://arxiv.org/abs/1702.01804
John H. Conway. Regular algebra and finite machines. Chapman and Hall Mathematics Series, 1971.
Zoltán Ésik and Laszlo Bernátsky. Equational properties of Kleene algebras of relations with conversion. TCS, 137(2):237-251, 1995. URL: http://dx.doi.org/10.1016/0304-3975(94)00041-G.
http://dx.doi.org/10.1016/0304-3975(94)00041-G
Peter J. Freyd and Andre Scedrov. Categories, Allegories. NH, 1990.
Martin Fürer. The complexity of the inequivalence problem for regular expressions with intersection. In Proc. ICALP, pages 234-245, 1980. URL: http://dx.doi.org/10.1007/3-540-10003-2_74.
http://dx.doi.org/10.1007/3-540-10003-2_74
Murdoch J. Gabbay and Vincenzo Ciancia. Freshness and name-restriction in sets of traces with names. In Proc. FoSSaCS, pages 365-380. Springer, 2011. URL: http://dx.doi.org/10.1007/978-3-642-19805-2_25.
http://dx.doi.org/10.1007/978-3-642-19805-2_25
Stephen C. Kleene. Representation of Events in Nerve Nets and Finite Automata. Memorandum. Rand Corporation, 1951.
Dexter Kozen. A completeness theorem for Kleene Algebras and the algebra of regular events. In Proc. LICS, pages 214-225. IEEE Computer Society, 1991. URL: http://dx.doi.org/10.1109/LICS.1991.151646.
http://dx.doi.org/10.1109/LICS.1991.151646
Dexter Kozen. Kleene algebra with tests. Transactions on Programming Languages and Systems, 19(3):427-443, 1997. URL: http://dx.doi.org/10.1145/256167.256195.
http://dx.doi.org/10.1145/256167.256195
Dexter Kozen, Konstantinos Mamouras, Daniela Petrisan, and Alexandra Silva. Nominal Kleene coalgebra. In Proc. ICALP, pages 286-298. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47666-6_23.
http://dx.doi.org/10.1007/978-3-662-47666-6_23
Dexter Kozen, Konstantinos Mamouras, and Alexandra Silva. Completeness and incompleteness in nominal Kleene algebra. In Proc. RAMiCS, pages 51-66, 2015. URL: http://dx.doi.org/10.1007/978-3-319-24704-5_4.
http://dx.doi.org/10.1007/978-3-319-24704-5_4
Albert R. Meyer and Larry J. Stockmeyer. The equivalence problem for regular expressions with squaring requires exponential space. In Proc. SWAT, pages 125-129, 1972. URL: http://dx.doi.org/10.1109/SWAT.1972.29.
http://dx.doi.org/10.1109/SWAT.1972.29
Volodimir N. Redko. On defining relations for the algebra of regular events. Ukrainskii Matematicheskii Zhurnal, pages 120-126, 1964.
Arto Salomaa. Two complete axiom systems for the algebra of regular events. J. ACM, 13(1):158-169, 1966. URL: http://dx.doi.org/10.1145/321312.321326.
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Walter J. Savitch. Relationships between nondeterministic and deterministic tape complexities. Journal of computer and system sciences, 4(2):177-192, 1970. URL: http://dx.doi.org/10.1016/S0022-0000(70)80006-X.
http://dx.doi.org/10.1016/S0022-0000(70)80006-X
Jacobo Valdes, Robert E. Tarjan, and Eugene L. Lawler. The recognition of series parallel digraphs. In Proc. STOC, STOC '79, pages 1-12. ACM, 1979. URL: http://dx.doi.org/10.1145/800135.804393.
http://dx.doi.org/10.1145/800135.804393
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Lossy Kernels for Hitting Subgraphs
In this paper, we study the Connected H-hitting Set and Dominating Set problems from the perspective of approximate kernelization, a framework recently introduced by Lokshtanov et al. [STOC 2017]. For the Connected H-hitting set problem, we obtain an \alpha-approximate kernel for every \alpha>1 and complement it with a lower bound for the natural weighted version. We then perform a refined analysis of the tradeoff between the approximation factor and kernel size for the Dominating Set problem on d-degenerate graphs and provide an interpolation of approximate kernels between the known d^2-approximate kernel of constant size and 1-approximate kernel of size k^{O(d^2)}.
parameterized algorithms
lossy kernelization
graph theory
67:1-67:14
Regular Paper
Eduard
Eiben
Eduard Eiben
Danny
Hermelin
Danny Hermelin
M. S.
Ramanujan
M. S. Ramanujan
10.4230/LIPIcs.MFCS.2017.67
Jochen Alber, Michael R. Fellows, and Rolf Niedermeier. Polynomial-time data reduction for dominating set. J. ACM, 51(3):363-384, 2004.
Noga Alon and Shai Gutner. Linear time algorithms for finding a dominating set of fixed size in degenerated graphs. Algorithmica, 54(4):544-556, 2009. URL: http://dx.doi.org/10.1007/s00453-008-9204-0.
http://dx.doi.org/10.1007/s00453-008-9204-0
Hans L. Bodlaender, Fedor V. Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, and Dimitrios M. Thilikos. (meta) kernelization. J. ACM, 63(5):44:1-44:69, 2016.
Al Borchers and Ding-Zhu Du. The k-steiner ratio in graphs. In Frank Thomson Leighton and Allan Borodin, editors, Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, 29 May-1 June 1995, Las Vegas, Nevada, USA, pages 641-649. ACM, 1995.
Jaroslaw Byrka, Fabrizio Grandoni, Thomas Rothvoß, and Laura Sanità. Steiner tree approximation via iterative randomized rounding. J. ACM, 60(1):6:1-6:33, 2013.
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
Erik D. Demaine, Fedor V. Fomin, Mohammadtaghi Hajiaghayi, and Dimitrios M. Thilikos. Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. J. ACM, 52(6):866-893, 2005.
Erik D. Demaine and MohammadTaghi Hajiaghayi. The bidimensionality theory and its algorithmic applications. The Computer Journal, 51(3):292-302, 2008.
Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012.
Michael Dom, Daniel Lokshtanov, and Saket Saurabh. Kernelization lower bounds through colors and ids. ACM Transactions on Algorithms, 11(2):13:1-13:20, 2014.
Rodney G Downey and Michael Ralph Fellows. Parameterized complexity. Springer Science &Business Media, 2012.
Pål Grønås Drange, Markus Sortland Dregi, Fedor V. Fomin, Stephan Kreutzer, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, Felix Reidl, Fernando Sánchez Villaamil, Saket Saurabh, Sebastian Siebertz, and Somnath Sikdar. Kernelization and sparseness: the case of dominating set. In 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016, February 17-20, 2016, Orléans, France, pages 31:1-31:14, 2016.
S. E. Dreyfus and R. A. Wagner. The steiner problem in graphs. Networks, 1(3):195-207, 1971.
P. Erdös and R. Rado. Intersection theorems for systems of sets. Journal of the London Mathematical Society, s1-35(1):85-90, 1960.
Fedor V. Fomin and Dimitrios M. Thilikos. Dominating sets in planar graphs: Branch-width and exponential speed-up. SIAM J. Comput., 36:281-309, 2006.
Mark Jones, Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh, and Ondrej Suchý. Parameterized complexity of directed steiner tree on sparse graphs. In Algorithms - ESA 2013 - 21st Annual European Symposium, Sophia Antipolis, France, September 2-4, 2013. Proceedings, pages 671-682, 2013.
Stefan Kratsch. Recent developments in kernelization: A survey. Bulletin of the EATCS, 113, 2014.
Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Kernelization-preprocessing with a guarantee. In The Multivariate Algorithmic Revolution and Beyond, pages 129-161. Springer, 2012.
Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh. Lossy kernelization. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 224-237. ACM, 2017.
Geevarghese Philip, Venkatesh Raman, and Somnath Sikdar. Polynomial kernels for dominating set in graphs of bounded degeneracy and beyond. ACM Trans. Algorithms, 9(1):11:1-11:23, 2012.
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Undecidable Problems for Probabilistic Network Programming
The software-defined networking language NetKAT is able to verify many useful properties of networks automatically via a PSPACE decision procedure for program equality. However, for its probabilistic extension ProbNetKAT, no such decision procedure is known. We show that several potentially useful properties of ProbNetKAT are in fact undecidable, including emptiness of support intersection and certain kinds of distribution bounds and program comparisons. We do so by embedding the Post Correspondence Problem in ProbNetKAT via direct product expressions, and by directly embedding probabilistic finite automata.
Software-defined networking
NetKAT
ProbNetKAT
Undecidability
Probabilistic finite automata
68:1-68:17
Regular Paper
David M.
Kahn
David M. Kahn
10.4230/LIPIcs.MFCS.2017.68
Carolyn Jane Anderson, Nate Foster, Arjun Guha, Jean-Baptiste Jeannin, Dexter Kozen, Cole Schlesinger, and David Walker. NetKAT: Semantic foundations for networks. In Proc. 41st ACM SIGPLAN-SIGACT Symp. Principles of Programming Languages (POPL'14), pages 113-126, San Diego, California, USA, January 2014. ACM.
Vincent D Blondel, Vincent Canterini, et al. Undecidable problems for probabilistic automata of fixed dimension. Theory of Computing systems, 36(3):231-245, 2003.
Corinna Cortes, Mehryar Mohri, and Ashish Rastogi. On the Computation of Some Standard Distances Between Probabilistic Automata, pages 137-149. Springer Berlin Heidelberg, Berlin, Heidelberg, 2006. URL: http://dx.doi.org/10.1007/11812128_14.
http://dx.doi.org/10.1007/11812128_14
Nate Foster, Dexter Kozen, Konstantinos Mamouras, Mark Reitblatt, and Alexandra Silva. Probabilistic NetKAT. In Peter Thiemann, editor, 25th European Symposium on Programming (ESOP 2016), volume 9632 of Lecture Notes in Computer Science, pages 282-309, Eindhoven, The Netherlands, April 2016. Springer.
Nate Foster, Dexter Kozen, Matthew Milano, Alexandra Silva, and Laure Thompson. A coalgebraic decision procedure for NetKAT. In Proc. 42nd ACM SIGPLAN-SIGACT Symp. Principles of Programming Languages (POPL'15), pages 343-355, Mumbai, India, January 2015. ACM.
Hugo Gimbert and Youssouf Oualhadj. Probabilistic Automata on Finite Words: Decidable and Undecidable Problems, pages 527-538. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010. URL: http://dx.doi.org/10.1007/978-3-642-14162-1_44,
http://dx.doi.org/10.1007/978-3-642-14162-1_44
Stefan Kiefer, Andrzej S. Murawski, Joël Ouaknine, Björn Wachter, and James Worrell. On the Complexity of the Equivalence Problem for Probabilistic Automata, pages 467-481. Springer Berlin Heidelberg, Berlin, Heidelberg, 2012. URL: http://dx.doi.org/10.1007/978-3-642-28729-9_31.
http://dx.doi.org/10.1007/978-3-642-28729-9_31
Dexter Kozen. Kleene algebra with tests and commutativity conditions. In T. Margaria and B. Steffen, editors, Proc. Second Int. Workshop Tools and Algorithms for the Construction and Analysis of Systems (TACAS'96), volume 1055 of Lecture Notes in Computer Science, pages 14-33, Passau, Germany, March 1996. Springer-Verlag.
Dexter Kozen. Kleene algebra with tests. Transactions on Programming Languages and Systems, 19(3):427-443, May 1997.
Dexter Kozen. NetKAT: A formal system for the verification of networks. In Jacques Garrigue, editor, Proc. 12th Asian Symposium on Programming Languages and Systems (APLAS 2014), volume 8858 of Lecture Notes in Computer Science, Singapore, November 17-19 2014. Asian Association for Foundation of Software (AAFS), Springer.
Steffen Smolka, David Kahn, Praveen Kumar, and Nate Foster. Deciding probabilistic program equivalence in NetKAT. http://www.cs.cornell.edu/~smolka/papers/mcnetkat.pdf, 2017.
http://www.cs.cornell.edu/~smolka/papers/mcnetkat.pdf
Steffen Smolka, Praveen Kumar, Nate Foster, Dexter Kozen, and Alexandra Silva. Cantor meets Scott: Domain-theoretic foundations for probabilistic network programming. In Proc. 44th ACM SIGPLAN-SIGACT Symp. Principles of Programming Languages (POPL'17), pages 557-571, Paris, France, January 2017. ACM.
Wen-Guey Tzeng. A polynomial-time algorithm for the equivalence of probabilistic automata. SIAM Journal on Computing, 21(2):216-227, 1992. URL: http://dx.doi.org/10.1137/0221017.
http://dx.doi.org/10.1137/0221017
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Computational Complexity of Graph Partition under Vertex-Compaction to an Irreflexive Hexagon
In this paper, we solve a long-standing graph partition problem under vertex-compaction that has been of interest since about 1999. The graph partition problem that we consider in this paper is to decide whether or not it is possible to partition the vertices of a graph into six distinct non-empty sets A, B, C, D, E, and F, such that the vertices in each set are independent, i.e., there is no edge within any set, and an edge is possible but not necessary only between the pairs of sets A and B, B and C, C and D, D and E, E and F, and F and A, and there is no edge between any other pair of sets. We study the problem as the vertex-compaction problem for an irreflexive hexagon (6-cycle). Determining the computational complexity of this problem has been a long-standing problem of interest since about 1999, especially after the results of open problems obtained by the author on a related compaction problem appeared in 1999. We show in this paper that the vertex-compaction problem for an irreflexive hexagon is NP-complete. Our proof can be extended for larger even irreflexive cycles, showing that the vertex-compaction problem for an irreflexive even k-cycle is NP-complete, for all even k \geq 6.
computational complexity
algorithms
graph
partition
colouring
homomorphism
retraction
compaction
vertex-compaction
69:1-69:14
Regular Paper
Narayan
Vikas
Narayan Vikas
10.4230/LIPIcs.MFCS.2017.69
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Recognizing Graphs Close to Bipartite Graphs
We continue research into a well-studied family of problems that ask if the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We let G be the class of k-degenerate graphs. The problem is known to be polynomial-time solvable if k=0 (bipartite graphs) and NP-complete if k=1 (near-bipartite graphs) even for graphs of diameter 4, as shown by Yang and Yuan, who also proved polynomial-time solvability for graphs of diameter 2. We show that recognizing near-bipartite graphs of diameter 3 is NP-complete resolving their open problem. To answer another open problem, we consider graphs of maximum degree D on n vertices. We show how to find A and B in O(n) time for k=1 and D=3, and in O(n^2) time for k >= 2 and D >= 4. These results also provide an algorithmic version of a result of Catlin [JCTB, 1979] and enable us to complete the complexity classification of another problem: finding a path in the vertex colouring reconfiguration graph between two given k-colourings of a graph of bounded maximum degree.
degenerate graphs
near-bipartite graphs
reconfiguration graphs
70:1-70:14
Regular Paper
Marthe
Bonamy
Marthe Bonamy
Konrad K.
Dabrowski
Konrad K. Dabrowski
Carl
Feghali
Carl Feghali
Matthew
Johnson
Matthew Johnson
Daniël
Paulusma
Daniël Paulusma
10.4230/LIPIcs.MFCS.2017.70
Bradley Baetz and David R. Wood. Brooks' vertex-colouring theorem in linear time. CoRR, abs/1401.8023, 2014.
Marthe Bonamy, Konrad K. Dabrowski, Carl Feghali, Matthew Johnson, and Daniël Paulusma. Independent feedback vertex set for P₅-free graphs. Manuscript, 2017.
Paul S. Bonsma and Luis Cereceda. Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theoretical Computer Science, 410(50):5215-5226, 2009.
Andreas Brandstädt, Synara Brito, Sulamita Klein, Loana Tito Nogueira, and Fábio Protti. Cycle transversals in perfect graphs and cographs. Theoretical Computer Science, 469:15-23, 2013.
Andreas Brandstädt, Peter L. Hammer, Van Bang Le, and Vadim V. Lozin. Bisplit graphs. Discrete Mathematics, 299(1-3):11-32, 2005.
Andreas Brandstädt, Van Bang Le, and Thomas Szymczak. The complexity of some problems related to graph 3-colorability. Discrete Applied Mathematics, 89(1-3):59-73, 1998.
Leizhen Cai and Derek G. Corneil. A generalization of perfect graphs - i-perfect graphs. Journal of Graph Theory, 23(1):87-103, 1996.
Paul A. Catlin. Brooks' graph-coloring theorem and the independence number. Journal of Combinatorial Theory, Series B, 27(1):42-48, 1979.
Paul A. Catlin and Hong-Jian Lai. Vertex arboricity and maximum degree. Discrete Mathematics, 141(1-3):37-46, 1995.
Luis Cereceda. Mixing graph colourings. PhD thesis, London School of Economics, 2007.
Konrad K. Dabrowski, Vadim V. Lozin, and Juraj Stacho. Stable-Π partitions of graphs. Discrete Applied Mathematics, 182:104-114, 2015.
François Dross, Mickaël Montassier, and Alexandre Pinlou. Partitioning sparse graphs into an independent set and a forest of bounded degree. CoRR, abs/1606.04394, 2016.
Tomás Feder, Pavol Hell, Sulamita Klein, and Rajeev Motwani. List partitions. SIAM Journal on Discrete Mathematics, 16(3):449-478, 2003.
Carl Feghali, Matthew Johnson, and Daniël Paulusma. A reconfigurations analogue of Brooks' Theorem and its consequences. Journal of Graph Theory, 83(4):340-358, 2016.
Michael Randolph Garey, David S. Johnson, and Larry J. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, 1(3):237-267, 1976.
Martin Grötschel, László Lovász, and Alexander Schrijver. Polynomial algorithms for perfect graphs. Annals of Discrete Mathematics, 21:325-356, 1984.
Chính T. Hoàng and Van Bang Le. On P₄-transversals of perfect graphs. Discrete Mathematics, 216(1-3):195-210, 2000.
Matthew Johnson, Dieter Kratsch, Stefan Kratsch, Viresh Patel, and Daniël Paulusma. Finding shortest paths between graph colourings. Algorithmica, 75(2):295-321, 2016.
Jan Kratochvíl and Ingo Schiermeyer. On the computational complexity of (𝒪,𝒫)-partition problems. Discussiones Mathematicae Graph Theory, 17(2):253-258, 1997.
László Lovász. Coverings and coloring of hypergraphs. Congressus Numerantium, VIII:3-12, 1973.
Vadim V. Lozin. Between 2- and 3-colorability. Information Processing Letters, 94(4):179-182, 2005.
Nadimpalli V. R. Mahadev and Uri N. Peled. Threshold graphs and related topics, volume 56 of Annals of Discrete Mathematics. North-Holland, Amsterdam, 1995.
Martín Matamala. Vertex partitions and maximum degenerate subgraphs. Journal of Graph Theory, 55(3):227-232, 2007.
Colin McDiarmid and Nikola Yolov. Recognition of unipolar and generalised split graphs. Algorithms, 8(1):46-59, 2015.
George B. Mertzios and Paul G. Spirakis. Algorithms and almost tight results for 3-colorability of small diameter graphs. Algorithmica, 74(1):385-414, 2016.
Neeldhara Misra, Geevarghese Philip, Venkatesh Raman, and Saket Saurabh. On parameterized independent feedback vertex set. Theoretical Computer Science, 461:65-75, 2012.
Yuma Tamura, Takehiro Ito, and Xiao Zhou. Algorithms for the independent feedback vertex set problem. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E98-A(6):1179-1188, 2015.
Yongqi Wu, Jinjiang Yuan, and Yongcheng Zhao. Partition a graph into two induced forests. Journal of Mathematical Study, 29:1-6, 1996.
Aifeng Yang and Jinjiang Yuan. Partition the vertices of a graph into one independent set and one acyclic set. Discrete Mathematics, 306(12):1207-1216, 2006.
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Parameterized Algorithms and Kernels for Rainbow Matching
In this paper, we study the NP-complete colorful variant of the classical Matching problem, namely, the Rainbow Matching problem. Given an edge-colored graph G and a positive integer k, this problem asks whether there exists a matching of size at least k such that all the edges in the matching have distinct colors. We first develop a deterministic algorithm that solves Rainbow Matching on paths in time O*(((1+\sqrt{5})/2)^k) and polynomial space. This algorithm is based on a curious combination of the method of bounded search trees and a "divide-and-conquer-like" approach, where the branching process is guided by the maintenance of an auxiliary bipartite graph where one side captures "divided-and-conquered" pieces of the path. Our second result is a randomized algorithm that solves Rainbow Matching on general graphs in time O*(2^k) and polynomial space. Here, we show how a result by Björklund et al. [JCSS, 2017] can be invoked as a black box, wrapped by a probability-based analysis tailored to our problem. We also complement our two main results by designing kernels for Rainbow Matching on general and bounded-degree graphs.
Rainbow Matching
Parameterized Algorithm
Bounded Search Trees
Divide-and-Conquer
3-Set Packing
3-Dimensional Matching
71:1-71:13
Regular Paper
Sushmita
Gupta
Sushmita Gupta
Sanjukta
Roy
Sanjukta Roy
Saket
Saurabh
Saket Saurabh
Meirav
Zehavi
Meirav Zehavi
10.4230/LIPIcs.MFCS.2017.71
F. N. Abu-Khzam. An improved kernelization algorithm for r-set packing. Information Processing Letters, 110:621-624, 2010.
Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Narrow sieves for parameterized paths and packings. Journal of Computer and System Sciences, 87:119-139, 2017.
M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms. Springer, 2015.
H. Dell and D. Marx. Kernelization of packing problems. In SODA'12, 2012.
R. G. Downey and M. R. Fellows. Fundamentals of parameterized complexity. Springer, 2013.
Jack Edmonds. Paths, trees, and flowers. Canadian Journal of mathematics, 17(3):449-467, 1965.
Fedor V. Fomin, Fabrizio Grandoni, and Dieter Kratsch. A measure & conquer approach for the analysis of exact algorithms. J. ACM, 56(5):25:1-25:32, 2009.
Fedor V. Fomin and Dieter Kratsch. Exact Exponential Algorithms. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2010.
M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979.
J. E. Hopcroft and R. M. Karp. An n^5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Computing, 2:225-231, 1973.
Alon Itai, Michael Rodeh, and Steven L. Tanimoto. Some matching problems for bipartite graphs. J. ACM, 25(4):517-525, 1978.
Mikio Kano and Xueliang Li. Monochromatic and heterochromatic subgraphs in edge-colored graphs-a survey. Graphs and Combinatorics, 24(4):237-263, 2008.
Van Bang Le and Florian Pfender. Complexity results for rainbow matchings. Theor. Comput. Sci., 524:27-33, 2014.
László Lovász and Michael D Plummer. Matching theory, volume 367. American Mathematical Soc., 2009.
Silvio Micali and Vijay V. Vazirani. An O(√|V| |E|) algorithm for finding maximum matching in general graphs. In 21st Annual Symposium on Foundations of Computer Science, Syracuse, New York, USA, 13-15 October 1980, pages 17-27, 1980.
Herbert J Ryser. Neuere probleme der kombinatorik. Vorträge über Kombinatorik, Oberwolfach, pages 69-91, 1967.
Larry J. Stockmeyer and Vijay V. Vazirani. Np-completeness of some generalizations of the maximum matching problem. Inf. Process. Lett., 15(1):14-19, 1982.
Mihalis Yannakakis and Fanica Gavril. Edge dominating sets in graphs. SIAM Journal on Applied Mathematics, 38(3):364-372, 1980.
Meirav Zehavi. Mixing color coding-related techniques. In Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, volume 9294 of Lecture Notes in Computer Science, pages 1037-1049, 2015.
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Compositional Weak Metrics for Group Key Update
We investigate the compositionality of both weak bisimilarity metric and weak similarity quasi- metric semantics with respect to a variety of standard operators, in the context of probabilistic process algebra. We show how compositionality with respect to nondeterministic and probabilistic choice requires to resort to rooted semantics. As a main application, we demonstrate how our results can be successfully used to conduct compositional reasonings to estimate the performances of group key update protocols in a multicast setting.
Behavioural metric
compositional reasoning
group key update
72:1-72:16
Regular Paper
Ruggero
Lanotte
Ruggero Lanotte
Massimo
Merro
Massimo Merro
Simone
Tini
Simone Tini
10.4230/LIPIcs.MFCS.2017.72
Luca Aceto, Bard Bloom, and Fritz W. Vaandrager. Turning SOS Rules into Equations. Information &Computation, 111(1):1-52, 1994. URL: http://dx.doi.org/10.1006/inco.1994.1040.
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http://dx.doi.org/10.4230/LIPIcs.CONCUR.2016.20
Pedro R. D'Argenio, Daniel Gebler, and Matias D. Lee. Axiomatizing Bisimulation Equivalences and Metrics from Probabilistic SOS Rules. In 17th International Conference on Foundations of Software Science and Computation Structures, volume 8412 of LNCS, pages 289-303. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-642-54830-7_19.
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Pedro R. D'Argenio, Daniel Gebler, and Matias D. Lee. A General SOS Theory for the Specification of Probabilistic Transition Systems. Information &Computation, 249:76-109, 2016. URL: http://dx.doi.org/10.1016/j.ic.2016.03.009.
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http://dx.doi.org/10.1109/LICS.2007.22
Yuxin Deng, Tom Chothia, Catuscia Palamidessi, and Jun Pang. Metrics for Action-labelled Quantitative Transition Systems. ENTCS, 153(2):79-96, 2006. URL: http://dx.doi.org/10.1016/j.entcs.2005.10.033.
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Yuxin Deng and Wenjie Du. The Kantorovich Metric in Computer Science: A Brief Survey. ENTCS, 253(3):73-82, 2009. URL: http://dx.doi.org/10.1016/j.entcs.2009.10.006.
http://dx.doi.org/10.1016/j.entcs.2009.10.006
Yuxin Deng, Rob J. van Glabbeek, Matthew Hennessy, and Carroll Morgan. Characterising Testing Preorders for Finite Probabilistic Processes. Logical Methods in Computer Science, 4(4), 2008. URL: http://dx.doi.org/10.2168/LMCS-4(4:4)2008.
http://dx.doi.org/10.2168/LMCS-4(4:4)2008
Josée Desharnais, Vineet Gupta, Radha Jagadeesan, and Prakash Panangaden. Metrics for Labelled Markov Processes. Theoretical Computer Science, 318(3):323-354, 2004. URL: http://dx.doi.org/10.1016/j.tcs.2003.09.013.
http://dx.doi.org/10.1016/j.tcs.2003.09.013
Josée Desharnais, Radha Jagadeesan, Vineet Gupta, and Prakash Panangaden. The Metric Analogue of Weak Bisimulation for Probabilistic Processes. In 17th IEEE Symposium on Logic in Computer Science, pages 413-422. IEEE Computer Society, 2002. URL: http://dx.doi.org/10.1109/LICS.2002.1029849.
http://dx.doi.org/10.1109/LICS.2002.1029849
Uli Fahrenberg and Axel Legay. The Quantitative Linear-time-branching-time Spectrum. Theoretical Computer Science, 538:54-69, 2014. URL: http://dx.doi.org/10.1016/j.tcs.2013.07.030.
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Daniel Gebler, Kim G. Larsen, and Simone Tini. Compositional Bisimulation Metric Reasoning with Probabilistic Process Calculi. Logical Methods in Computer Science, 12(4), 2016. URL: http://dx.doi.org/10.2168/LMCS-12(4:12)2016.
http://dx.doi.org/10.2168/LMCS-12(4:12)2016
Daniel Gebler and Simone Tini. Fixed-point Characterization of Compositionality Properties of Probabilistic Processes Combinators. In Combined 21th International Workshop on Expressiveness in Concurrency and 11th Workshop on Structural Operational Semantics, volume 160 of EPTCS, pages 63-78. OPA, 2014. URL: http://dx.doi.org/10.4204/EPTCS.160.7.
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Daniel Gebler and Simone Tini. SOS Specifications of Probabilistic Systems by Uniformly Continuous Operators. In 26th Conference on Concurrency Theory, volume 42 of LIPIcs, pages 155-168. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015.
Hans Hansson and Bengt Jonsson. A Logic for Reasoning about Time and Reliability. Formal Aspects of Computing, 6(5):512-535, 1994. URL: http://dx.doi.org/10.1007/BF01211866.
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Bengt Jonsson, Kim G. Larsen, and Wang Yi. Probabilistic Extensions of Process Algebras. In Handbook of Process Algebra, pages 685-710. Elsevier, 2001.
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Ruggero Lanotte, Andrea Maggiolo-Schettini, and Angelo Troina. Weak bisimulation for Probabilistic Timed Automata. Theoretical Computer Science, 411(50):4291-4322, 2010. URL: http://dx.doi.org/10.1016/j.tcs.2010.09.003.
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http://dx.doi.org/10.1007/978-3-319-60225-7_10
Matteo Mio. Upper-Expectation Bisimilarity and Łukasiewicz μ-Calculus. In 17th International Conference on Foundations of Software Science and Computation Structures, volume 8412 of LNCS, pages 335-350. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-642-54830-7_22.
http://dx.doi.org/10.1007/978-3-642-54830-7_22
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Roberto Segala. Modeling and Verification of Randomized Distributed Real-Time Systems. PhD thesis, MIT, 1995.
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Rob J. van Glabbeek and Peter W. Weijland. Branching Time and Abstraction in Bisimulation Semantics. Journal of the ACM, 43(3):555-600, 1996. URL: http://dx.doi.org/10.1145/233551.233556.
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http://dx.doi.org/10.1007/978-3-540-71050-9
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Clique-Width for Graph Classes Closed under Complementation
Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set H of forbidden induced subgraphs. We initiate a systematic study into the boundedness of clique-width of hereditary graph classes closed under complementation. First, we extend the known classification for the |H|=1 case by classifying the boundedness of clique-width for every set H of self-complementary graphs. We then completely settle the |H|=2 case. In particular, we determine one new class of (H1, complement of H1)-free graphs of bounded clique-width (as a side effect, this leaves only six classes of (H1, H2)-free graphs, for which it is not known whether their clique-width is bounded).
Once we have obtained the classification of the |H|=2 case, we research the effect of forbidding self-complementary graphs on the boundedness of clique-width. Surprisingly, we show that for a set F of self-complementary graphs on at least five vertices, the classification of the boundedness of clique-width for ({H1, complement of H1} + F)-free graphs coincides with the one for the |H|=2 case if and only if F does not include the bull (the only non-empty self-complementary graphs on fewer than five vertices are P_1 and P_4, and P_4-free graphs have clique-width at most 2).
Finally, we discuss the consequences of our results for COLOURING.
clique-width
self-complementary graph
forbidden induced subgraph
73:1-73:14
Regular Paper
Alexandre
Blanché
Alexandre Blanché
Konrad K.
Dabrowski
Konrad K. Dabrowski
Matthew
Johnson
Matthew Johnson
Vadim V.
Lozin
Vadim V. Lozin
Daniël
Paulusma
Daniël Paulusma
Viktor
Zamaraev
Viktor Zamaraev
10.4230/LIPIcs.MFCS.2017.73
Lowell W. Beineke and Allen J. Schwenk. On a bipartite form of the Ramsey problem. Congressus Numerantium, XV:17-22, 1975.
Alexandre Blanché, Konrad K. Dabrowski, Matthew Johnson, and Daniël Paulusma. Hereditary graph classes: When the complexities of Colouring and Clique Cover coincide. CoRR, abs/1607.06757, 2016.
Rodica Boliac and Vadim V. Lozin. On the clique-width of graphs in hereditary classes. Proc. ISAAC 2002, LNCS, 2518:44-54, 2002.
Andreas Brandstädt, Konrad K. Dabrowski, Shenwei Huang, and Daniël Paulusma. Bounding the clique-width of H-free split graphs. Discrete Applied Mathematics, 211:30-39, 2016.
Andreas Brandstädt, Konrad K. Dabrowski, Shenwei Huang, and Daniël Paulusma. Bounding the clique-width of H-free chordal graphs. Journal of Graph Theory, (in press), 2017.
Andreas Brandstädt, Feodor F. Dragan, Hoàng-Oanh Le, and Raffaele Mosca. New graph classes of bounded clique-width. Theory of Computing Systems, 38(5):623-645, 2005.
Andreas Brandstädt, Joost Engelfriet, Hoàng-Oanh Le, and Vadim V. Lozin. Clique-width for 4-vertex forbidden subgraphs. Theory of Computing Systems, 39(4):561-590, 2006.
Andreas Brandstädt, Tilo Klembt, and Suhail Mahfud. P₆- and triangle-free graphs revisited: structure and bounded clique-width. Discrete Mathematics and Theoretical Computer Science, 8(1):173-188, 2006.
Andreas Brandstädt, Hoàng-Oanh Le, and Raffaele Mosca. Gem- and co-gem-free graphs have bounded clique-width. International Journal of Foundations of Computer Science, 15(1):163-185, 2004.
Andreas Brandstädt, Hoàng-Oanh Le, and Raffaele Mosca. Chordal co-gem-free and (P₅,gem)-free graphs have bounded clique-width. Discrete Applied Mathematics, 145(2):232-241, 2005.
Andreas Brandstädt and Suhail Mahfud. Maximum weight stable set on graphs without claw and co-claw (and similar graph classes) can be solved in linear time. Information Processing Letters, 84(5):251-259, 2002.
Julia Chuzhoy. Improved bounds for the flat wall theorem. Proc. SODA 2015, pages 256-275, 2015.
Derek G. Corneil, Michel Habib, Jean-Marc Lanlignel, Bruce A. Reed, and Udi Rotics. Polynomial-time recognition of clique-width ≤ 3 graphs. Discrete Applied Mathematics, 160(6):834-865, 2012.
Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, 33(2):125-150, 2000.
Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discrete Applied Mathematics, 101(1-3):77-114, 2000.
Konrad K. Dabrowski, François Dross, and Daniël Paulusma. Colouring diamond-free graphs. Journal of Computer and System Sciences, (to appear).
Konrad K. Dabrowski, Petr A. Golovach, and Daniël Paulusma. Colouring of graphs with Ramsey-type forbidden subgraphs. Theoretical Computer Science, 522:34-43, 2014.
Konrad K. Dabrowski, Shenwei Huang, and Daniël Paulusma. Bounding clique-width via perfect graphs. Journal of Computer and System Sciences, (in press).
Konrad K. Dabrowski, Vadim V. Lozin, and Daniël Paulusma. Clique-width and well-quasi ordering of triangle-free graph classes. Proc. WG 2017, LNCS, (to appear).
Konrad K. Dabrowski, Vadim V. Lozin, and Daniël Paulusma. Well-quasi-ordering versus clique-width: New results on bigenic classes. Order, (to appear).
Konrad K. Dabrowski, Vadim V. Lozin, Rajiv Raman, and Bernard Ries. Colouring vertices of triangle-free graphs without forests. Discrete Mathematics, 312(7):1372-1385, 2012.
Konrad K. Dabrowski and Daniël Paulusma. Classifying the clique-width of H-free bipartite graphs. Discrete Applied Mathematics, 200:43-51, 2016.
Konrad K. Dabrowski and Daniël Paulusma. Clique-width of graph classes defined by two forbidden induced subgraphs. The Computer Journal, 59(5):650-666, 2016.
H. N. de Ridder et al. Information System on Graph Classes and their Inclusions, 2001-2013. http://www.graphclasses.org.
Wolfgang Espelage, Frank Gurski, and Egon Wanke. How to solve NP-hard graph problems on clique-width bounded graphs in polynomial time. Proc. WG 2001, LNCS, 2204:117-128, 2001.
Alastair Farrugia. Self-complementary graphs and generalisations: a comprehensive reference manual. Master’s thesis, University of Malta, 1999.
Michael R. Fellows, Frances A. Rosamond, Udi Rotics, and Stefan Szeider. Clique-width is NP-Complete. SIAM Journal on Discrete Mathematics, 23(2):909-939, 2009.
Stéphane Földes and Peter Ladislaw Hammer. Split graphs. Congressus Numerantium, XIX:311-315, 1977.
Petr A. Golovach, Matthew Johnson, Daniël Paulusma, and Jian Song. A survey on the computational complexity of colouring graphs with forbidden subgraphs. Journal of Graph Theory, 84(4):331-363, 2017.
Martin Grohe and Pascal Schweitzer. Isomorphism testing for graphs of bounded rank width. Proc. FOCS 2015, pages 1010-1029, 2015.
Frank Gurski. The behavior of clique-width under graph operations and graph transformations. Theory of Computing Systems, 60(2):346-376, 2017.
András Gyárfás. Problems from the world surrounding perfect graphs. Applicationes Mathematicae, 19(3-4):413-441, 1987.
Öjvind Johansson. Clique-decomposition, NLC-decomposition, and modular decomposition - relationships and results for random graphs. Congressus Numerantium, 132:39-60, 1998.
Marcin Kamiński, Vadim V. Lozin, and Martin Milanič. Recent developments on graphs of bounded clique-width. Discrete Applied Mathematics, 157(12):2747-2761, 2009.
Daniel Kobler and Udi Rotics. Edge dominating set and colorings on graphs with fixed clique-width. Discrete Applied Mathematics, 126(2-3):197-221, 2003.
Vadim V. Lozin and Dieter Rautenbach. On the band-, tree-, and clique-width of graphs with bounded vertex degree. SIAM Journal on Discrete Mathematics, 18(1):195-206, 2004.
Vadim V. Lozin and Dieter Rautenbach. The tree- and clique-width of bipartite graphs in special classes. Australasian Journal of Combinatorics, 34:57-67, 2006.
Johann A. Makowsky and Udi Rotics. On the clique-width of graphs with few P₄’s. International Journal of Foundations of Computer Science, 10(03):329-348, 1999.
Sang-Il Oum and Paul D. Seymour. Approximating clique-width and branch-width. Journal of Combinatorial Theory, Series B, 96(4):514-528, 2006.
Michaël Rao. MSOL partitioning problems on graphs of bounded treewidth and clique-width. Theoretical Computer Science, 377(1-3):260-267, 2007.
Ronald C. Read. On the number of self-complementary graphs and digraphs. Journal of the London Mathematical Society, s1-38(1):99-104, 1963.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Computing the Maximum using (min, +) Formulas
We study computation by formulas over (min,+). We consider the
computation of max{x_1,...,x_n} over N as a difference of
(min,+) formulas, and show that size n + n \log n is sufficient
and necessary. Our proof also shows that any (min,+) formula
computing the minimum of all sums of n-1 out of n variables must
have n \log n leaves; this too is tight. Our proofs use a
complexity measure for (min,+) functions based on minterm-like
behaviour and on the entropy of an associated graph.
Formulas
Circuits
Lower bounds
Tropical semiring
74:1-74:11
Regular Paper
Meena
Mahajan
Meena Mahajan
Prajakta
Nimbhorkar
Prajakta Nimbhorkar
Anuj
Tawari
Anuj Tawari
10.4230/LIPIcs.MFCS.2017.74
Eric Allender. Arithmetic circuits and counting complexity classes. In Jan Krajicek, editor, Complexity of Computations and Proofs, Quaderni di Matematica Vol. 13, pages 33-72. Seconda Universita di Napoli, 2004. An earlier version appeared in the Complexity Theory Column, SIGACT News 28, 4 (Dec. 1997) pp. 2-15.
Richard Bellman. On a routing problem. Quarterly of Applied Mathematics, 16:87-90, 1956.
Imre Csiszár, János Körner, László Lovász, Katalin Marton, and Gábor Simonyi. Entropy splitting for antiblocking corners and perfect graphs. Combinatorica, 10(1):27-40, 1990.
Robert W Floyd. Algorithm 97: shortest path. Communications of the ACM, 5(6):345, 1962.
Lester R Ford Jr. Network flow theory. Technical Report P-923, Rand Corporation, 1956.
Michael Held and Richard M Karp. A dynamic programming approach to sequencing problems. Journal of the Society for Industrial and Applied Mathematics, 10(1):196-210, 1962.
Mark Jerrum and Marc Snir. Some exact complexity results for straight-line computations over semirings. Journal of the ACM (JACM), 29(3):874-897, 1982.
Stasys Jukna. Lower bounds for tropical circuits and dynamic programs. Theory of Computing Systems, 57(1):160-194, 2015.
Stasys Jukna. Tropical complexity, Sidon sets, and dynamic programming. SIAM Journal on Discrete Mathematics, 30(4):2064-2085, 2016.
Stasys Jukna and Georg Schnitger. On the optimality of Bellman-Ford-Moore shortest path algorithm. Theoretical Computer Science, 628:101-109, 2016.
János Körner. Coding of an information source having ambiguous alphabet and the entropy of graphs. In Transactions of 6th Prague Conference on Information Theory, pages 411-425. Academia, Prague, 1973.
János Körner. Fredman-Komlós bounds and information theory. SIAM. J. on Algebraic and Discrete Methods, 7(4):560-570, 1986.
János Körner and Katalin Marton. New bounds for perfect hashing via information theory. European Journal of Combinatorics, 9(6):523-530, 1988.
Edward F Moore. The shortest path through a maze. Bell Telephone System., 1959.
Ilan Newman and Avi Wigderson. Lower bounds on formula size of boolean functions using hypergraph entropy. SIAM Journal on Discrete Mathematics, 8(4):536-542, 1995.
Gábor Simonyi. Graph entropy: A survey. Combinatorial Optimization, 20:399-441, 1995.
Gábor Simonyi. Perfect graphs and graph entropy: An updated survey. In Perfect Graphs, pages 293-328. John Wiley and Sons, 2001.
Stephen Warshall. A theorem on Boolean matrices. Journal of the ACM (JACM), 9(1):11-12, 1962.
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Selecting Nodes and Buying Links to Maximize the Information Diffusion in a Network
The Independent Cascade Model (ICM) is a widely studied model that aims to capture the dynamics of the information diffusion in social networks and in general complex networks. In this model, we can distinguish between active nodes which spread the information and inactive ones. The process starts from a set of initially active nodes called seeds. Recursively, currently active nodes can activate their neighbours according to a probability distribution on the set of edges. After a certain number of these recursive cycles, a large number of nodes might become active. The process terminates when no further node gets activated.
Starting from the work of Domingos and Richardson [Domingos et al. 2001], several studies have been conducted with the aim of shaping a given diffusion process so as to maximize the number of activated nodes at the end of the process. One of the most studied problems has been formalized by Kempe et al. and consists in finding a set of initial seeds that maximizes the expected number of active nodes under a budget constraint [Kempe et al. 2003].
In this paper we study a generalization of the problem of Kempe et al. in which we are allowed to spend part of the budget to create new edges incident to the seeds. That is, the budget can be spent to buy seeds or edges according to a cost function. The problem does not admin a PTAS, unless P=NP. We propose two approximation algorithms: the former one gives an approximation ratio that depends on the edge costs and increases when these costs are high; the latter algorithm gives a constant approximation guarantee which is greater than that of the first algorithm when the edge costs can be small.
Approximation algorithms
information diffusion
complex networks
independent cascade model
network augmentation
75:1-75:14
Regular Paper
Gianlorenzo
D'Angelo
Gianlorenzo D'Angelo
Lorenzo
Severini
Lorenzo Severini
Yllka
Velaj
Yllka Velaj
10.4230/LIPIcs.MFCS.2017.75
C. Asavathiratham, S. Roy, B. Lesieutre, and G. Verghese. The influence model. IEEE Control Systems, 21(6):52-64, 2001.
Eytan Bakshy, Jake M. Hofman, Winter A. Mason, and Duncan J. Watts. Everyone’s an influencer: Quantifying influence on twitter. In Proc. of the 4th ACM International Conference on Web Search and Data Mining, WSDM11, pages 65-74. ACM, 2011.
Frank M Bass. A new product growth for model consumer durables. Management science, 15(5):215-227, 1969.
Wei Chen, Chi Wang, and Yajun Wang. Scalable influence maximization for prevalent viral marketing in large-scale social networks. In Proc. of the 16th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD10, 2010.
James S. Coleman, Elihu Katz, and Herbert Menzel. Medical innovation: A diffusion study. The Bobbs-Merrill Company, 1966.
Pierluigi Crescenzi, Gianlorenzo D'Angelo, Lorenzo Severini, and Yllka Velaj. Greedily improving our own closeness centrality in a network. ACM Trans. Knowl. Discov. Data, 11(1):9:1-9:32, 2016.
Gianlorenzo D'Angelo, Lorenzo Severini, and Yllka Velaj. Influence maximization in the independent cascade model. In Proceedings of the 17th Italian Conference on Theoretical Computer Science, Lecce, Italy, September 7-9, 2016., pages 269-274, 2016.
Gianlorenzo D'Angelo, Lorenzo Severini, and Yllka Velaj. Recommending links through influence maximization. arXiv preprint, 2017. URL: http://arxiv.org/abs/1706.04368.
http://arxiv.org/abs/1706.04368
Gianlorenzo D'Angelo, Lorenzo Severini, and Yllka Velaj. Selecting nodes and buying links to maximize the information diffusion in a network. arXiv preprint, 2017. URL: https://arxiv.org/abs/1706.06466.
https://arxiv.org/abs/1706.06466
Pedro Domingos and Matt Richardson. Mining the network value of customers. In Proceedings of the 7th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD01, pages 57-66. ACM, 2001.
Jacob Goldenberg, Barak Libai, and Eitan Muller. Talk of the network: A complex systems look at the underlying process of word-of-mouth. Marketing letters, 12(3):211-223, 2001.
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K4-free Graphs as a Free Algebra
Graphs of treewidth at most two are the ones excluding the clique with four vertices as a minor. Equivalently, they are the graphs whose biconnected components are series-parallel.
We turn those graphs into a free algebra, answering positively a question by Courcelle and Engelfriet, in the case of treewidth two.
First we propose a syntax for denoting them: in addition to series and parallel compositions, it suffices to consider the neutral elements of those operations and a unary transpose operation. Then we give a finite equational presentation and we prove it complete: two terms from the syntax are congruent if and only if they denote the same graph.
Universal Algebra
Graph theory
Axiomatisation
Tree decompositions
Graph minors
76:1-76:14
Regular Paper
Enric
Cosme Llópez
Enric Cosme Llópez
Damien
Pous
Damien Pous
10.4230/LIPIcs.MFCS.2017.76
H. Andréka and D. A. Bredikhin. The equational theory of union-free algebras of relations. Algebra Universalis, 33(4):516-532, 1995. URL: http://dx.doi.org/10.1007/BF01225472.
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S. Arnborg, B. Courcelle, A. Proskurowski, and D. Seese. An algebraic theory of graph reduction. Journal of the ACM, 40(5):1134-1164, 1993. URL: http://dx.doi.org/10.1145/174147.169807.
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Michel Bauderon and Bruno Courcelle. Graph expressions and graph rewritings. Mathematical Systems Theory, 20(2-3):83-127, 1987. URL: http://dx.doi.org/10.1007/BF01692060.
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Mikołaj Bojańczyk and Michal Pilipczuk. Definability equals recognizability for graphs of bounded treewidth. In LICS, pages 407-416. ACM, 2016. URL: http://dx.doi.org/10.1145/2933575.2934508.
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Chandra Chekuri and Anand Rajaraman. Conjunctive query containment revisited. Theoretical Computer Science, 239(2):211-229, 2000. URL: http://dx.doi.org/10.1016/S0304-3975(99)00220-0.
http://dx.doi.org/10.1016/S0304-3975(99)00220-0
Enric Cosme-Llópez and Damien Pous. K₄-free graphs as a free algebra, 2017. Full version of this extended abstract, with all proofs, available at URL: https://hal.archives-ouvertes.fr/hal-01515752/.
https://hal.archives-ouvertes.fr/hal-01515752/
B. Courcelle. The monadic second-order logic of graphs. I: Recognizable sets of finite graphs. Information and Computation, 85(1):12-75, 1990. URL: http://dx.doi.org/10.1016/0890-5401(90)90043-H.
http://dx.doi.org/10.1016/0890-5401(90)90043-H
B. Courcelle. The monadic second-order logic of graphs V: on closing the gap between definability and recognizability. Theoretical Computer Science, 80(2):153-202, 1991. URL: http://dx.doi.org/10.1016/0304-3975(91)90387-H.
http://dx.doi.org/10.1016/0304-3975(91)90387-H
B. Courcelle. Recognizable sets of graphs: equivalent definitions and closure properties. Mathematical Structures in Computer Science, 4(1):1-32, 1994.
B. Courcelle. The monadic second-order logic of graphs XI: Hierarchical decompositions of connected graphs. Theoretical Computer Science, 224(1):35-58, 1999. URL: http://dx.doi.org/10.1016/S0304-3975(98)00306-5.
http://dx.doi.org/10.1016/S0304-3975(98)00306-5
B. Courcelle and J. Engelfriet. Graph Structure and Monadic Second-Order Logic - A Language-Theoretic Approach, volume 138 of Encyclopedia of mathematics and its applications. Cambridge University Press, 2012.
B. Courcelle and J. Lagergren. Equivalent definitions of recognizability for sets of graphs of bounded tree-width. Mathematical Structures in Computer Science, 6(2):141-165, 1996. URL: http://dx.doi.org/10.1017/S096012950000092X.
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Making Metric Temporal Logic Rational
We study an extension of MTL in pointwise time with regular expression guarded modality Reg_I(re) where re is a rational expression over subformulae. We study the decidability and expressiveness of this extension (MTL+Ureg+Reg), called RegMTL, as well as its fragment SfrMTL where only star-free rational expressions are allowed. Using the technique of temporal projections, we show that RegMTL has decidable satisfiability by giving an equisatisfiable reduction to MTL. We also identify a subclass MITL+UReg of RegMTL for which our equisatisfiable reduction gives rise to formulae of MITL, yielding elementary decidability. As our second main result, we show a tight automaton-logic connection between SfrMTL and partially ordered (or very weak) 1-clock alternating timed automata.
Metric Temporal Logic
Timed Automata
Regular Expression
Equisatisfiability
Expressiveness
77:1-77:14
Regular Paper
Shankara Narayanan
Krishna
Shankara Narayanan Krishna
Khushraj
Madnani
Khushraj Madnani
Paritosh K.
Pandya
Paritosh K. Pandya
10.4230/LIPIcs.MFCS.2017.77
R. Alur, T. Feder, and T. Henzinger. The benefits of relaxing punctuality. J.ACM, 43(1):116-146, 1996.
Rajeev Alur and Thomas A. Henzinger. Real-time logics: Complexity and expressiveness. Inf. Comput., 104(1):35-77, 1993. URL: http://dx.doi.org/10.1006/inco.1993.1025.
http://dx.doi.org/10.1006/inco.1993.1025
Eugene Asarin, Paul Caspi, and Oded Maler. Timed regular expressions. J. ACM, 49(2):172-206, 2002. URL: http://dx.doi.org/10.1145/506147.506151.
http://dx.doi.org/10.1145/506147.506151
Augustin Baziramwabo, Pierre McKenzie, and Denis Thérien. Modular temporal logic. In 14th Annual IEEE Symposium on Logic in Computer Science, Trento, Italy, July 2-5, 1999, pages 344-351, 1999. URL: http://dx.doi.org/10.1109/LICS.1999.782629.
http://dx.doi.org/10.1109/LICS.1999.782629
Patricia Bouyer, Fabrice Chevalier, and Nicolas Markey. On the expressiveness of TPTL and MTL. In FSTTCS 2005: Foundations of Software Technology and Theoretical Computer Science, 25th International Conference, Hyderabad, India, December 15-18, 2005, Proceedings, pages 432-443, 2005. URL: http://dx.doi.org/10.1007/11590156_35.
http://dx.doi.org/10.1007/11590156_35
Cindy Eisner and Dana Fisman. A Practical Introduction to PSL. Springer, 2006.
IEEE P1850-Standard for PSL-Property Specification Language, 2005.
Jesper G. Henriksen and P. S. Thiagarajan. Dynamic linear time temporal logic. Ann. Pure Appl. Logic, 96(1-3):187-207, 1999. URL: http://dx.doi.org/10.1016/S0168-0072(98)00039-6.
http://dx.doi.org/10.1016/S0168-0072(98)00039-6
Philippe Herrmann. Renaming is necessary in timed regular expressions. In Foundations of Software Technology and Theoretical Computer Science, 19th Conference, Chennai, India, December 13-15, 1999, Proceedings, pages 47-59, 1999. URL: http://dx.doi.org/10.1007/3-540-46691-6_4.
http://dx.doi.org/10.1007/3-540-46691-6_4
P. Hunter. When is metric temporal logic expressively complete? In CSL, pages 380-394, 2013.
S. N. Krishna K. Madnani and P. K. Pandya. Partially punctual metric temporal logic is decidable. In TIME, pages 174-183, 2014.
Shankara Narayanan Krishna, Khushraj Madnani, and Paritosh K. Pandya. Metric temporal logic with counting. In Foundations of Software Science and Computation Structures - 19th International Conference, FOSSACS 2016, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2016, Eindhoven, The Netherlands, April 2-8, 2016, Proceedings, pages 335-352, 2016.
F. Laroussinie, A. Meyer, and E. Petonnet. Counting ltl. In TIME, pages 51-58, 2010.
K. Lodaya and A. V. Sreejith. Ltl can be more succinct. In ATVA, pages 245-258, 2010.
J. Ouaknine and J. Worrell. On the decidability of metric temporal logic. In LICS, pages 188-197, 2005.
A. Rabinovich. Complexity of metric temporal logic with counting and pnueli modalities. In FORMATS, pages 93-108, 2008.
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Complexity of Restricted Variants of Skolem and Related Problems
Given a linear recurrence sequence (LRS), the Skolem problem, asks whether it ever becomes zero. The decidability of this problem has been open for several decades. Currently decidability is known only for LRS of order upto 4. For arbitrary orders (i.e., number of terms the n-th depends on), the only known complexity result is NP-hardness by a result of Blondel and Portier from 2002.
In this paper, we give a different proof of this hardness result, which is arguably simpler and pinpoints the source of hardness. To demonstrate this, we identify a subclass of LRS for which the Skolem problem is in fact NP-complete. We show the generic nature of our lower-bound technique by adapting it to show stronger lower bounds of a related problem that encompasses many known decision problems on linear recurrent sequences.
Linear recurrence sequences
Skolem problem
NP-completeness
Program termination
78:1-78:14
Regular Paper
Akshay
S.
Akshay S.
Nikhil
Balaji
Nikhil Balaji
Nikhil
Vyas
Nikhil Vyas
10.4230/LIPIcs.MFCS.2017.78
Manindra Agrawal, S. Akshay, Blaise Genest, and P. S. Thiagarajan. Approximate verification of the symbolic dynamics of markov chains. J. ACM, 62(1):2:1-2:34, 2015.
S. Akshay, Timos Antonopoulos, Joël Ouaknine, and James Worrell. Reachability problems for markov chains. Inf. Process. Lett., 115(2):155-158, 2015.
S. Akshay, Blaise Genest, Bruno Karelovic, and Nikhil Vyas. On regularity of unary probabilistic automata. In 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016, February 17-20, 2016, Orléans, France, pages 8:1-8:14, 2016.
Sanjeev Arora and Boaz Barak. Computational Complexity - A Modern Approach. Cambridge University Press, 2009. URL: http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521424264.
http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521424264
M. Artin. Algebra. Pearson Prentice Hall, 2011. URL: https://books.google.de/books?id=QsOfPwAACAAJ.
https://books.google.de/books?id=QsOfPwAACAAJ
Amir M. Ben-Amram, Samir Genaim, and Abu Naser Masud. On the termination of integer loops. ACM Trans. Program. Lang. Syst., 34(4):16:1-16:24, December 2012.
V. D. Blondel and N. Portier. The presence of a zero in an integer linear recurrent sequence is NP-hard to decide. In Linear Algebra and its Applications, pages 351-352. Elsevier, 2002.
Mark Braverman. Termination of integer linear programs. In Computer Aided Verification, 18th International Conference, CAV 2006, Seattle, WA, USA, August 17-20, 2006, Proceedings, pages 372-385, 2006.
Ventsislav Chonev, Joël Ouaknine, and James Worrell. The polyhedron-hitting problem. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 940-956, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.64.
http://dx.doi.org/10.1137/1.9781611973730.64
Ventsislav Chonev, Joël Ouaknine, and James Worrell. On the complexity of the orbit problem. J. ACM, 63(3):23:1-23:18, 2016. URL: http://dx.doi.org/10.1145/2857050.
http://dx.doi.org/10.1145/2857050
Henri Cohen. A course in computational algebraic number theory, volume 138. Springer Science &Business Media, 2013.
Graham Everest, Alfred J. van der Poorten, Igor E. Shparlinski, and Thomas Ward. Recurrence Sequences, volume 104 of Mathematical surveys and monographs. American Mathematical Society, 2003. URL: http://www.ams.org/bookstore?fn=20&arg1=survseries&item=SURV-104.
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Godfrey H. Hardy and Edward M. Wright. An introduction to the theory of numbers (5. ed.). Clarendon Press, 1995.
Dan Kalman. The generalized vandermonde matrix. Mathematics Magazine, 57(1):15-21, 1984.
Richard M Karp. Reducibility among combinatorial problems. In Complexity of computer computations, pages 85-103. Springer, 1972.
Joël Ouaknine, João Sousa Pinto, and James Worrell. On termination of integer linear loops. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 957-969, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.65.
http://dx.doi.org/10.1137/1.9781611973730.65
Joël Ouaknine and James Worrell. Decision problems for linear recurrence sequences. In Reachability Problems - 6th International Workshop, RP 2012, Bordeaux, France, September 17-19, 2012. Proceedings, pages 21-28, 2012. URL: http://dx.doi.org/10.1007/978-3-642-33512-9_3.
http://dx.doi.org/10.1007/978-3-642-33512-9_3
Joël Ouaknine and James Worrell. On the positivity problem for simple linear recurrence sequences,. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part II, pages 318-329, 2014. URL: http://dx.doi.org/10.1007/978-3-662-43951-7_27.
http://dx.doi.org/10.1007/978-3-662-43951-7_27
Joël Ouaknine and James Worrell. Positivity problems for low-order linear recurrence sequences. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 366-379, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.27.
http://dx.doi.org/10.1137/1.9781611973402.27
Joël Ouaknine and James Worrell. Ultimate positivity is decidable for simple linear recurrence sequences. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part II, pages 330-341, 2014. URL: http://dx.doi.org/10.1007/978-3-662-43951-7_28.
http://dx.doi.org/10.1007/978-3-662-43951-7_28
Joël Ouaknine and James Worrell. On linear recurrence sequences and loop termination. SIGLOG News, 2(2):4-13, 2015.
Marcus Schaefer and Christopher Umans. Completeness in the polynomial-time hierarchy: A compendium. SIGACT news, 33(3):32-49, 2002.
Ashish Tiwari. Termination of linear programs. In Computer Aided Verification, 16th International Conference, CAV 2004, Boston, MA, USA, July 13-17, 2004, Proceedings, pages 70-82, 2004.
Prasoon Tiwari. A problem that is easier to solve on the unit-cost algebraic RAM. J. Complexity, 8(4):393-397, 1992. URL: http://dx.doi.org/10.1016/0885-064X(92)90003-T.
http://dx.doi.org/10.1016/0885-064X(92)90003-T
M.Hirvensalo V.Halava, T.Harju and J.Karhumäki. Skolem’s problem on the border between decidability and undecidability. In TUCS Technical Report Number 683, 2005.
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Being Even Slightly Shallow Makes Life Hard
We study the computational complexity of identifying dense substructures, namely r/2-shallow topological minors and r-subdivisions. Of particular interest is the case r = 1, when these substructures correspond to very localized relaxations of subgraphs. Since Densest Subgraph can be solved in polynomial time, we ask whether these slight relaxations also admit efficient algorithms.
In the following, we provide a negative answer: Dense r/2-Shallow Topological Minor and Dense r-Subdivsion are already NP-hard for r = 1 in very sparse graphs. Further, they do not admit algorithms with running time 2^(o(tw^2)) n^O(1) when parameterized by the treewidth of the input graph for r > 2 unless ETH fails.
Topological minors
NP Completeness
Treewidth
ETH
FPT algorithms
79:1-79:13
Regular Paper
Irene
Muzi
Irene Muzi
Michael P.
O'Brien
Michael P. O'Brien
Felix
Reidl
Felix Reidl
Blair D.
Sullivan
Blair D. Sullivan
10.4230/LIPIcs.MFCS.2017.79
S. Arnborg, J. Lagergren, and D. Seese. Easy problems for tree-decomposable graphs. Journal of Algorithms, 12(2):308-340, 1991.
H. L. Bodlaender, M. Cygan, S. Kratsch, and J. Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. In International Colloquium on Automata, Languages, and Programming, pages 196-207. Springer, 2013.
H. L. Bodlaender, T. Wolle, and A. Koster. Contraction and treewidth lower bounds. J. Graph Algorithms Appl., 10(1):5-49, 2006.
J. Chen, X. Huang, I. A. Kanj, and G. Xia. Strong computational lower bounds via parameterized complexity. Journal of Computer and System Sciences, 72(8):1346-1367, 2006.
M. Cygan, F.V. Fomin, Ł. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Lower bounds based on the exponential-time hypothesis. In Parameterized Algorithms, pages 467-521. Springer, 2015.
M. Cygan, J. Nederlof, M. Pilipczuk, M. Pilipczuk, van J.M.M. Rooij, and J. O. Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In \FOCS52nd, pages 150-159. IEEE Computer Society, 2011.
E. D. Demaine and M. Hajiaghayi. The bidimensionality theory and its algorithmic applications. The Computer Journal, 51(3):292-302, 2008.
F. Dorn, F. V. Fomin, and D. M. Thilikos. Subexponential parameterized algorithms. Computer Science Review, 2(1):29-39, 2008.
P. G. Drange, M. Dregi, F.V. Fomin, S. Kreutzer, D. Lokshtanov, M. Pilipczuk, M. Pilipczuk, F. Reidl, S. Saurabh, F. Sánchez Villaamil, S. Siebertz, and S. Sikdar. Kernelization and sparseness: the case of dominating set. In 33rd Symposium on Theoretical Aspects of Computer Science, 2016.
Z. Dvořák. Asymptotical Structure of Combinatorial Objects. PhD thesis, Charles University, Faculty of Mathematics and Physics, 2007.
Z. Dvořák, D. Král, and R. Thomas. Deciding first-order properties for sparse graphs. In \FOCS51st, pages 133-142. IEEE Computer Society, 2010.
F. V. Fomin, D. Lokshtanov, S. Saurabh, and D. M. Thilikos. Linear kernels for (connected) dominating set on H-minor-free graphs. In \SODA23rd, pages 82-93. SIAM, 2012.
J. Gajarský, P. Hliněný, J. Obdržálek, S. Ordyniak, F. Reidl, P. Rossmanith, F. Sánchez Villaamil, and S. Sikdar. Kernelization using structural parameters on sparse graph classes. To appear in Journal of Computer and System Sciences, 2016.
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Walrasian Pricing in Multi-Unit Auctions
Multi-unit auctions are a paradigmatic model, where a seller brings multiple units of a good, while several buyers bring monetary endowments. It is well known that Walrasian equilibria do not always exist in this model, however compelling relaxations such as Walrasian envy-free pricing do. In this paper we design an optimal envy-free mechanism for multi-unit auctions with budgets. When the market is even mildly competitive, the approximation ratios of this mechanism are small constants for both the revenue and welfare objectives, and in fact for welfare the approximation converges to 1 as the market becomes fully competitive. We also give an impossibility theorem, showing that truthfulness requires discarding resources, and in particular, is incompatible with (Pareto) efficiency.
mechanism design
multi-unit auctions
Walrasian pricing
market share
80:1-80:14
Regular Paper
Simina
Brânzei
Simina Brânzei
Aris
Filos-Ratsikas
Aris Filos-Ratsikas
Peter Bro
Miltersen
Peter Bro Miltersen
Yulong
Zeng
Yulong Zeng
10.4230/LIPIcs.MFCS.2017.80
E. Anderson and D. Simester. Price stickiness and customer antagonism. Available at SSRN 1273647, 2008.
L. M. Ausubel. An efficient ascending-bid auction for multiple objects. The American Economic Review, 94(5):1452-1475, 2004.
M. Babaioff, B. Lucier, N. Nisan, and R. Paes Leme. On the efficiency of the walrasian mechanism. In ACM EC, pages 783-800. ACM, 2014.
M. F. Balcan, A. Blum, and Y. Mansour. Item pricing for revenue maximization. In ACM EC, pages 50-59. ACM, 2008.
Y. Bartal, R. Gonen, and N. Nisan. Incentive compatible multi unit combinatorial auctions. In TARK, pages 72-87, 2003.
C. Borgs, J. Chayes, N. Immorlica, M. Mahdian, and A. Saberi. Multi-unit auctions with budget-constrained bidders. In ACM EC, pages 44-51, 2005.
A. Borodin, O. Lev, and T. Strangway. Budgetary effects on pricing equilibrium in online markets. In AAMAS, 2016.
S. Brânzei, Y. Chen, X. Deng, A. Filos-Ratsikas, S. K. S. Frederiksen, and J. Zhang. The fisher market game: Equilibrium and welfare. In AAAI, pages 587-593, 2014.
Simina Brânzei and Ariel D. Procaccia. Verifiably truthful mechanisms. In ITCS, pages 297-306, 2015.
Y. Cai, C. Daskalakis, and M. Weinberg. Optimal multi-dimensional mechanism design: Reducing revenue to welfare maximization. In FOCS, pages 130-139, 2012.
Y. Cai, C. Daskalakis, and M. Weinberg. Reducing revenue to welfare maximization: Approximation algorithms and other generalizations. In SODA, pages 578-595, 2013.
N. Chen, X. Deng, and J. Zhang. How profitable are strategic behaviors in a market? In ESA, pages 106-118. Springer, 2011.
M. Cheung and C. Swamy. Approximation algorithms for single-minded envy-free profit-maximization problems with limited supply. In FOCS, pages 35-44, 2008.
V. Cohen-Addad, A. Eden, M. Feldman, and A. Fiat. The invisible hand of dynamic market pricing. In ACM EC, pages 383-400, 2016.
R. Colini-Baldeschi, S. Leonardi, P. Sankowski, and Q. Zhang. Revenue maximizing envy-free fixed-price auctions with budgets. In WINE, pages 233-246. Springer, 2014.
S. Dobzinski, R. Lavi, and N. Nisan. Multi-unit auctions with budget limits. GEB, 74(2):486-503, 2012.
S. Dobzinski and R. P. Leme. Efficiency guarantees in auctions with budgets. In ICALP, pages 392-404. Springer, 2014.
S. Dobzinski and N. Nisan. Mechanisms for multi-unit auctions. In ACM EC, pages 346-351, 2007.
S. Dobzinski and N. Nisan. Multi-unit auctions: beyond roberts. JET, 156:14-44, 2015.
A. Eden, M. Feldman, O. Friedler, I. Talgam-Cohen, and M. Weinberg. The competition complexity of auctions: A bulow-klemperer result for multi-dimensional bidders. In ACM EC, 2017.
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M. Feldman, A. Fiat, S. Leonardi, and P. Sankowski. Revenue maximizing envy-free multi-unit auctions with budgets. In ACM EC, pages 532-549, 2012.
M. Feldman, N. Gravin, and B. Lucier. Combinatorial auctions via posted prices. In SODA, pages 123-135, 2015.
F. Gul and E. Stacchetti. Walrasian equilibrium with gross substitutes. Journal of Economic Theory, 87(1):95-124, 1999.
V. Guruswami, J. Hartline, A. Karlin, D. Kempe, C. Kenyon, and F. McSherry. On profit-maximizing envy-free pricing. In SODA, pages 1164-1173, 2005.
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J. Hartline and Q. Yan. Envy, truth, and profit. In ACM EC, pages 243-252, 2011.
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Distributed Strategies Made Easy
Distributed/concurrent strategies have been introduced as special maps of event structures. As such they factor through their "rigid images," themselves strategies. By concentrating on such "rigid image" strategies we are able to give an elementary account of distributed strategies and their composition, resulting in a category of games and strategies. This is in contrast to the usual development where composition involves the pullback of event structures explicitly and results in a bicategory. It is shown how, in this simpler setting, to extend strategies to probabilistic strategies; and indicated how through probability we can track nondeterministic branching behaviour, that one might otherwise think lost irrevocably in restricting attention to "rigid image" strategies.
Games
Strategies
Event Structures
Probability
81:1-81:13
Regular Paper
Simon
Castellan
Simon Castellan
Pierre
Clairambault
Pierre Clairambault
Glynn
Winskel
Glynn Winskel
10.4230/LIPIcs.MFCS.2017.81
Samson Abramsky and Paul-André Melliès. Concurrent games and full completeness. In LICS '99. IEEE Computer Society, 1999.
Simon Castellan and Pierre Clairambault. Causality vs interleavings in concurrent games semantics. In CONCUR'16, 2016.
Simon Castellan, Jonathan Hayman, Marc Lasson, and Glynn Winskel. Strategies as concurrent processes. ENTCS, 308, 2014.
Gian Luca Cattani and Glynn Winskel. Profunctors, open maps and bisimulation. Mathematical Structures in Computer Science, 15(3):553-614, 2005.
Pierre Clairambault, Julian Gutierrez, and Glynn Winskel. The winning ways of concurrent games. In LICS 2012: 235-244, 2012.
Pierre Clairambault and Glynn Winskel. On concurrent games with payoff. Electr. Notes Theor. Comput. Sci. 298: 71-92, 2013.
John Conway. On Numbers and Games. Wellesley, MA: A K Peters, 2000.
Claudia Faggian and Mauro Piccolo. Partial orders, event structures and linear strategies. In TLCA '09, volume 5608 of LNCS. Springer, 2009.
Tom Hirschowitz and Damien Pous. Innocent strategies as presheaves and interactive equivalences for CCS. Sci. Ann. Comp. Sci., 22(1):147-199, 2012.
Martin Hyland. Some reasons for generalising domain theory. Mathematical Structures in Computer Science, 20(2):239-265, 2010.
Claire Jones and Gordon Plotkin. A probabilistic powerdomain of valuations. In LICS '89. IEEE Computer Society, 1989.
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Paul-André Melliès and Samuel Mimram. Asynchronous games: Innocence without alternation. In CONCUR, volume 4703 of LNCS, pages 395-411, 2007.
Silvain Rideau and Glynn Winskel. Concurrent strategies. In LICS 2011.
Peter Selinger and Benoît Valiron. Quantum lambda calculus. In Simon Gay and Ian Mackie, editors, Semantic Techniques in Quantum Computation, chapter 4, pages 135-172. Cambridge University Press, 2009.
Jonathan Hayman Simon Castellan, Pierre Clairambault and Glynn Winskel. Non-angelic concurrent game semantics. 2016.
Takeshi Tsukada and C.-H. Luke Ong. Nondeterminism in game semantics via sheaves. In LICS 2015. IEEE Computer Society, 2015.
Glynn Winskel. Event structure semantics for CCS and related languages. In ICALP'82, LNCS 140, 1982.
Glynn Winskel. Event structures as presheaves -two representation theorems. In CONCUR '99, 1999.
Glynn Winskel. Event structures with symmetry. Electr. Notes Theor. Comput. Sci. 172: 611-652, 2007.
Glynn Winskel. Winning, losing and drawing in concurrent games with perfect or imperfect information. In Festschrift for Dexter Kozen, volume 7230 of LNCS. Springer, 2012.
Glynn Winskel. Distributed probabilistic and quantum strategies. ENTCS 298, 2013.
Glynn Winskel. Strategies as profunctors. In FOSSACS 2013, volume 7794 of LNCS. Springer, 2013.
Glynn Winskel. Probabilistic and quantum event structures. In Festschrift for Prakash Panangaden, volume 8464 of LNCS. Springer, 2014.
Glynn Winskel. ECSYM Notes: Event Structures, Stable Families and Concurrent Games. http://www.cl.cam.ac.uk/∼gw104/ecsym-notes.pdf, 2016.
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On Definable and Recognizable Properties of Graphs of Bounded Treewidth (Invited Talk)
This is an overview of the invited talk delivered at the 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017).
monadic second-order logic
treewidth
recognizability
82:1-82:2
Invited Talk
Michal
Pilipczuk
Michal Pilipczuk
10.4230/LIPIcs.MFCS.2017.82
Mikołaj Bojańczyk and Michał Pilipczuk. Definability equals recognizability for graphs of bounded treewidth. In LICS'16, pages 407-416. ACM, 2016. Full version available as arXiv preprint 1605.03045.
Bruno Courcelle. The Monadic Second-Order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990.
Valentine Kabanets. Recognizability equals definability for partial k-paths. In ICALP'97, volume 1256 of LNCS, pages 805-815. Springer, 1997.
Denis Lapoire. Recognizability equals Monadic Second-Order definability for sets of graphs of bounded tree-width. In STACS'98, volume 1373 of LNCS, pages 618-628. Springer, 1998.
Imre Simon. Factorization forests of finite height. Theor. Comput. Sci., 72(1):65-94, 1990.
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Hardness and Approximation of High-Dimensional Search Problems (Invited Talk)
Hardness and Approximation of High-Dimensional Search Problems.
Hardness
high-dimensional search
83:1-83:1
Invited Talk
Rasmus
Pagh
Rasmus Pagh
10.4230/LIPIcs.MFCS.2017.83
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Temporal Logics for Multi-Agent Systems (Invited Talk)
This is an overview of an invited talk delivered during the 42nd International Conference on Mathematical Foundations of Computer Science (MFCS 2017).
Temporal logics
verification
game theory
strategic reasoning.
84:1-84:3
Invited Talk
Nicolas
Markey
Nicolas Markey
10.4230/LIPIcs.MFCS.2017.84
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Ideal-Based Algorithms for the Symbolic Verification of Well-Structured Systems (Invited Talk)
We explain how the downward-closed subsets of a well-quasi-ordering (X,\leq) can be represented via the ideals of X and how this leads to simple and efficient algorithms for the verification of well-structured systems.
Well-structured systems and verification
Order theory
85:1-85:4
Invited Talk
Philippe
Schnoebelen
Philippe Schnoebelen
10.4230/LIPIcs.MFCS.2017.85
P. A. Abdulla. Well (and better) quasi-ordered transition systems. Bull. Symbolic Logic, 16(4):457-515, 2010. URL: http://dx.doi.org/10.2178/bsl/1294171129.
http://dx.doi.org/10.2178/bsl/1294171129
P. A. Abdulla, K. Čerāns, B. Jonsson, and Yih-Kuen Tsay. Algorithmic analysis of programs with well quasi-ordered domains. Information and Computation, 160(1/2):109-127, 2000. URL: http://dx.doi.org/10.1006/inco.1999.2843.
http://dx.doi.org/10.1006/inco.1999.2843
P. A. Abdulla, A. Collomb-Annichini, A. Bouajjani, and B. Jonsson. Using forward reachability analysis for verification of lossy channel systems. Formal Methods in System Design, 25(1):39-65, 2004. URL: http://dx.doi.org/10.1023/B:FORM.0000033962.51898.1a.
http://dx.doi.org/10.1023/B:FORM.0000033962.51898.1a
P. A. Abdulla and B. Jonsson. Verifying programs with unreliable channels. Information and Computation, 127(2):91-101, 1996. URL: http://dx.doi.org/10.1006/inco.1996.0053.
http://dx.doi.org/10.1006/inco.1996.0053
B. Bérard, F. Cassez, S. Haddad, D. Lime, and O. H. Roux. The expressive power of time Petri nets. Theoretical Computer Science, 474, 2012. URL: http://dx.doi.org/10.1016/j.tcs.2012.12.005.
http://dx.doi.org/10.1016/j.tcs.2012.12.005
M. Blondin, A. Finkel, and P. McKenzie. Well behaved transition systems. arXiv:1608.02636 [cs.LO], August 2016. To appear in Logical Meth. Comp. Sci. URL: http://arxiv.org/abs/1608.02636.
http://arxiv.org/abs/1608.02636
G. Cécé, A. Finkel, and S. Purushothaman Iyer. Unreliable channels are easier to verify than perfect channels. Information and Computation, 124(1):20-31, 1996. URL: http://dx.doi.org/10.1006/inco.1996.0003.
http://dx.doi.org/10.1006/inco.1996.0003
C. Dufourd, P. Jančar, and Ph. Schnoebelen. Boundedness of Reset P/T nets. In CONCUR '99, LNCS 1644, pages 301-310. Springer, 1999. URL: http://dx.doi.org/10.1007/3-540-48523-6_27.
http://dx.doi.org/10.1007/3-540-48523-6_27
A. Finkel. The ideal theory for WSTS. In RP 2016, LNCS 9899, pages 1-22. Springer, 2016. URL: http://dx.doi.org/10.1007/978-3-319-45994-3_1.
http://dx.doi.org/10.1007/978-3-319-45994-3_1
A. Finkel and J. Goubault-Larrecq. Forward analysis for WSTS, part I: Completions. In STACS 2009, LIPIcs 3, pages 433-444. Leibniz-Zentrum für Informatik, 2009. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2009.1844.
http://dx.doi.org/10.4230/LIPIcs.STACS.2009.1844
A. Finkel and J. Goubault-Larrecq. Forward analysis for WSTS, part II: Complete WSTS. Logical Methods in Comp. Science, 8(4), 2012. URL: http://dx.doi.org/10.2168/LMCS-8(3:28)2012.
http://dx.doi.org/10.2168/LMCS-8(3:28)2012
A. Finkel, P. McKenzie, and C. Picaronny. A well-structured framework for analysing Petri nets extensions. Information and Computation, 195(1-2):1-29, 2004. URL: http://dx.doi.org/10.1016/j.ic.2004.01.005.
http://dx.doi.org/10.1016/j.ic.2004.01.005
A. Finkel and Ph. Schnoebelen. Well-structured transition systems everywhere! Theoretical Computer Science, 256(1-2):63-92, 2001. URL: http://dx.doi.org/10.1016/S0304-3975(00)00102-X.
http://dx.doi.org/10.1016/S0304-3975(00)00102-X
G. Geeraerts, J.-F. Raskin, and L. Van Begin. Expand, enlarge and check: New algorithms for the coverability problem of WSTS. Journal of Computer and System Sciences, 72(1):180-203, 2006. URL: http://dx.doi.org/10.1016/j.jcss.2005.09.001.
http://dx.doi.org/10.1016/j.jcss.2005.09.001
J. Goubault-Larrecq, S. Halfon, P. Karandikar, K. Narayan Kumar, and Ph. Schnoebelen. The ideal approach to computing closed subsets in well-quasi-orderings. In preparation, 2017.
J. Goubault-Larrecq and S. Schmitz. Deciding piecewise testable separability for regular tree languages. In ICALP 2016, LIPIcs 55, pages 97:1-97:15. Leibniz-Zentrum für Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.97.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.97
Ch. Haase, S. Schmitz, and Ph. Schnoebelen. The power of priority channel systems. Logical Methods in Comp. Science, 10(4:4), 2014. URL: http://dx.doi.org/10.2168/LMCS-10(4:4)2014.
http://dx.doi.org/10.2168/LMCS-10(4:4)2014
T. A. Henzinger, R. Majumdar, and J.-F. Raskin. A classification of symbolic transition systems. ACM Trans. Computational Logic, 6(1):1-32, 2005. URL: http://dx.doi.org/10.1145/1042038.1042039.
http://dx.doi.org/10.1145/1042038.1042039
R. M. Karp and R. E. Miller. Parallel program schemata. Journal of Computer and System Sciences, 3(2):147-195, 1969. URL: http://dx.doi.org/10.1016/S0022-0000(69)80011-5.
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R. Lazić, T. Newcomb, J. Ouaknine, A. W. Roscoe, and J. Worrell. Nets with tokens which carry data. Fundamenta Informaticae, 88(3):251-274, 2008.
R. Lazić and S. Schmitz. The complexity of coverability in ν-Petri nets. In LICS 2016, pages 467-476. ACM Press, 2016. URL: http://dx.doi.org/10.1145/2933575.2933593.
http://dx.doi.org/10.1145/2933575.2933593
J. Leroux and S. Schmitz. Ideal decompositions for vector addition systems. In STACS 2016, LIPIcs 47, pages 1:1-1:13. Leibniz-Zentrum für Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2016.1.
http://dx.doi.org/10.4230/LIPIcs.STACS.2016.1
S. Schmitz and Ph. Schnoebelen. The power of well-structured systems. In CONCUR 2013, LNCS 8052, pages 5-24. Springer, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40184-8_2.
http://dx.doi.org/10.1007/978-3-642-40184-8_2
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