10th International Symposium on Parameterized and Exact Computation (IPEC 2015), IPEC 2015, September 16-18, 2015, Patras, Greece
IPEC 2015
September 16-18, 2015
Patras, Greece
International Symposium on Parameterized and Exact Computation
IPEC
http://fpt.wikidot.com/ipec
https://dblp.org/db/conf/iwpec
Leibniz International Proceedings in Informatics
LIPIcs
https://www.dagstuhl.de/dagpub/1868-8969
https://dblp.org/db/series/lipics
1868-8969
Thore
Husfeldt
Thore Husfeldt
Iyad
Kanj
Iyad Kanj
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
43
2015
978-3-939897-92-7
https://www.dagstuhl.de/dagpub/978-3-939897-92-7
Front Matter, Table of Contents, Preface, Program Committee, External Reviewers, List of Authors
Front Matter, Table of Contents, Preface, Program Committee, External Reviewers, List of Authors
IPEC
i-xiv
Front Matter
Thore
Husfeldt
Thore Husfeldt
Iyad
Kanj
Iyad Kanj
10.4230/LIPIcs.IPEC.2015.i
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Bidimensionality and Parameterized Algorithms (Invited Talk)
We provide an exposition of the main results of the theory of bidimensionality in parameterized algorithm design. This theory applies to graph problems that are bidimensional in the sense that i) their solution value is not increasing when we take minors or contractions of the input graph and ii) their solution value for the (triangulated) (k x k)-grid graph grows as a quadratic function of k. Under certain additional conditions, mainly of logical and combinatorial nature, such problems admit subexponential parameterized algorithms and linear kernels when their inputs are restricted to certain topologically defined graph classes. We provide all formal definitions and concepts in order to present these results in a rigorous way and in their latest update.
Parameterized algorithms
Subexponential FPT-algorithms
Kernelization
Linear kenrels
Bidimensionality
Graph Minors
1-16
Invited Talk
Dimitrios M.
Thilikos
Dimitrios M. Thilikos
10.4230/LIPIcs.IPEC.2015.1
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Hardness of Easy Problems: Basing Hardness on Popular Conjectures such as the Strong Exponential Time Hypothesis (Invited Talk)
Algorithmic research strives to develop fast algorithms for fundamental problems. Despite its many successes, however, many problems still do not have very efficient algorithms. For years researchers have explained the hardness for key problems by proving NP-hardness, utilizing polynomial time reductions to base the hardness of key problems on the famous conjecture P != NP. For problems that already have polynomial time algorithms, however, it does not seem that one can show any sort of hardness based on P != NP. Nevertheless, we would like to provide evidence that a problem $A$ with a running time O(n^k) that has not been improved in decades, also requires n^{k-o(1)} time, thus explaining the lack of progress on the problem. Such unconditional time lower bounds seem very difficult to obtain, unfortunately. Recent work has concentrated on an approach mimicking NP-hardness: (1) select a few key problems that are conjectured to require T(n) time to solve, (2) use special, fine-grained reductions to prove time lower bounds for many diverse problems in P based on the conjectured hardness of the key problems. In this abstract we outline the approach, give some examples of hardness results based on the Strong Exponential Time Hypothesis, and present an overview of some of the recent work on the topic.
reductions
satisfiability
strong exponential time hypothesis
shortest paths
3SUM
equivalences
fine-grained complexity
17-29
Invited Talk
Virginia
Vassilevska Williams
Virginia Vassilevska Williams
10.4230/LIPIcs.IPEC.2015.17
Amir Abboud, Arturs Backurs, and Virginia V. Williams. If the current clique algorithms are optimal, so is Valiant’s parser. In 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, October 17-20, 2015, page to appear, 2015.
Amir Abboud, Arturs Backurs, and Virginia V. Williams. Quadratic-time hardness of LCS and other sequence similarity measures. In 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, October 17-20, 2015, page to appear, 2015.
Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1681-1697, 2015.
Amir Abboud and Kevin Lewi. Exact weight subgraphs and the k-sum conjecture. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, pages 1-12, 2013.
Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 434-443, 2014.
Amir Abboud, Virginia Vassilevska Williams, and Joshua R. Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, page to appear, 2016.
Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of faster alignment of sequences. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, pages 39-51, 2014.
Amir Abboud, Virginia Vassilevska Williams, and Huacheng Yu. Matching triangles and basing hardness on an extremely popular conjecture. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 41-50, 2015.
Amir Abboud, Richard Ryan Williams, and Huacheng Yu. More applications of the polynomial method to algorithm design. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 218-230, 2015.
Manuel Abellanas, Ferran Hurtado, Christian Icking, Rolf Klein, Elmar Langetepe, Lihong Ma, Belén Palop, and Vera Sacristán. Smallest color-spanning objects. In Algorithms - ESA 2001, 9th Annual European Symposium, Aarhus, Denmark, August 28-31, 2001, Proceedings, pages 278-289, 2001.
Amihood Amir, Timothy M. Chan, Moshe Lewenstein, and Noa Lewenstein. On hardness of jumbled indexing. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, pages 114-125, 2014.
Boris Aronov and Sariel Har-Peled. On approximating the depth and related problems. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, Vancouver, British Columbia, Canada, January 23-25, 2005, pages 886-894, 2005.
Arturs Backurs and Piotr Indyk. Edit distance cannot be computed in strongly subquadratic time (unless SETH is false). In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 51-58, 2015.
I. Baran, E.D. Demaine, and M. Pǎtraşcu. Subquadratic algorithms for 3SUM. Algorithmica, 50(4):584-596, 2008.
Gill Barequet and Sariel Har-Peled. Polygon-containment and translational min-Hausdorff-distance between segment sets are 3SUM-hard. In Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, 17-19 January 1999, Baltimore, Maryland., pages 862-863, 1999.
Philip Bille and Martin Farach-Colton. Fast and compact regular expression matching. Theoretical Computer Science, 409(3):486-496, 2008.
Karl Bringmann and Marvin Künnemann. Quadratic conditional lower bounds for string problems and dynamic time warping. In 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, October 17-20, 2015, page to appear, 2015.
Chris Calabro, Russell Impagliazzo, and Ramamohan Paturi. The complexity of satisfiability of small depth circuits. In Parameterized and Exact Computation, pages 75-85. Springer, 2009.
Marco L. Carmosino, Jiawei Gao, Russell Impagliazzo, Ivan Mikhailin, Ramamohan Paturi, and Stefan Schneider. Nondeterministic extensions of the strong exponential time hypothesis and consequences for non-reducibility. Technical Report TR15-148, Electronic Colloquium on Computational Complexity (ECCC), 2015.
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, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 31-40, 2015.
Shiri Chechik, Daniel Larkin, Liam Roditty, Grant Schoenebeck, Robert Endre Tarjan, and Virginia Vassilevska Williams. Better approximation algorithms for the graph diameter. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1041-1052, 2014.
Otfried Cheong, Alon Efrat, and Sariel Har-Peled. On finding a guard that sees most and a shop that sells most. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, New Orleans, Louisiana, USA, January 11-14, 2004, pages 1098-1107, 2004.
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Variants of Plane Diameter Completion
The Plane Diameter Completion problem asks, given a plane graph G and a positive integer d, if it is a spanning subgraph of a plane graph H that has diameter at most d. We examine two variants of this problem where the input comes with another parameter k. In the first variant, called BPDC, k upper bounds the total number of edges to be added and in the second, called BFPDC, k upper bounds the number of additional edges per face. We prove that both problems are NP-complete, the first even for 3-connected graphs of face-degree at most 4 and the second even when k=1 on 3-connected graphs of face-degree at most 5. In this paper we give parameterized algorithms for both problems that run in O(n^{3})+2^{2^{O((kd)^2\log d)}} * n steps.
Planar graphs
graph modification problems
parameterized algorithms
dynamic programming
branchwidth
30-42
Regular Paper
Petr A.
Golovach
Petr A. Golovach
Clément
Requilé
Clément Requilé
Dimitrios M.
Thilikos
Dimitrios M. Thilikos
10.4230/LIPIcs.IPEC.2015.30
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Parameterized and Approximation Algorithms for the Load Coloring Problem
Let c, k be two positive integers. Given a graph G=(V,E), the c-Load Coloring problem asks whether there is a c-coloring varphi: V => [c] such that for every i in [c], there are at least k edges with both endvertices colored i. Gutin and Jones (IPL 2014) studied this problem with c=2. They showed 2-Load Coloring to be fixed-parameter tractable (FPT) with parameter k by obtaining a kernel with at most 7k vertices. In this paper, we extend the study to any fixed c by giving both a linear-vertex and a linear-edge kernel. In the particular case of c=2, we obtain a kernel with less than 4k vertices and less than 8k edges. These results imply that for any fixed c >= 2, c-Load Coloring is FPT and the optimization version of c-Load Coloring (where k is to be maximized) has an approximation algorithm with a constant ratio.
Load Coloring
fixed-parameter tractability
kernelization
43-54
Regular Paper
Florian
Barbero
Florian Barbero
Gregory
Gutin
Gregory Gutin
Mark
Jones
Mark Jones
Bin
Sheng
Bin Sheng
10.4230/LIPIcs.IPEC.2015.43
N. Ahuja, A. Baltz, B. Doerr, A. Prívetivý, and A. Srivastav. On the minimum load coloring problem. J. Discrete Algorithms, 5(3):533-545, 2007.
J.-P. Allouche and J. Shallit. The ring of k-regular sequences, II. Theor. Comput. Sci., 307(1):3-29, 2003.
H. L. Bodlaender, F. V. Fomin, D. Lokshtanov, E. Penninkx, S. Saurabh, and D. M. Thilikos. (meta) kernelization. In Foundations of Computer Science, FOCS 2009, pages 629-638. IEEE Computer Society, 2009.
M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms. Springer, 2015.
E. D. Demaine, F. V. Fomin, M. T. Hajiaghayi, and D. M. Thilikos. Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. J. ACM, 52(6):866-893, 2005.
E. D. Demaine and M. T. Hajiaghayi. The bidimensionality theory and its algorithmic applications. Comput. J., 51(3):292-302, 2008.
R. G. Downey and M. R. Fellows. Fundamentals of Parameterized Complexity. Springer, 2013.
F. V. Fomin, D. Lokshtanov, N. Misra, G. Philip, and S. Saurabh. Hitting forbidden minors: Approximation and kernelization. In Symposium on Theoretical Aspects of Computer Science, STACS 2011, volume 9 of LIPIcs, pages 189-200. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2011.
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S. Kratsch. Recent developments in kernelization: A survey. Bulletin of the EATCS, 113, 2014.
D. Lokshtanov, N. Misra, and S. Saurabh. Kernelization - preprocessing with a guarantee. In The Multivariate Algorithmic Revolution and Beyond, volume 7370 of Lecture Notes in Computer Science, pages 129-161. Springer, 2012.
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http://oeis.org/A073121
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Scheduling Two Competing Agents When One Agent Has Significantly Fewer Jobs
We study a scheduling problem where two agents (each equipped with a private set of jobs) compete to perform their respective jobs on a common single machine. Each agent wants to keep the weighted sum of completion times of his jobs below a given (agent-dependent) bound. This problem is known to be NP-hard, even for quite restrictive settings of the problem parameters.
We consider parameterized versions of the problem where one of the agents has a small number of jobs (and where this small number constitutes the parameter). The problem becomes much more tangible in this case, and we present three positive algorithmic results for it. Our study is complemented by showing that the general problem is NP-complete even when one agent only has a single job.
Parameterized Complexity
Multiagent Scheduling
55-65
Regular Paper
Danny
Hermelin
Danny Hermelin
Judith-Madeleine
Kubitza
Judith-Madeleine Kubitza
Dvir
Shabtay
Dvir Shabtay
Nimrod
Talmon
Nimrod Talmon
Gerhard
Woeginger
Gerhard Woeginger
10.4230/LIPIcs.IPEC.2015.55
Alessandro Agnetis, Jean-Charles Billaut, Stanisław Gawiejnowicz, Dario Pacciarelli, and Ameur Soukhal. Multiagent Scheduling: Models and Algorithms. Springer, 2014.
Allesandro Agnetis, Pitu B. Mirchandani, Dario Pacciarelli, and Andrea Pacifici. Scheduling problems with two competing agents. Operations Research, 52(2):229-242, 2004.
Kenneth R. Baker and J. Cole Smith. A multiple-criterion model for machine scheduling. Journal of Scheduling, 6(1):7-16, 2003.
René Bevernvan Bevern, Matthias Mnich, Rolf Niedermeier, and Mathias Weller. Interval scheduling and colorful independent sets. In Proceedings of the 23th International Symposium on Algorithms and Computation (ISAAC'12), volume 7676 of LNCS, pages 247-256. Springer, 2012.
René Bevernvan Bevern, Rolf Niedermeier, and Ondrej Suchý. A parameterized complexity view on non-preemptively scheduling interval-constrained jobs: few machines, small looseness, and small slack. arXiv preprint arXiv:1005.4159, 2015.
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Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Springer, 2013.
Michael R. Fellows and Catherine McCartin. On the parametric complexity of schedules to minimize tardy tasks. Theoretical computer science, 298(2):317-324, 2003.
Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Springer, 2006.
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On the Workflow Satisfiability Problem with Class-independent Constraints
A workflow specification defines sets of steps and users. An authorization policy determines for each user a subset of steps the user is allowed to perform. Other security requirements, such as separation-of-duty, impose constraints on which subsets of users may perform certain subsets of steps. The workflow satisfiability problem (WSP) is the problem of determining whether there exists an assignment of users to workflow steps that satisfies all such authorizations and constraints. An algorithm for solving WSP is important, both as a static analysis tool for workflow specifications, and for the construction of run-time reference monitors for workflow management systems. Given the computational difficulty of WSP, it is important, particularly for the second application, that such algorithms are as efficient as possible.
We introduce class-independent constraints, enabling us to model scenarios where the set of users is partitioned into groups, and the identities of the user groups are irrelevant to the satisfaction of the constraint. We prove that solving WSP is fixed-parameter tractable (FPT) for this class of constraints and develop an FPT algorithm that is useful in practice. We compare the performance of the FPT algorithm with that of SAT4J (a pseudo-Boolean SAT solver) in computational experiments, which show that our algorithm significantly outperforms SAT4J for many instances of WSP. User-independent constraints, a large class of constraints including many practical ones, are a special case of class-independent constraints for which WSP was proved to be FPT (Cohen et al., J. Artif. Intel. Res. 2014). Thus our results considerably extend our knowledge of the fixed-parameter tractability of WSP.
Workflow Satisfiability Problem; Constraint Satisfaction Problem; fixed-parameter tractability; user-independent constraints
66-77
Regular Paper
Jason
Crampton
Jason Crampton
Andrei
Gagarin
Andrei Gagarin
Gregory
Gutin
Gregory Gutin
Mark
Jones
Mark Jones
10.4230/LIPIcs.IPEC.2015.66
American National Standards Institute. ANSI INCITS 359-2004 for Role Based Access Control, 2004.
D. A. Basin, S. J. Burri, and G. Karjoth. Obstruction-free authorization enforcement: Aligning security and business objectives. J. Comput. Security, 22(5):661-698, 2014.
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D. Cohen, J. Crampton, A. Gagarin, G. Gutin, and M. Jones. Engineering algorithms for workflow satisfiability problem with user-independent constraints. In J. Chen, J.E. Hopcroft, and J. Wang, editors, Frontiers in Algorithmics, FAW 2014, volume 8497 of Lecture Notes in Computer Science, pages 48-59. Springer, 2014.
D. Cohen, J. Crampton, A. Gagarin, G. Gutin, and M. Jones. Iterative plan construction for the workflow satisfiability problem. J. Artif. Intel. Res., 51:555-577, 2014.
D. Cohen, J. Crampton, A. Gagarin, G. Gutin, and M. Jones. Algorithms for the workflow satisfiability problem engineered for counting constraints. J. Comb. Optim., to appear, 2015. (DOI: 10.1007/s10878-015-9877-7).
J. Crampton. A reference monitor for workflow systems with constrained task execution. In E. Ferrari and G.-J. Ahn, editors, SACMAT, pages 38-47. ACM, 2005.
J. Crampton, A. V. Gagarin, G. Gutin, and M. Jones. On the workflow satisfiability problem with class-independent constraints. CoRR, abs/1504.03561, 2015.
J. Crampton, G. Gutin, and A. Yeo. On the parameterized complexity and kernelization of the workflow satisfiability problem. ACM Trans. Inf. Syst. Secur., 16(1):4, 2013.
A. Gagarin, J. Crampton, G. Gutin, and M. Jones. Implementation of the pattern-backtracking FPT algorithm and experimental data set for the WSP with class-independent constraints. http://dx.doi.org/10.6084/m9.figshare.1502692, Aug 2015.
http://dx.doi.org/10.6084/m9.figshare.1502692
D. Karapetyan, A. Gagarin, and G. Gutin. Pattern backtracking algorithm for the workflow satisfiability problem. In Frontiers in Algorithmics 2015, volume 9130 of Lect. Notes Comput. Sci., pages 138-149. Springer, 2015.
D. Le Berre and A. Parrain. The SAT4J library, release 2.2. J. Satisf. Bool. Model. Comput., 7:59-64, 2010.
W. Myrvold and W. Kocay. Errors in graph embedding algorithms. J. Comput. Syst. Sci., 77(2):430-438, 2011.
Q. Wang and N. Li. Satisfiability and resiliency in workflow authorization systems. ACM Trans. Inf. Syst. Secur., 13(4):40, 2010.
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Parameterized Algorithms for Min-Max Multiway Cut and List Digraph Homomorphism
In this paper we design FPT-algorithms for two parameterized problems. The first is List Digraph Homomorphism: given two digraphs G and H and a list of allowed vertices of H for every vertex of G, the question is whether there exists a homomorphism from G to H respecting the list constraints. The second problem is a variant of Multiway Cut, namely Min-Max Multiway Cut: given a graph G, a non-negative integer l, and a set T of r terminals, the question is whether we can partition the vertices of G into r parts such that (a) each part contains one terminal and (b) there are at most l edges with only one endpoint in this part. We parameterize List Digraph Homomorphism by the number w of edges of G that are mapped to non-loop edges of H and we give a time 2^{O(l * log(h) + l^{2 * log(l)}} * n^{4} * log(n) algorithm, where h is the order of the host graph H.We also prove that Min-Max Multiway Cut can be solved in time 2^{O((l * r)^2 * log(l *r))} * n^{4} * log(n). Our approach introduces a general problem, called List Allocation, whose expressive power permits the design of parameterized reductions of both aforementioned problems to it. Then our results are based on an FPT-algorithm for the List Allocation problem that is designed using a suitable adaptation of the randomized contractions technique (introduced by [Chitnis, Cygan, Hajiaghayi, Pilipczuk, and Pilipczuk, FOCS 2012]).
Parameterized complexity
Fixed-Parameter Tractable algorithm
Multiway Cut
Digraph homomorphism
78-89
Regular Paper
Eun Jung
Kim
Eun Jung Kim
Christophe
Paul
Christophe Paul
Ignasi
Sau
Ignasi Sau
Dimitrios M.
Thilikos
Dimitrios M. Thilikos
10.4230/LIPIcs.IPEC.2015.78
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Improved Exact Algorithms for Mildly Sparse Instances of Max SAT
We present improved exponential time exact algorithms for Max SAT. Our algorithms run in time of the form O(2^{(1-mu(c))n}) for instances with n variables and m=cn clauses. In this setting, there are three incomparable currently best algorithms: a deterministic exponential space algorithm with mu(c)=1/O(c * log(c)) due to Dantsin and Wolpert [SAT 2006], a randomized polynomial space algorithm with mu(c)=1/O(c * log^3(c)) and a deterministic polynomial space algorithm with mu(c)=1/O(c^2 * log^2(c)) due to Sakai, Seto and Tamaki [Theory Comput. Syst., 2015]. Our first result is a deterministic polynomial space algorithm with mu(c)=1/O(c * log(c)) that achieves the previous best time complexity without exponential space or randomization. Furthermore, this algorithm can handle instances with exponentially large weights and hard constraints. The previous algorithms and our deterministic polynomial space algorithm run super-polynomially faster than 2^n only if m=O(n^2).
Our second results are deterministic exponential space algorithms for Max SAT with mu(c)=1/O((c * log(c))^{2/3}) and for Max 3-SAT with mu(c)=1/O(c^{1/2}) that run super-polynomially faster than 2^n when m=o(n^{5/2}/log^{5/2}(n)) and m=o(n^3/log^2(n)) respectively.
maximum satisfiability
weighted
polynomial space
exponential space
90-101
Regular Paper
Takayuki
Sakai
Takayuki Sakai
Kazuhisa
Seto
Kazuhisa Seto
Suguru
Tamaki
Suguru Tamaki
Junichi
Teruyama
Junichi Teruyama
10.4230/LIPIcs.IPEC.2015.90
Nikhil Bansal and Venkatesh Raman. Upper bounds for MaxSAT: Further improved. In Proceedings of the 10th International Symposium on Algorithms and Computation (ISAAC), volume 1741 of Lecture Notes in Computer Science, pages 247-258. Springer, 1999.
Daniel Binkele-Raible and Henning Fernau. A new upper bound for Max-2-SAT: A graph-theoretic approach. J. Discrete Algorithms, 8(4):388-401, 2010.
Ivan Bliznets and Alexander Golovnev. A new algorithm for parameterized MAX-SAT. In Proceedings of the 7th International Symposium on Parameterized and Exact Computation (IPEC), volume 7535 of Lecture Notes in Computer Science, pages 37-48. Springer, 2012.
Chris Calabro, Russell Impagliazzo, and Ramamohan Paturi. A duality between clause width and clause density for SAT. In Proceedings of the 21st Annual IEEE Conference on Computational Complexity (CCC), pages 252-260, 2006.
Jianer Chen and Iyad A. Kanj. Improved exact algorithms for MAX-SAT. Discrete Applied Mathematics, 142(1-3):17-27, 2004.
Ruiwen Chen, Valentine Kabanets, Antonina Kolokolova, Ronen Shaltiel, and David Zuckerman. Mining circuit lower bound proofs for meta-algorithms. Computational Complexity, 24(2):333-392, 2015.
Ruiwen Chen and Rahul Santhanam. Improved algorithms for sparse max-sat and max-k-csp. In Proceedings of the 18th International Conference on Theory and Applications of Satisfiability Testing (SAT), 2015. To appear.
Evgeny Dantsin and Edward A. Hirsch. Worst-case upper bounds. In Handbook of Satisfiability, volume 185 of Frontiers in Artificial Intelligence and Applications, pages 403-424. IOS Press, 2009.
Evgeny Dantsin and Alexander Wolpert. MAX-SAT for formulas with constant clause density can be solved faster than in O(2ⁿ) time. In Proceedings of the 9th International Conference on Theory and Applications of Satisfiability Testing (SAT), volume 4121 of Lecture Notes in Computer Science, pages 266-276. Springer, 2006.
Serge Gaspers and Gregory B. Sorkin. A universally fastest algorithm for Max 2-SAT, Max 2-CSP, and everything in between. J. Comput. Syst. Sci., 78(1):305-335, 2012.
Serge Gaspers and Gregory B. Sorkin. Separate, measure and conquer: Faster polynomial-space algorithms for Max 2-CSP and counting dominating sets. In Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming (ICALP), Part I, pages 567-579, 2015.
Alexander Golovnev and Konstantin Kutzkov. New exact algorithms for the 2-constraint satisfaction problem. Theor. Comput. Sci., 526:18-27, 2014.
Jens Gramm, Edward A. Hirsch, Rolf Niedermeier, and Peter Rossmanith. Worst-case upper bounds for MAX-2-SAT with an application to MAX-CUT. Discrete Applied Mathematics, 130(2):139-155, 2003.
Jens Gramm and Rolf Niedermeier. Faster exact solutions for MAX2SAT. In Proceedings of the 4th Italian Conference on Algorithms and Complexity (CIAC), volume 1767 of Lecture Notes in Computer Science, pages 174-186. Springer, 2000.
Gregory Gutin and Anders Yeo. Constraint satisfaction problems parameterized above or below tight bounds: A survey. In The Multivariate Algorithmic Revolution and Beyond, volume 7370 of Lecture Notes in Computer Science, pages 257-286. Springer, 2012.
Edward A. Hirsch. A new algorithm for MAX-2-SAT. In Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science (STACS), volume 1770 of Lecture Notes in Computer Science, pages 65-73. Springer, 2000.
Edward A. Hirsch. Worst-case study of local search for MAX-k-SAT. Discrete Applied Mathematics, 130(2):173-184, 2003.
Russell Impagliazzo, Ramamohan Paturi, and Stefan Schneider. A satisfiability algorithm for sparse depth two threshold circuits. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 479-488, 2013.
Joachim Kneis, Daniel Mölle, Stefan Richter, and Peter Rossmanith. On the parameterized complexity of exact satisfiability problems. In Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science (MFCS), volume 3618 of Lecture Notes in Computer Science, pages 568-579. Springer, 2005.
Joachim Kneis, Daniel Mölle, Stefan Richter, and Peter Rossmanith. A bound on the pathwidth of sparse graphs with applications to exact algorithms. SIAM J. Discrete Math., 23(1):407-427, 2009.
Mikko Koivisto. Optimal 2-constraint satisfaction via sum-product algorithms. Inf. Process. Lett., 98(1):24-28, 2006.
Arist Kojevnikov and Alexander S. Kulikov. A new approach to proving upper bounds for MAX-2-SAT. In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 11-17, 2006.
Alexander S. Kulikov. Automated generation of simplification rules for SAT and MAXSAT. In Proceedings of the 8th International Conference on Theory and Applications of Satisfiability Testing (SAT), volume 3569 of Lecture Notes in Computer Science, pages 430-436. Springer, 2005.
Alexander S. Kulikov and Konstantin Kutzkov. New bounds for MAX-SAT by clause learning. In Proceedings of the 8th International Computer Science Symposium in Russia (CSR), volume 4649 of Lecture Notes in Computer Science, pages 194-204. Springer, 2007.
François Le Gall. Powers of tensors and fast matrix multiplication. In Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation (ISSAC), pages 296-303, 2014.
Meena Mahajan and Venkatesh Raman. Parameterizing above guaranteed values: MaxSAT and MaxCUT. J. Algorithms, 31(2):335-354, 1999.
Rolf Niedermeier and Peter Rossmanith. New upper bounds for maximum satisfiability. J. Algorithms, 36(1):63-88, 2000.
Takayuki Sakai, Kazuhisa Seto, and Suguru Tamaki. Solving sparse instances of max SAT via width reduction and greedy restriction. Theory Comput. Syst., 57(2):426-443, 2015.
Takayuki Sakai, Kazuhisa Seto, Suguru Tamaki, and Junichi Teruyama. A satisfiability algorithm for depth-2 circuits with a symmetric gate at the top and AND gates at the bottom. Electronic Colloquium on Computational Complexity (ECCC), TR15-136, 2015.
Rahul Santhanam. Fighting perebor: New and improved algorithms for formula and QBF satisfiability. In Proceedings of the 51th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 183-192, 2010.
Rainer Schuler. An algorithm for the satisfiability problem of formulas in conjunctive normal form. J. Algorithms, 54(1):40-44, 2005.
Alex D. Scott and Gregory B. Sorkin. Faster algorithms for MAX CUT and MAX CSP, with polynomial expected time for sparse instances. In Proceedings of the 6th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX) and the 7th International Workshop on Randomization and Computation (RANDOM), volume 2764 of Lecture Notes in Computer Science, pages 382-395. Springer, 2003.
Alexander D. Scott and Gregory B. Sorkin. Linear-programming design and analysis of fast algorithms for Max 2-CSP. Discrete Optimization, 4(3-4):260-287, 2007.
Kazuhisa Seto and Suguru Tamaki. A satisfiability algorithm and average-case hardness for formulas over the full binary basis. Computational Complexity, 22(2):245-274, 2013.
Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci., 348(2-3):357-365, 2005.
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Polynomial Fixed-parameter Algorithms: A Case Study for Longest Path on Interval Graphs
We study the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time.
The main motivation is to get more efficient algorithms for problems with unattractive polynomial running times. Here, we focus on a fundamental graph problem: Longest Path; it is NP-hard in general but known to be solvable in O(n^4) time on n-vertex interval graphs. We show how to solve Longest Path on Interval Graphs, parameterized by vertex deletion number k to proper interval graphs, in O(k^9n) time. Notably, Longest Path is trivially solvable in linear time on proper interval graphs, and the parameter value k can be approximated up to a factor of 4 in linear time. From a more general perspective, we believe that using parameterized complexity analysis for polynomial-time solvable problems offers a very fertile ground for future studies for all sorts of algorithmic problems. It may enable a refined understanding of efficiency aspects for polynomial-time solvable problems, similarly to what classical parameterized complexity analysis does for NP-hard problems.
fixed-parameter algorithm
preprocessing
data reduction
polynomial-time algorithm
longest path problem
interval graphs
proper interval vertex del
102-113
Regular Paper
Archontia C.
Giannopoulou
Archontia C. Giannopoulou
George B.
Mertzios
George B. Mertzios
Rolf
Niedermeier
Rolf Niedermeier
10.4230/LIPIcs.IPEC.2015.102
Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1681-1697, 2015.
Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In Proceedings of the 55th IEEE Symposium on Foundations of Computer Science (FOCS), pages 434-443, 2014.
Amir Abboud, Virginia Vassilevska Williams, and Joshua Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), 2016. To appear.
Ittai Abraham, Amos Fiat, Andrew V. Goldberg, and Renato Fonseca F. Werneck. Highway dimension, shortest paths, and provably efficient algorithms. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 782-793, 2010.
Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. J. of ACM, 42(4):844-856, 1995.
Amihood Amir, Moshe Lewenstein, and Ely Porat. Faster algorithms for string matching with k mismatches. Journal of Algorithms, 50(2):257-275, 2004.
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Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. on Comp., 25(6):1305-1317, 1996.
Karl Bringmann. Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless SETH fails. In Proceedings of the 55th IEEE Symp. on Foundations of Computer Science (FOCS), pages 661-670, 2014.
Sergio Cabello and Christian Knauer. Algorithms for graphs of bounded treewidth via orthogonal range searching. Computational Geometry, 42(9):815-824, 2009.
Leizhen Cai. Fixed-parameter tractability of graph modification problems for hereditary properties. Information Processing Letters, 58(4):171-176, 1996.
Erin W. Chambers, Jeff Erickson, and Amir Nayyeri. Homology flows, cohomology cuts. SIAM J. on Comput., 41(6):1605-1634, 2012.
Jianer Chen, Songjian Lu, Sing-Hoi Sze, and Fenghui Zhang. Improved algorithms for path, matching, and packing problems. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 298-307, 2007.
Maria Chudnovsky, Gérard Cornuéjols, Xinming Liu, Paul Seymour, and Kristina Vušković. Recognizing Berge graphs. Combinatorica, 25(2):143-186, 2005.
Peter Damaschke. Paths in interval graphs and circular arc graphs. Discrete Mathematics, 112(1-3):49-64, 1993.
Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013.
Michael R. Fellows, Bart M. P. Jansen, and Frances A. Rosamond. Towards fully multivariate algorithmics: Parameter ecology and the deconstruction of computational complexity. European Journal of Combinatorics, 34(3):541-566, 2013.
Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Springer, 2006.
Anka Gajentaan and Mark H. Overmars. On a class of O(n²) problems in computational geometry. Computational Geometry, 5:165-185, 1995.
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), pages 162-173, 2004.
Torben Hagerup, Jyrki Katajainen, Naomi Nishimura, and Prabhakar Ragde. Characterizing multiterminal flow networks and computing flows in networks of small treewidth. Journal of Computer and System Sciences, 57(3):366-375, 1998.
Jan M. Hochstein and Karsten Weihe. Maximum s-t-flow with k crossings in O(k³ n log n) time. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 843-847, 2007.
K. Ioannidou, George B. Mertzios, and Stavros D. Nikolopoulos. The longest path problem has a polynomial solution on interval graphs. Algorithmica, 61(2):320-341, 2011.
Ken-ichi Kawarabayashi and Bruce A. Reed. Computing crossing number in linear time. In Proceedings of the 39th ACM Symp. on Th. of Comp. (STOC), pages 382-390, 2007.
J. M. Keil. Finding Hamiltonian circuits in interval graphs. Information Processing Letters, 20:201-206, 1985.
Joachim Kneis, Daniel Mölle, Stefan Richter, and Peter Rossmanith. Divide-and-color. In Proceedings of the 32nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG), pages 58-67, 2006.
Ioannis Koutis, Gary L. Miller, and Richard Peng. A nearly-m log n time solver for SDD linear systems. In Proceedings of the IEEE 52nd Symposium on Foundations of Computer Science (FOCS), pages 590-598, 2011.
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George B. Mertzios and Derek G. Corneil. A simple polynomial algorithm for the longest path problem on cocomparability graphs. SIAM J. on Discr. Math., 26(3):940-963, 2012.
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James B. Orlin, Kamesh Madduri, K. Subramani, and Matthew D. Williamson. A faster algorithm for the single source shortest path problem with few distinct positive lengths. Journal of Discrete Algorithms, 8(2):189-198, 2010.
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René van Bevern. Fixed-Parameter Linear-Time Algorithms for NP-hard Graph and Hypergraph Problems Arising in Industrial Applications. PhD thesis, Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Germany, 2014.
Ryan Williams. Finding paths of length k in O^*(2^k) time. Information Processing Letters, 109(6):315-318, 2009.
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Meta-kernelization using Well-structured Modulators
Kernelization investigates exact preprocessing algorithms with performance guarantees. The most prevalent type of parameters used in kernelization is the solution size for optimization problems; however, also structural parameters have been successfully used to obtain polynomial kernels for a wide range of problems. Many of these parameters can be defined as the size of a smallest modulator of the given graph into a fixed graph class (i.e., a set of vertices whose deletion puts the graph into the graph class). Such parameters admit the construction of polynomial kernels even when the solution size is large or not applicable. This work follows up on the research on meta-kernelization frameworks in terms of structural parameters.
We develop a class of parameters which are based on a more general view on modulators: instead of size, the parameters employ a combination of rank-width and split decompositions to measure structure inside the modulator. This allows us to lift kernelization results from modulator-size to more general parameters, hence providing smaller kernels. We show (i) how such large but well-structured modulators can be efficiently approximated, (ii) how they can be used to obtain polynomial kernels for any graph problem expressible in Monadic Second Order logic, and (iii) how they allow the extension of previous results in the area of structural meta-kernelization.
Kernelization
Parameterized complexity
Structural parameters
Rank-width
Split decompositions
114-126
Regular Paper
Eduard
Eiben
Eduard Eiben
Robert
Ganian
Robert Ganian
Stefan
Szeider
Stefan Szeider
10.4230/LIPIcs.IPEC.2015.114
Noga Alon, Gregory Gutin, Eun Jung Kim, Stefan Szeider, and Anders Yeo. Solving max-r-sat above a tight lower bound. Algorithmica, 61(3):638-655, 2011.
Stefan Arnborg, Bruno Courcelle, Andrzej Proskurowski, and Detlef Seese. An algebraic theory of graph reduction. J. of the ACM, 40(5):1134-1164, 1993.
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. In FOCS 2009, pages 629-638. IEEE Computer Society, 2009.
Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernel bounds for path and cycle problems. Theor. Comput. Sci., 511:117-136, 2013.
Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Preprocessing for treewidth: A combinatorial analysis through kernelization. SIAM J. Discrete Math., 27(4):2108-2142, 2013.
Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM J. Discrete Math., 28(1):277-305, 2014.
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. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer Verlag, 2013.
Eduard Eiben, Robert Ganian, and Stefan Szeider. Solving problems on graphs of high rank-width. In Algorithms and Data Structures - 14th International Symposium, volume 9214 of Lecture Notes in Computer Science, pages 314-326. Springer, 2015.
Fedor V. Fomin, Bart M. P. Jansen, and Michal Pilipczuk. Preprocessing subgraph and minor problems: When does a small vertex cover help? J. Comput. Syst. Sci., 80(2):468-495, 2014.
Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Bidimensionality and kernels. In SODA, pages 503-510, 2010.
Jakub Gajarský, Petr Hliněný, Jan Obdržálek, Sebastian Ordyniak, Felix Reidl, Peter Rossmanith, Fernando Sanchez Villaamil, and Somnath Sikdar. Kernelization using structural parameters on sparse graph classes. In ESA 2013, volume 8125 of Lecture Notes in Computer Science, pages 529-540. Springer, 2013.
Robert Ganian and Petr Hliněný. On parse trees and Myhill-Nerode-type tools for handling graphs of bounded rank-width. Discr. Appl. Math., 158(7):851-867, 2010.
Robert Ganian, Friedrich Slivovsky, and Stefan Szeider. Meta-kernelization with structural parameters. In MFCS, pages 457-468, 2013.
Petr Hliněný and Sang il Oum. Finding branch-decompositions and rank-decompositions. SIAM J. Comput., 38(3):1012-1032, 2008.
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.
Eun Jung Kim, Alexander Langer, Christophe Paul, Felix Reidl, Peter Rossmanith, Ignasi Sau, and Somnath Sikdar. Linear kernels and single-exponential algorithms via protrusion decompositions. In ICALP (1), pages 613-624, 2013.
Leonid Libkin. Elements of Finite Model Theory. Springer, 2004.
Jaroslav Nesetril and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and Combinatorics. Springer, 2012.
Sang-il Oum and P. Seymour. Approximating clique-width and branch-width. J. Combin. Theory Ser. B, 96(4):514-528, 2006.
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Parameter Compilation
In resolving instances of a computational problem, if multiple instances of interest share a feature in common,it may be fruitful to compile this feature into a format that allows for more efficient resolution, even if the compilation is relatively expensive. In this article, we introduce a formal framework for classifying problems according to their compilability. The basic object in our framework is that of a parameterized problem, which here is a language along with a parameterization—a map which provides, for each instance, a so-called parameter on which compilation may be performed. Our framework is positioned within the paradigm of parameterized complexity, and our notions are relatable to established concepts in the theory of parameterized complexity. Indeed, we view our framework as playing a unifying role, integrating together parameterized complexity and compilability theory.
compilation
parameterized complexity
127-137
Regular Paper
Hubie
Chen
Hubie Chen
10.4230/LIPIcs.IPEC.2015.127
Simone Bova, Florent Capelli, Stefan Mengel, and Friedrich Slivovsky. Expander cnfs have exponential DNNF size. CoRR, abs/1411.1995, 2014.
Marco Cadoli, Francesco M. Donini, Paolo Liberatore, and Marco Schaerf. Preprocessing of intractable problems. Information and Computation, 176(2):89-120, 2002.
Hubie Chen. A theory of average-case compilability in knowledge representation. In IJCAI-03, Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence, Acapulco, Mexico, August 9-15, 2003, pages 455-460, 2003.
Hubie Chen. Parameterized compilability. In IJCAI-05, Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence, Edinburgh, Scotland, UK, July 30-August 5, 2005, pages 412-417, 2005.
Hubie Chen. On the complexity of existential positive queries. ACM Trans. Comput. Log., 15(1), 2014.
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Wenfei Fan, Floris Geerts, and Frank Neven. Making queries tractable on big data with preprocessing. PVLDB, 6(9):685-696, 2013.
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An FPT Algorithm and a Polynomial Kernel for Linear Rankwidth-1 Vertex Deletion
Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B, 96(4):514-528, 2006.], and it is similar to pathwidth, which is the linearized variant of treewidth. Motivated from the results on graph modification problems into graphs of bounded treewidth or pathwidth, we investigate a graph modification problem into the class of graphs having linear rankwidth at most one, called the Linear Rankwidth-1 Vertex Deletion (shortly, LRW1-Vertex Deletion). In this problem, given an n-vertex graph G and a positive integer k, we want to decide whether there is a set of at most k vertices whose removal turns G into a graph of linear rankwidth at most one and if one exists, find such a vertex set. While the meta-theorem of Courcelle, Makowsky, and Rotics implies thatLRW1-Vertex Deletion can be solved in time f(k) * n^3 for some function f, it is not clear whether this problem allows a runtime with a modest exponential function. We establish that LRW1-Vertex Deletion can be solved in time 8^k * n^{O(1)}. The major obstacle to this end is how to handle a long induced cycle as an obstruction. To fix this issue, we define the necklace graphs and investigate their structural properties.
We also show that the LRW1-Vertex Deletion has a polynomial kernel.
(linear) rankwidth
distance-hereditary graphs
thread graphs
parameterized complexity
kernelization
138-150
Regular Paper
Mamadou Moustapha
Kanté
Mamadou Moustapha Kanté
Eun Jung
Kim
Eun Jung Kim
O-joung
Kwon
O-joung Kwon
Christophe
Paul
Christophe Paul
10.4230/LIPIcs.IPEC.2015.138
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Extending the Kernel for Planar Steiner Tree to the Number of Steiner Vertices
In the Steiner Tree problem one is given an undirected graph, a subset T of its vertices, and an integer k and the question is whether there is a connected subgraph of the given graph containing all the vertices of T and at most k other vertices. The vertices in the subset T are called terminals and the other vertices are called Steiner vertices. Recently, Pilipczuk, Pilipczuk, Sankowski, and van Leeuwen [FOCS 2014] gave a polynomial kernel for Steiner Tree in planar graphs, when parameterized by |T|+k, the total number of vertices in the constructed subgraph.
In this paper we present several polynomial time applicable reduction rules for Planar Steiner Tree. In an instance reduced with respect to the presented reduction rules, the number of terminals |T| is at most quadratic in the number of other vertices k in the subgraph. Hence, using and improving the result of Pilipczuk et al., we give a polynomial kernel for Steiner Tree in planar graphs for the parameterization by the number k of Steiner vertices in the solution.
Steiner Tree
polynomial kernel
planar graphs
polynomial-time preprocessing
network sparsification
151-162
Regular Paper
Ondrj
Suchý
Ondrj Suchý
10.4230/LIPIcs.IPEC.2015.151
J. Alber, M. R. Fellows, and R. Niedermeier. Polynomial-time data reduction for dominating set. J. ACM, 51(3):363-384, May 2004.
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R. van BevernBevern, S. Hartung, F. Kammer, R. Niedermeier, and M. Weller. Linear-time computation of a linear problem kernel for dominating set on planar graphs. In Parameterized and Exact Computation - 6th International Symposium, IPEC 2011, Revised Selected Papers, volume 7112 of LNCS, pages 194-206. Springer, 2011.
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H. L. Bodlaender, M. Cygan, S. Kratsch, and J. Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput., 243:86-111, 2015.
H. L. Bodlaender, R. G. Downey, M. R. Fellows, and D. Hermelin. On problems without polynomial kernels. J. Comput. Syst. Sci., 75(8):423-434, 2009.
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M. Cygan, J. Nederlof, M. Pilipczuk, M. Pilipczuk, J. M. M. van Rooij, and J. O. Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, pages 150-159. IEEE Computer Society, 2011.
M. Dom, D. Lokshtanov, and S. Saurabh. Kernelization lower bounds through colors and IDs. ACM Transactions on Algorithms, 11(2):13:1-13:20, 2014.
R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer-Verlag, New York, 1999.
S. E. Dreyfus and R. A. Wagner. The Steiner problem in graphs. Networks, 1:195-207, 1972.
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F. V. Fomin, F. Grandoni, D. Kratsch, D. Lokshtanov, and S. Saurabh. Computing optimal Steiner trees in polynomial space. Algorithmica, 65(3):584-604, 2013.
F. V. Fomin, P. Kaski, D. Lokshtanov, F. Panolan, and S. Saurabh. Parameterized single-exponential time polynomial space algorithm for Steiner tree. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Proceedings, Part I, volume 9134 of LNCS, pages 494-505. Springer, 2015.
F. V. Fomin, D. Lokshtanov, and S. 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 2014, pages 142-151. SIAM, 2014.
D. S. R. Frank K. Hwang and P. Winter. The Steiner Tree Problem, volume 53 of Annals of Discrete Mathematics. Elsevier, 1992.
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M. Hauptmann and M. Karpinski. A compendium on Steiner tree problems. online. URL: http://theory.cs.uni-bonn.de/info5/steinerkompendium/.
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M. Jones, D. Lokshtanov, M. S. Ramanujan, S. Saurabh, and O. Suchý. Parameterized complexity of directed Steiner tree on sparse graphs. In Algorithms - ESA 2013 - 21st Annual European Symposium, Proceedings, volume 8125 of LNCS, pages 671-682. Springer, 2013.
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J. Nederlof. Fast polynomial-space algorithms using inclusion-exclusion. Algorithmica, 65(4):868-884, 2013.
M. Patrascu and R. Williams. On the possibility of faster SAT algorithms. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pages 1065-1075. SIAM, 2010.
M. Pilipczuk, M. Pilipczuk, P. Sankowski, and E. J. van Leeuwen. Subexponential-time parameterized algorithm for Steiner tree on planar graphs. In 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, volume 20 of LIPIcs, pages 353-364. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2013.
M. Pilipczuk, M. Pilipczuk, P. Sankowski, and E. J. van Leeuwen. Network sparsification for Steiner problems on planar and bounded-genus graphs. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, pages 276-285. IEEE Computer Society, 2014.
H. J. Prömel and A. Steger. The Steiner Tree Problem; a Tour through Graphs, Algorithms, and Complexity. Vieweg, 2002.
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Sparsification Upper and Lower Bounds for Graphs Problems and Not-All-Equal SAT
We present several sparsification lower and upper bounds for classic problems in graph theory and logic. For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time algorithm that reduces any n-vertex input to an equivalent instance, of an arbitrary problem, with bitsize O(n^{2-epsilon}) for epsilon > 0, unless NP is a subset of coNP/poly and the polynomial-time hierarchy collapses. These results imply that existing linear-vertex kernels for k-Nonblocker and k-Max Leaf Spanning Tree (the parametric duals of (Connected) Dominating Set) cannot be improved to have O(k^{2-epsilon}) edges, unless NP is a subset of NP/poly. We also present a positive result and exhibit a non-trivial sparsification algorithm for d-Not-All-Equal-SAT. We give an algorithm that reduces an n-variable input with clauses of size at most d to an equivalent input with O(n^{d-1}) clauses, for any fixed d. Our algorithm is based on a linear-algebraic proof of Lovász that bounds the number of hyperedges in critically 3-chromatic d-uniform n-vertex hypergraphs by binom{n}{d-1}. We show that our kernel is tight under the assumption that NP is not a subset of NP/poly.
sparsification
graph coloring
Hamiltonian cycle
satisfiability
163-174
Regular Paper
Bart M. P.
Jansen
Bart M. P. Jansen
Astrid
Pieterse
Astrid Pieterse
10.4230/LIPIcs.IPEC.2015.163
Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. J. Comput. Syst. Sci., 75(8):423-434, 2009.
Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernel bounds for path and cycle problems. Theor. Comput. Sci., 511:117-136, 2013.
Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM J. Discrete Math., 28(1):277-305, 2014.
Hans L. Bodlaender, Stéphan Thomassé, and Anders Yeo. Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci., 412(35):4570-4578, 2011.
Marek Cygan, Fabrizio Grandoni, and Danny Hermelin. Tight kernel bounds for problems on graphs with small degeneracy. In Proc. 21st ESA, pages 361-372, 2013.
Frank K. H. A. Dehne, Michael R. Fellows, Henning Fernau, Elena Prieto, and Frances A. Rosamond. NONBLOCKER: parameterized algorithmics for minimum dominating set. In Proc. 32nd SOFSEM, pages 237-245, 2006.
Holger Dell and Dániel Marx. Kernelization of packing problems. In Proc. 23rd SODA, pages 68-81, 2012.
Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM, 61(4):23:1-23:27, 2014.
Michael Dom, Daniel Lokshtanov, and Saket Saurabh. Kernelization lower bounds through colors and IDs. ACM Trans. Algorithms, 11(2):13:1-13:20, 2014.
David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Amnon Nissenzweig. Sparsification - a technique for speeding up dynamic graph algorithms. J. ACM, 44(5):669-696, 1997.
Vladimir Estivill-Castro, Michael Fellows, Michael Langston, and Frances Rosamond. FPT is P-time extremal structure I. In Proc. 1st ACiD, pages 1-41, 2005.
Michael R. Fellows and Bart M. P. Jansen. FPT is characterized by useful obstruction sets: Connecting algorithms, kernels, and quasi-orders. TOCT, 6(4):16, 2014.
Lance Fortnow and Rahul Santhanam. Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci., 77(1):91-106, 2011.
Michael R. Garey and David S. Johnson. Computers and Intractability. W.H. Freeman, 1979.
Danny Hermelin and Xi Wu. Weak compositions and their applications to polynomial lower bounds for kernelization. In Proc. 23rd SODA, pages 104-113, 2012.
Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001.
Bart M. P. Jansen. On sparsification for computing treewidth. Algorithmica, 71(3):605-635, 2015.
Bart M. P. Jansen and Stefan Kratsch. Data reduction for graph coloring problems. Information and Computation, 231:70-88, 2013.
Bart M. P. Jansen and Astrid Pieterse. Sparsification upper and lower bounds for graphs problems and not-all-equal SAT. CoRR, abs/1509.07437, 2015.
Richard M. Karp. Reducibility Among Combinatorial Problems. In Complexity of Computer Computations, pages 85-103. Plenum Press, 1972.
Lásló Lovász. Chromatic number of hypergraphs and linear algebra. In Studia Scientiarum Mathematicarum Hungarica 11, pages 113-114, 1976.
George L. Nemhauser and Leslie E. Trotter Jr. Vertex packings: structural properties and algorithms. Math. Program., 8:232-248, 1975.
Bjarne Toft. On the maximal number of edges of critical k-chromatic graphs. Studia Scientiarum Mathematicarum Hungarica, 5:461-470, 1970.
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Definability Equals Recognizability for k-Outerplanar Graphs
One of the most famous algorithmic meta-theorems states that every graph property that can be defined by a sentence in counting monadic second order logic (CMSOL) can be checked in linear time for graphs of bounded treewidth, which is known as Courcelle's Theorem. These algorithms are constructed as finite state tree automata, and hence every CMSOL-definable graph property is recognizable. Courcelle also conjectured that the converse holds, i.e., every recognizable graph property is definable in CMSOL for graphs of bounded treewidth. We prove this conjecture for k-outerplanar graphs, which are known to have treewidth at most 3k-1.
treewidth
monadic second order logic of graphs
finite state tree automata
$k$-outerplanar graphs
175-186
Regular Paper
Lars
Jaffke
Lars Jaffke
Hans L.
Bodlaender
Hans L. Bodlaender
10.4230/LIPIcs.IPEC.2015.175
Stefan Arnborg, Jens Lagergren, and Detlef Seese. Easy problems for tree-decomposable graphs. Journal of Algorithms, 12(2):308-340, 1991.
Hans L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science, 209(1-2):1-45, 1998.
Hans L. Bodlaender, Pinar Heggernes, and Jan Arne Telle. Recognizability equals definability for graphs of bounded treewidth and bounded chordality. In Proceedings EUROCOMB 2015, Electronic Notes in Discrete Mathematics. Elsevier, 2015.
Richard B. Borie, R. Gary Parker, and Craig A. Tovey. Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families. Algorithmica, 7(1-6):555-581, 1992.
J. Richard Büchi. Weak second-order arithmetic and finite automata. Mathematical Logic Quarterly, 6(1-6):66-92, 1960.
Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Information and Computation, 85(1):12-75, 1990.
Bruno 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.
Bruno Courcelle. The monadic second order logic of graphs VI: On several representations of graphs by relational structures. Discrete Applied Mathematics, 54(2-3):117-149, 1994.
Bruno Courcelle. The monadic second-order logic of graphs VIII: Orientations. Annals of Pure and Applied Logic, 72(2):103-143, 1995.
Bruno Courcelle. The monadic second-order logic of graphs XI: Hierarchical decompositions of connected graphs. Theoretical Computer Science, 224(1-2):38-53, 1999.
Bruno Courcelle. The monadic second-order logic of graphs XII: Planar graphs and planar maps. Theoretical Computer Science, 237(1-2):1-32, 2000.
Reinhard Diestel. Graph Theory. Number 173 in Graduate Texts in Mathematics. Springer, 4th edition, 2012. Corrected reprint.
Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013.
Lars Jaffke and Hans L. Bodlaender. Definability equals recognizability for k-outerplanar graphs. ArXiv e-prints, 2015. http://arxiv.org/abs/1509.08315.
Valentine Kabanets. Recognizability equals definability for partial k-paths. In Proceedings ICALP 1997, volume 1256 of LNCS, pages 805-815. Springer, 1997.
Damon Kaller. Definability equals recognizability of partial 3-trees and k-connected partial k-trees. Algorithmica, 27(3-4):348-381, 2000.
Ioannis Katsikarelis. Computing bounded-width tree and branch decompositions of k-outerplanar graphs. ArXiv e-prints, 2013. http://arxiv.org/abs/1301.5896.
Denis Lapoire. Recognizability equals monadic second-order definability for sets of graphs of bounded tree-width. In Proceedings STACS 1998, volume 1373 of LNCS, pages 618-628. Springer, 1998.
William T. Tutte. Connectivity in Graphs. University of Toronto Press, 1966.
William T. Tutte. Graph Theory, volume 21 of Encyclopedia of Mathematics and its Applications. Addison-Wesley, 1984.
Hassler Whitney. Congruent graphs and the connectivity of graphs. American Journal of Mathematics, 54:150-168, 1932.
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Practical Algorithms for Linear Boolean-width
In this paper, we give a number of new exact algorithms and heuristics to compute linear boolean decompositions, and experimentally evaluate these algorithms. The experimental evaluation shows that significant improvements can be made with respect to running time without increasing the width of the generated decompositions. We also evaluated dynamic programming algorithms on linear boolean decompositions for several vertex subset problems. This evaluation shows that such algorithms are often much faster (up to several orders of magnitude) compared to theoretical worst case bounds.
graph decomposition
boolean-width
heuristics
exact algorithms
vertex subset problems
187-198
Regular Paper
Chiel B.
ten Brinke
Chiel B. ten Brinke
Frank J. P.
van Houten
Frank J. P. van Houten
Hans L.
Bodlaender
Hans L. Bodlaender
10.4230/LIPIcs.IPEC.2015.187
R. Belmonte and M. Vatshelle. Graph classes with structured neighborhoods and algorithmic applications. Theoretical Computer Science, 511:54-65, 2013. Exact and Parameterized Computation.
B.-M. Bui-Xuan, J. A. Telle, and M. Vatshelle. Boolean-width of graphs. In IWPEC 2009, volume 5917 of LNCS, pages 61-74. Springer, 2009.
B.-M. Bui-Xuan, J. A. Telle, and M. Vatshelle. Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems. Theoretical Computer Science, 511:66-76, 2013.
P. Erdös and A. Rényi. On random graphs. Publicationes Mathematicae 6: 290–297, 1959.
E. M. Hvidevold, S. Sharmin, J. A. Telle, and M. Vatshelle. Finding good decompositions for dynamic programming on dense graphs. In IWPEC 2012, volume 7112 of LNCS, pages 219-231. Springer, 2012.
K. H. Kim. Boolean Matrix Theory and its Applications. Marcel Dekker, 1982.
F. Manne and S. Sharmin. Efficient counting of maximal independent sets in sparse graphs. In Experimental Algorithms, volume 7933 of LNCS, pages 103-114. Springer, 2013.
Y. Rabinovich, J. A. Telle, and M. Vatshelle. Upper bounds on boolean-width with applications to exact algorithms. In IWPEC 2013, volume 8246 of LNCS, pages 308-320. Springer, 2013.
N. Robertson and P. D. Seymour. Graph minors. I. Excluding a forest. Journal of Combinatorial Theory, Series B, 35(1):39-61, 1983.
S. Sharmin. Practical Aspects of the Graph Parameter Boolean-width. PhD thesis, University of Bergen, Norway, 2014.
J. A. Telle. Complexity of domination-type problems in graphs. Nordic Journal of Computing, 1(1):157-171, 1994.
Ch. B. Ten Brinke, F. J. P. van Houten, and H. L. Bodlaender. Practical Algorithms for Linear Boolean-width. ArXiv e-prints ArXiV:1509.07687, 2015.
Treewidthlib. http://www.staff.science.uu.nl/∼bodla101/treewidthlib/. A benchmark for algorithms for treewidth and related graph problems.
F. J. P. van Houten. Experimental research and algorithmic improvements involving the graph parameter boolean-width. Master’s thesis, Utrecht University, The Netherlands, 2015.
J. M. M. van Rooij, H. L. Bodlaender, and P. Rossmanith. Dynamic programming on tree decompositions using generalised fast subset convolution. In Algorithms - ESA 2009, volume 5757 of LNCS, pages 566-577. Springer, 2009.
M. Vatshelle. New Width Parameters of Graphs. PhD thesis, University of Bergen, Norway, 2012.
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Linear Kernels for Outbranching Problems in Sparse Digraphs
In the k-Leaf Out-Branching and k-Internal Out-Branching problems we are given a directed graph D with a designated root r and a nonnegative integer k. The question is to determine the existence of an outbranching rooted at r that has at least k leaves, or at least k internal vertices, respectively. Both these problems were intensively studied from the points of view of parameterized complexity and kernelization, and in particular for both of them kernels with O(k^2) vertices are known on general graphs. In this work we show that k-Leaf Out-Branching admits a kernel with O(k) vertices on H-minor-free graphs, for any fixed H, whereas k-Internal Out-Branching admits a kernel with O(k) vertices on any graph class of bounded expansion.
FPT algorithm
kernelization
outbranching
sparse graphs
199-211
Regular Paper
Marthe
Bonamy
Marthe Bonamy
Lukasz
Kowalik
Lukasz Kowalik
Michal
Pilipczuk
Michal Pilipczuk
Arkadiusz
Socala
Arkadiusz Socala
10.4230/LIPIcs.IPEC.2015.199
Marthe Bonamy, Łukasz Kowalik, Michał Pilipczuk, and Arkadiusz Socała. Linear kernels for outbranching problems in sparse digraphs. CoRR, abs/1509.01675, 2015.
Jean Daligault and Stéphan Thomassé. On finding directed trees with many leaves. In Parameterized and Exact Computation, pages 86-97. Springer, 2009.
Erik D. Demaine, MohammadTaghi Hajiaghayi, and Ken-ichi Kawarabayashi. Contraction decomposition in H-minor-free graphs and algorithmic applications. In Proc. STOC'11, pages 441-450. ACM, 2011.
Frederic Dorn, Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Beyond bidimensionality: Parameterized subexponential algorithms on directed graphs. Inf. Comput., 233:60-70, 2013.
Pål Grønås Drange, Markus S. Dregi, Fedor V. Fomin, Stephan Kreutzer, Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, Felix Reidl, Saket Saurabh, Fernando Sánchez Villaamil, and Somnath Sikdar. Kernelization and sparseness: the case of dominating set. CoRR, abs/1411.4575, 2014.
Mike Fellows, Pinar Heggernes, Frances A. Rosamond, Christian Sloper, and Jan Arne Telle. Finding k disjoint triangles in an arbitrary graph. In WG'04, pages 235-244, 2004.
Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Bidimensionality and kernels. In Proc. SODA'10, pages 503-510, 2010.
Jakub Gajarský, Petr Hlinený, Jan Obdrzálek, Sebastian Ordyniak, Felix Reidl, Peter Rossmanith, Fernando Sanchez Villaamil, and Somnath Sikdar. Kernelization using structural parameters on sparse graph classes. In ESA 2013, pages 529-540. Springer, 2013.
Gregory Gutin, Igor Razgon, and Eun Jung Kim. Minimum leaf out-branching and related problems. Theor. Comput. Sci., 410(45):4571-4579, 2009.
Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity: Graphs, Structures, and Algorithms, volume 28 of Algorithms and Combinatorics. Springer, 2012.
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Maximum Matching Width: New Characterizations and a Fast Algorithm for Dominating Set
We give alternative definitions for maximum matching width, e.g., a graph G has mmw(G) <= k if and only if it is a subgraph of a chordal graph H and for every maximal clique X of H there exists A,B,C \subseteq X with A \cup B \cup C=X and |A|,|B|,|C| <= k such that any subset of X that is a minimal separator of H is a subset of either A, B or C. Treewidth and branchwidth have alternative definitions through intersections of subtrees, where treewidth focuses on nodes and branchwidth focuses on edges. We show that mm-width combines both aspects, focusing on nodes and on edges. Based on this we prove that given a graph G and a branch decomposition of mm-width k we can solve Dominating Set in time O^*(8^k), thereby beating O^*(3^{tw(G)}) whenever tw(G) > log_3(8) * k ~ 1.893 k. Note that mmw(G) <= tw(G)+1 <= 3 mmw(G) and these inequalities are tight. Given only the graph G and using the best known algorithms to find decompositions, maximum matching width will be better for solving Dominating Set whenever tw(G) > 1.549 * mmw(G).
FPT algorithms
treewidth
dominating set
212-223
Regular Paper
Jisu
Jeong
Jisu Jeong
Sigve Hortemo
Sæther
Sigve Hortemo Sæther
Jan Arne
Telle
Jan Arne Telle
10.4230/LIPIcs.IPEC.2015.212
Eyal Amir. Approximation algorithms for treewidth. Algorithmica, 56(4):448-479, 2010.
Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Fourier meets Möbius: fast subset convolution. In STOC'07 - Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pages 67-74. ACM, New York, 2007.
Hans L Bodlaender, Pal Gronas Drange, Markus S Dregi, Fedor V Fomin, Daniel Lokshtanov, and Michal Pilipczuk. An o(c^k n) 5-approximation algorithm for treewidth. In Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium on, pages 499-508. IEEE, 2013.
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Hans L. Bodlaender, Erik Jan van Leeuwen, Johan M. M. van Rooij, and Martin Vatshelle. Faster algorithms on branch and clique decompositions. In Mathematical foundations of computer science 2010, volume 6281 of Lecture Notes in Comput. Sci., pages 174-185. Springer, Berlin, 2010.
Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer International Publishing, New York, 2016.
Reinhard Diestel. Graph theory, volume 173 of Graduate Texts in Mathematics. Springer, Heidelberg, fourth edition, 2010.
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Jisu Jeong, Sigve Hortemo Sæther, and Jan Arne Telle. An FPT algorithm computing a decomposition of optimal mm-width. in preparation, 2015.
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Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Known algorithms on graphs of bounded treewidth are probably optimal. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pages 777-789. SIAM, Philadelphia, PA, 2011.
Sang-il Oum and Paul Seymour. Approximating clique-width and branch-width. J. Combin. Theory Ser. B, 96(4):514-528, 2006.
Christophe Paul and Jan Arne Telle. Edge-maximal graphs of branchwidth k: the k-branches. Discrete Math., 309(6):1467-1475, 2009.
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Sigve Hortemo Sæther and Jan Arne Telle. Between treewidth and clique-width. In Graph-theoretic concepts in computer science, volume 8747 of Lecture Notes in Comput. Sci., pages 396-407. Springer, Cham, 2014.
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Martin Vatshelle. New Width Parameters of Graphs. PhD thesis, University of Bergen, 2012.
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Fast Parallel Fixed-parameter Algorithms via Color Coding
Fixed-parameter algorithms have been successfully applied to solve numerous difficult problems within acceptable time bounds on large inputs. However, most fixed-parameter algorithms are inherently sequential and, thus, make no use of the parallel hardware present in modern computers. We show that parallel fixed-parameter algorithms do not only exist for numerous parameterized problems from the literature - including vertex cover, packing problems, cluster editing, cutting vertices, finding embeddings, or finding matchings - but that there are parallel algorithms working in constant time or at least in time depending only on the parameter (and not on the size of the input) for these problems. Phrased in terms of complexity classes, we place numerous natural parameterized problems in parameterized versions of AC^0. On a more technical level, we show how the color coding method can be implemented in constant time and apply it to embedding problems for graphs of bounded tree-width or tree-depth and to model checking first-order formulas in graphs of bounded degree.
color coding
parallel computation
fixed-parameter tractability
graph packing
cutting $ell$ vertices
cluster editing
tree-width
tree-depth,
224-235
Regular Paper
Max
Bannach
Max Bannach
Christoph
Stockhusen
Christoph Stockhusen
Till
Tantau
Till Tantau
10.4230/LIPIcs.IPEC.2015.224
F. N. Abu-Khzam, M. A. Langston, P. Shanbhag, and C. T. Symons. Scalable Parallel Algorithms for FPT Problems. Algorithmica, 45(3):269-284, 2006.
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Kazuyuki Amano. k-Subgraph Isomorphism on AC⁰ Circuits. Computational Complexity, 19(2):1016-3328, 2010.
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S. Böcker and J. Baumbach. Cluster Editing. In Proceedings of the Ninth Conference on Computability in Europe, volume 7921 of CiE'13, pages 33-44. Springer Berlin Heidelberg, 2013.
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M. Elberfeld, C. Stockhusen, and T. Tantau. On the Space Complexity of Parameterized Problems: Classes and Completness. Algorithmica, 71(3):661-701, 2014.
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J. Flum and M. Grohe. Parameterized Complexity Theory. Springer, 2006.
F. V. Fomin, P. A. Golovach, and J. H. Korhonen. On the Parameterized Complexity of Cutting a Few Vertices from a Graph. In Proceedings of the Thirty-Eight International Symposium of Mathematical Foundations of Computer Science, volume 8087 of MFCS '13, pages 421-432. Springer, Heidelberg, Germany, 2013.
F. V. Fomin, S. Kratsch, M. Pilipczuk, M. Pilipczuk, and Y. Villanger. Tight Bounds for Parameterized Complexity of Cluster Editing. In Proceedings of the Thirtieth International Symposium on Theoretical Aspects of Computer Science, volume 20 of STACS'13, pages 30-43. International Symposium on Theoretical Aspects of Computer Science, 2013.
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J. Gramm, J. Guo, F. Hüffner, and R. Niedermeier. Graph-Modeled Data Clustering: Fixed-Parameter Algorithms for Clique Generation. In Proceedings of the Fifth Italian Conference of Algorithms and Complexity, volume 2653 of CIAC'03, pages 108-119. Springer Berlin Heidelberg, 2003.
D. Marx. Parameterized Graph Separation Problems. Theoretical Computer Science, 351(3):394-406, 2006.
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H. Vollmer. Introduction to Circuit Complexity. Springer, 1999.
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Fixed-parameter Tractable Distances to Sparse Graph Classes
We show that for various classes C of sparse graphs, and several measures of distance to such classes (such as edit distance and elimination distance), the problem of determining the distance of a given graph G to C is fixed-parameter tractable. The results are based on two general techniques. The first of these, building on recent work of Grohe et al. establishes that any class of graphs that is slicewise nowhere dense and slicewise first-order definable is FPT. The second shows that determining the elimination distance of a graph G to a minor-closed class C is FPT.
parameterized complexity
fixed-parameter tractable
distance
graph theory
sparse graphs
graph minor
nowhere dense
236-247
Regular Paper
Jannis
Bulian
Jannis Bulian
Anuj
Dawar
Anuj Dawar
10.4230/LIPIcs.IPEC.2015.236
I. Adler, M. Grohe, and S. Kreutzer. Computing excluded minors. In SODA'08: Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms. SIAM, January 2008.
J. Bulian and A. Dawar. Graph isomorphism parameterized by elimination distance to bounded degree. In Parameterized and Exact Computation - 9th International Symposium, IPEC 2014, Wroclaw, Poland, September 10-12, 2014. Revised Selected Papers, pages 135-146, 2014.
L. Cai. Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett., 58:171-176, 1996.
R. Diestel. Graph Theory. Springer, January 2000.
R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer, October 2012.
J. Flum and M. Grohe. Fixed-Parameter Tractability, Definability, and Model-Checking. SIAM J. Comput., 31(1):113-145, 2001.
J. Flum and M. Grohe. Parameterized Complexity Theory. Springer, May 2006.
F. V. Fomin, D. Lokshtanov, N. Misra, and S. Saurabh. Planar F-deletion: Approximation, kernelization and optimal FPT algorithms. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 470-479, 2012.
J. Gajarský, P. Hlinený, J. Obdrzálek, S. Ordyniak, F. Reidl, P. Rossmanith, F. Sanchez Villaamil, and S. Sikdar. Kernelization using structural parameters on sparse graph classes. In Algorithms - ESA 2013 - 21st Annual European Symposium, pages 529-540, 2013.
P. A. Golovach. Editing to a Graph of Given Degrees. In Parameterized and Exact Computation, pages 196-207. Springer, September 2014.
M. Grohe, S. Kreutzer, and S. Siebertz. Deciding first-order properties of nowhere dense graphs. In STOC'14: Proceedings of the 46th Annual ACM Symposium on Theory of Computing. ACM, May 2014.
J. Guo, F. Hüffner, and R. Niedermeier. A Structural View on Parameterizing Problems: Distance from Triviality. In Parameterized and Exact Computation, pages 162-173. Springer, 2004.
W. Hodges. A Shorter Model Theory. Cambridge University Press, 1997.
F. Hüffner, C. Komusiewicz, H. Moser, and R. Niedermeier. Fixed-parameter algorithms for cluster vertex deletion. Theory Comput. Syst., 47:196-217, 2010.
D. Marx. Parameterized coloring problems on chordal graphs. Theor. Comput. Sci., 351:407-424, 2006.
J. Nesetril and P. Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms. Springer, 2012.
R. Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford University Press, February 2006.
N. Robertson and P. D. Seymour. Graph minors. XX. wagner’s conjecture. J. Comb. Theory, Ser. B, 92:325-357, 2004.
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Strong ETH and Resolution via Games and the Multiplicity of Strategies
We consider a restriction of the Resolution proof system in which at most a fixed number of variables can be resolved more than once along each refutation path. This system lies between regular Resolution, in which no variable can be resolved more than once along any path, and general Resolution where there is no restriction on the number of such variables. We show that when the number of re-resolved variables is not too large, this proof system is consistent with the Strong Exponential Time Hypothesis (SETH). More precisely for large n and k we show that there are unsatisfiable k-CNF formulas which require Resolution refutations of size 2^{(1 - epsilon_k)n}, where n is the number of variables and epsilon_k=~O(k^{-1/5}), whenever in each refutation path we only allow at most ~O(k^{-1/5})n variables to be resolved multiple times. However, these re-resolved variables along different paths do not need to be the same. Prior to this work, the strongest proof system shown to be consistent with SETH was regular Resolution [Beck and Impagliazzo, STOC'13]. This work strengthens that result and gives a different and conceptually simpler game-theoretic proof for the case of regular Resolution.
Strong Exponential Time Hypothesis
resolution
proof systems
248-257
Regular Paper
Ilario
Bonacina
Ilario Bonacina
Navid
Talebanfard
Navid Talebanfard
10.4230/LIPIcs.IPEC.2015.248
Albert Atserias and Víctor Dalmau. A combinatorial characterization of resolution width. J. Comput. Syst. Sci., 74(3):323-334, 2008.
Albert Atserias, Johannes Klaus Fichte, and Marc Thurley. Clause-learning algorithms with many restarts and bounded-width resolution. J. Artif. Intell. Res. (JAIR), 40:353-373, 2011.
Paul Beame, Christopher Beck, and Russell Impagliazzo. Time-space tradeoffs in resolution: superpolynomial lower bounds for superlinear space. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19-22, 2012, pages 213-232. ACM, 2012.
Christopher Beck and Russell Impagliazzo. Strong ETH Holds for Regular Resolution. In Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing, STOC'13, pages 487-494. ACM, 2013.
Eli Ben-Sasson and Avi Wigderson. Short proofs are narrow - resolution made simple. J. ACM, 48(2):149-169, 2001.
Archie Blake. Canonical Expressions in Boolean Algebra. PhD thesis, University of Chicago, 1937.
Ruiwen Chen, Valentine Kabanets, Antonina Kolokolova, Ronen Shaltiel, and David Zuckerman. Mining circuit lower bound proofs for meta-algorithms. In IEEE 29th Conference on Computational Complexity, CCC, pages 262-273, 2014.
Ruiwen Chen, Valentine Kabanets, and Nitin Saurabh. An improved deterministic #SAT algorithm for small de morgan formulas. In Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS, pages 165-176, 2014.
Shiteng Chen, Dominik Scheder, Navid Talebanfard, and Bangsheng Tang. Exponential Lower Bounds for the PPSZ k-SAT Algorithm. In SODA, pages 1253-1263, 2013.
Stefan S. Dantchev. Relativisation provides natural separations for resolution-based proof systems. In Computer Science - Theory and Applications, First International Computer Science Symposium in Russia, CSR 2006, St. Petersburg, Russia, June 8-12, 2006, Proceedings, pages 147-158, 2006.
Evgeny Dantsin, Andreas Goerdt, Edward A. Hirsch, Ravi Kannan, Jon M. Kleinberg, Christos H. Papadimitriou, Prabhakar Raghavan, and Uwe Schöning. A deterministic (2-2/(k+1))ⁿ algorithm for k-SAT based on local search. Theor. Comput. Sci., 289(1):69-83, 2002.
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Russell Impagliazzo and Ramamohan Paturi. On the Complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001.
Roberto J. Bayardo Jr. and Robert Schrag. Using CSP look-back techniques to solve real-world SAT instances. In Benjamin Kuipers and Bonnie L. Webber, editors, Proceedings of the Fourteenth National Conference on Artificial Intelligence and Ninth Innovative Applications of Artificial Intelligence Conference, AAAI 97, IAAI 97, July 27-31, 1997, Providence, Rhode Island., pages 203-208. AAAI Press / The MIT Press, 1997.
Matthew W. Moskewicz, Conor F. Madigan, Ying Zhao, Lintao Zhang, and Sharad Malik. Chaff: Engineering an efficient SAT solver. In Proceedings of the 38th Design Automation Conference, DAC 2001, Las Vegas, NV, USA, June 18-22, 2001, pages 530-535. ACM, 2001.
Ramamohan Paturi, Pavel Pudlák, Michael E. Saks, and Francis Zane. An improved exponential-time algorithm for k-SAT. J. ACM, 52(3):337-364, 2005.
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Knot Pipatsrisawat and Adnan Darwiche. On the power of clause-learning SAT solvers as resolution engines. Artif. Intell., 175(2):512-525, 2011.
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Pavel Pudlák and Russell Impagliazzo. A lower bound for DLL algorithms for k-SAT (preliminary version). In SODA, pages 128-136, 2000.
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Alasdair Urquhart. Hard examples for resolution. J. ACM, 34(1):209-219, 1987.
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Ryan Williams. Non-uniform ACC circuit lower bounds. In Proceedings of the 26th Annual IEEE Conference on Computational Complexity, CCC, pages 115-125, 2011.
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Quick but Odd Growth of Cacti
Let F be a family of graphs. Given an input graph G and a positive integer k, testing whether G has a k-sized subset of vertices S, such that G\S belongs to F, is a prototype vertex deletion problem. These type of problems have attracted a lot of attention in recent times in the domain of parameterized complexity. In this paper, we study two such problems; when F is either a family of cactus graphs or a family of odd-cactus graphs. A graph H is called a cactus graph if every pair of cycles in H intersect on at most one vertex. Furthermore, a cactus graph H is called an odd cactus, if every cycle of H is of odd length. Let us denote by C and C_{odd}, families of cactus and odd cactus, respectively. The vertex deletion problems corresponding to C and C_{odd} are called Diamond Hitting Set and Even Cycle Transversal, respectively. In this paper we design randomized algorithms with running time 12^{k}*n^{O(1)} for both these problems. Our algorithms considerably improve the running time for Diamond Hitting Set and Even Cycle Transversal, compared to what is known about them.
Even Cycle Transversal
Diamond Hitting Set
Randomized Algorithms
258-269
Regular Paper
Sudeshna
Kolay
Sudeshna Kolay
Daniel
Lokshtanov
Daniel Lokshtanov
Fahad
Panolan
Fahad Panolan
Saket
Saurabh
Saket Saurabh
10.4230/LIPIcs.IPEC.2015.258
Ann Becker, Reuven Bar-Yehuda, and Dan Geiger. Randomized algorithms for the loop cutset problem. (JAIR), 12:219-234, 2000.
Hans L. Bodlaender, Fedor V. Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, and Dimitrios M. Thilikos. (meta) kernelization. In FOCS 2009, pages 629-638, 2009.
Yixin Cao. Unit interval editing is fixed-parameter tractable. In ICALP 2015, volume 9134 of LNCS, pages 306-317. Springer, 2015.
Yixin Cao and Dániel Marx. Interval deletion is fixed-parameter tractable. ACM Transactions on Algorithms, 11(3):21:1-21:35, 2015.
R. Diestel. Graph Theory. Springer, Berlin, second ed., electronic edition, February 2000.
Samuel Fiorini, Gwenaël Joret, and Ugo Pietropaoli. Hitting diamonds and growing cacti. In IPCO 2010, pages 191-204, 2010.
Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, and Saket Saurabh. Hitting forbidden minors: Approximation and Kernelization. In Thomas Schwentick and Christoph Dürr, editors, STACS 2011, volume 9 of (LIPIcs), pages 189-200, Dagstuhl, Germany, 2011. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.
Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar f-deletion: Approximation, kernelization and optimal FPT algorithms. In FOCS 2012, pages 470-479, 2012.
Fedor V. Fomin and Yngve Villanger. Subexponential parameterized algorithm for minimum fill-in. SIAM J. Comput., 42(6):2197-2216, 2013.
Archontia C. Giannopoulou, Bart M. P. Jansen, Daniel Lokshtanov, and Saket Saurabh. Uniform kernelization complexity of hitting forbidden minors. In ICALP 2015, volume 9134 of LNCS, pages 629-641. Springer, 2015.
Gwenaël Joret, Christophe Paul, Ignasi Sau, Saket Saurabh, and Stéphan Thomassé. Hitting and harvesting pumpkins. SIAM J. Discrete Math., 28(3):1363-1390, 2014.
Eun Jung Kim, Alexander Langer, Christophe Paul, Felix Reidl, Peter Rossmanith, Ignasi Sau, and Somnath Sikdar. Linear kernels and single-exponential algorithms via protrusion decompositions. In ICALP 2013, volume 7965 of LNCS, pages 613-624. Springer, 2013.
John M. Lewis and Mihalis Yannakakis. The node-deletion problem for hereditary properties is np-complete. J. Comput. Syst. Sci., 20(2):219-230, 1980.
Daniel Lokshtanov and M. S. Ramanujan. Parameterized tractability of multiway cut with parity constraints. In ICALP 2012, pages 750-761, 2012.
Pranabendu Misra, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. Parameterized algorithms for even cycle transversal. In WG 2012, pages 172-183, 2012.
Carsten Thomassen. On the presence of disjoint subgraphs of a specified type. Journal of Graph Theory, 12(1):101-111, 1988.
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A Polynomial Kernel for Block Graph Deletion
In the Block Graph Deletion problem, we are given a graph G on n vertices and a positive integer k, and the objective is to check whether it is possible to delete at most k vertices from G to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel with O(k^{6}) vertices for the Block Graph Deletion problem. This is a first step to investigate polynomial kernels for deletion problems into non-trivial classes of graphs of bounded rank-width, but unbounded tree-width. Our result also implies that Chordal Vertex Deletion admits a polynomial-size kernel on diamond-free graphs. For the kernelization and its analysis, we introduce the notion of 'complete degree' of a vertex. We believe that the underlying idea can be potentially applied to other problems. We also prove that the Block Graph Deletion problem can be solved in time 10^{k} * n^{O(1)}.
block graph
polynomial kernel
single-exponential FPT algorithm
270-281
Regular Paper
Eun Jung
Kim
Eun Jung Kim
O-joung
Kwon
O-joung Kwon
10.4230/LIPIcs.IPEC.2015.270
Hans L. Bodlaender. On disjoint cycles. In Proceedings of the 17th International Workshop, WG'91, pages 230-238, London, UK, UK, 1992. Springer-Verlag.
Yixin Cao, Jianer Chen, and Yang Liu. On feedback vertex set: New measure and new structures. Algorithmica, pages 1-24, 2014.
Yixin Cao and Dániel Marx. Chordal editing is fixed-parameter tractable. In 31st International Symposium on Theoretical Aspects of Computer Science, volume 25 of LIPIcs. Leibniz Int. Proc. Inform., pages 214-225. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2014.
Jianer Chen, Fedor V. Fomin, Yang Liu, Songjian Lu, and Yngve Villanger. Improved algorithms for feedback vertex set problems. J. Comput. System Sci., 74(7):1188-1198, 2008.
Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inform. and Comput., 85(1):12-75, 1990.
Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000.
Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michał Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time (extended abstract). In 2011 IEEE 52nd Annual Symp. on Foundations of Computer Science (FOCS'11), pages 150-159. IEEE CS, 2011.
Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk, and Jakub Onufry Wojtaszczyk. Subset feedback vertex set is fixed-parameter tractable. SIAM Journal on Discrete Mathematics, 27(1):290-309, 2013.
Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk, and JakubOnufry Wojtaszczyk. An improved FPT algorithm and quadratic kernel for pathwidth one vertex deletion. In Venkatesh Raman and Saket Saurabh, editors, Parameterized and Exact Computation, volume 6478 of Lecture Notes in Computer Science, pages 95-106. Springer Berlin Heidelberg, 2010.
Frank Dehne, Michael Fellows, Michael Langston, Frances Rosamond, and Kim Stevens. An O(2^O(k)n³) fpt algorithm for the undirected feedback vertex set problem. Theory of Computing Systems, 41(3):479-492, 2007.
Rod Downey and Michael Fellows. Fixed-parameter tractability and completeness. III. Some structural aspects of the W hierarchy. In Complexity theory, pages 191-225. Cambridge Univ. Press, Cambridge, 1993.
Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar F-deletion: Approximation, kernelization and optimal FPT algorithms. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 470-479, 2012.
T. Gallai. Maximum-minimum Sätze und verallgemeinerte Faktoren von Graphen. Acta Math. Acad. Sci. Hungar., 12:131-173, 1961.
Jiong Guo, Jens Gramm, Falk Hüffner, Rolf Niedermeier, and Sebastian Wernicke. Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization. Journal of Computer and System Sciences, 72(8):1386-1396, 2006.
John Hopcroft and Robert Tarjan. Algorithm 447: Efficient algorithms for graph manipulation. Commun. ACM, 16(6):372-378, June 1973.
Iyad Kanj, Michael Pelsmajer, and Marcus Schaefer. Parameterized algorithms for feedback vertex set. In Parameterized and Exact Computation, volume 3162 of LNCS, pages 235-247. Springer, 2004.
Eun Jung Kim and O-joung Kwon. A polynomial kernel for Block Graph Vertex Deletion, 2015. arXiv.org:abs:1506.08477.
Eun Jung Kim, Alexander Langer, Christophe Paul, Felix Reidl, Peter Rossmanith, Ignasi Sau, and Somnath Sikdar. Linear kernels and single-exponential algorithms via protrusion decompositions. In Automata, Languages, and Programming - 40th Int'l Colloquium, ICALP 2013, Proceedings, Part I, pages 613-624, 2013.
Tomasz Kociumaka and Marcin Pilipczuk. Faster deterministic feedback vertex set. Information Processing Letters, 114(10):556-560, 2014.
Dániel Marx. Chordal deletion is fixed-parameter tractable. Algorithmica, 57(4):747-768, 2010.
Neeldhara Misra, Geevarghese Philip, Venkatesh Raman, and Saket Saurabh. On parameterized independent feedback vertex set. Theoretical Computer Science, 461(0):65-75, 2012. 17th International Computing and Combinatorics Conference (COCOON 2011).
Sang-il Oum. Rank-width and vertex-minors. J. Combin. Theory Ser. B, 95(1):79-100, 2005.
Venkatesh Raman, Saket Saurabh, and C. R. Subramanian. Faster fixed parameter tractable algorithms for undirected feedback vertex set. In Algorithms and computation, volume 2518 of Lecture Notes in Comput. Sci., pages 241-248. Springer, Berlin, 2002.
Bruce Reed, Kaleigh Smith, and Adrian Vetta. Finding odd cycle transversals. Operations Research Letters, 32(4):299-301, 2004.
Neil Robertson and P. D. Seymour. Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms, 7(3):309-322, 1986.
Stéphan Thomassé. A quadratic kernel for feedback vertex set. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 115-119. ACM/SIAM, 2009.
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Parameterized Complexity of Graph Constraint Logic
Graph constraint logic is a framework introduced by Hearn and Demaine, which provides several problems that are often a convenient starting point for reductions. We study the parameterized complexity of Constraint Graph Satisfiability and both bounded and unbounded versions of Nondeterministic Constraint Logic (NCL) with respect to solution length, treewidth and maximum degree of the underlying constraint graph as parameters. As a main result we show that restricted NCL remains PSPACE-complete on graphs of bounded bandwidth, strengthening Hearn and Demaine's framework. This allows us to improve upon existing results obtained by reduction from NCL. We show that reconfiguration versions of several classical graph problems (including independent set, feedback vertex set and dominating set) are PSPACE-complete on planar graphs of bounded bandwidth and that Rush Hour, generalized to k*n boards, is PSPACE-complete even when k is at most a constant.
Nondeterministic Constraint Logic
Reconfiguration Problems
Parameterized Complexity
Treewidth
Bandwidth
282-293
Regular Paper
Tom C.
van der Zanden
Tom C. van der Zanden
10.4230/LIPIcs.IPEC.2015.282
Hans L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theoretical computer science, 209(1):1-45, 1998.
Paul Bonsma and Luis Cereceda. Finding paths between graph colourings: Pspace-completeness and superpolynomial distances. Theoretical Computer Science, 410(50):5215-5226, 2009.
Erik D. Demaine, Martin L. Demaine, Eli Fox-Epstein, Duc A Hoang, Takehiro Ito, Hirotaka Ono, Yota Otachi, Ryuhei Uehara, and Takeshi Yamada. Polynomial-time algorithm for sliding tokens on trees. In Algorithms and Computation, pages 389-400. Springer, 2014.
Gary William Flake and Eric B. Baum. Rush hour is pspace-complete, or "why you should generously tip parking lot attendants". Theoretical Computer Science, 270(1):895-911, 2002.
Arash Haddadan, Takehiro Ito, Amer E. Mouawad, Naomi Nishimura, Hirotaka Ono, Akira Suzuki, and Youcef Tebbal. The complexity of dominating set reconfiguration. arXiv preprint arXiv:1503.00833, 2015.
Robert A. Hearn and Erik D. Demaine. The nondeterministic constraint logic model of computation: Reductions and applications. In Automata, Languages and Programming, pages 401-413. Springer, 2002.
Robert A. Hearn and Erik D. Demaine. Pspace-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theoretical Computer Science, 343(1):72-96, 2005.
Robert A. Hearn and Erik D. Demaine. Games, puzzles, and computation. CRC Press, 2009.
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. In Algorithms and Computation, pages 28-39. Springer, 2008.
Marcin Kamiński, Paul Medvedev, and Martin Milanič. Complexity of independent set reconfigurability problems. Theoretical computer science, 439:9-15, 2012.
Amer E. Mouawad, Naomi Nishimura, Venkatesh Raman, and Marcin Wrochna. Reconfiguration over tree decompositions. In Parameterized and Exact Computation, pages 246-257. Springer, 2014.
Bala Ravikumar. Peg-solitaire, string rewriting systems and finite automata. In Algorithms and Computation, pages 233-242. Springer, 1997.
Tom C. van der Zanden. Parameterized complexity of graph constraint logic. arXiv preprint:1509.02683, 2015.
Marcin Wrochna. Reconfiguration in bounded bandwidth and treedepth. arXiv preprint arXiv:1405.0847, 2014.
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Complexity and Approximability of Parameterized MAX-CSPs
We study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard CSPs, we want to decide whether this fraction equals one. The parameters we investigate are structural measures, such as the treewidth or the clique-width of the variable–constraint incidence graph of the CSP instance.
We consider Max-CSPs with the constraint types AND, OR, PARITY, and MAJORITY, and with various parameters k. We attempt to fully classify them into the following three cases:
1. The exact optimum can be computed in FPT-time.
2. It is W[1]-hard to compute the exact optimum, but there is a randomized FPT approximation scheme (FPT-AS), which computes a (1-epsilon)-approximation in time f(k,epsilon) * poly(n).
3. There is no FPT-AS unless FPT=W[1].
For the corresponding standard CSPs, we establish FPT vs. W[1]-hardness results.
Approximation
Structural Parameters
Constraint Satisfaction
294-306
Regular Paper
Holger
Dell
Holger Dell
Eun Jung
Kim
Eun Jung Kim
Michael
Lampis
Michael Lampis
Valia
Mitsou
Valia Mitsou
Tobias
Mömke
Tobias Mömke
10.4230/LIPIcs.IPEC.2015.294
Michael Alekhnovich and Alexander A. Razborov. Satisfiability, branch-width and Tseitin tautologies. Computational Complexity, 20(4):649-678, 2011.
Per Austrin and Subhash Khot. A characterization of approximation resistance for even k-partite csps. In Robert D. Kleinberg, editor, Innovations in Theoretical Computer Science, ITCS'13, Berkeley, CA, USA, January 9-12, 2013, pages 187-196. ACM, 2013.
Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000.
Nadia Creignou. A dichotomy theorem for maximum generalized satisfiability problems. Journal of Computer and System Sciences, 51(3):511-522, 1995.
Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013.
Serge Gaspers and Stefan Szeider. Kernels for global constraints. In Toby Walsh, editor, IJCAI 2011, Proceedings of the 22nd International Joint Conference on Artificial Intelligence, Barcelona, Catalonia, Spain, July 16-22, 2011, pages 540-545. IJCAI/AAAI, 2011.
Serge Gaspers and Stefan Szeider. Backdoors to acyclic SAT. In Artur Czumaj, Kurt Mehlhorn, Andrew M. Pitts, and Roger Wattenhofer, editors, Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part I, volume 7391 of Lecture Notes in Computer Science, pages 363-374. Springer, 2012.
Serge Gaspers and Stefan Szeider. Guarantees and limits of preprocessing in constraint satisfaction and reasoning. Artif. Intell., 216:1-19, 2014.
Martin Grohe. The structure of tractable constraint satisfaction problems. In Rastislav Kralovic and Pawel Urzyczyn, editors, MFCS 2006, Stará Lesná, Slovakia, August 28-September 1, 2006, Proceedings, volume 4162 of Lecture Notes in Computer Science, pages 58-72. Springer, 2006.
Frank Gurski and Egon Wanke. The tree-width of clique-width bounded graphs without K_n, n. In Ulrik Brandes and Dorothea Wagner, editors, Graph-Theoretic Concepts in Computer Science, 26th International Workshop, WG 2000, Konstanz, Germany, June 15-17, 2000, Proceedings, volume 1928 of Lecture Notes in Computer Science, pages 196-205. Springer, 2000.
Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, 2001.
Sanjeev Khanna, Madhu Sudan, and David P. Williamson. A complete classification of the approximability of maximization problems derived from boolean constraint satisfaction. In Frank Thomson Leighton and Peter W. Shor, editors, Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, El Paso, Texas, USA, May 4-6, 1997, pages 11-20. ACM, 1997.
Subhash Khot and Rishi Saket. Approximating csps using LP relaxation. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, volume 9134 of Lecture Notes in Computer Science, pages 822-833. Springer, 2015.
Michael Lampis. Algorithmic meta-theorems for restrictions of treewidth. Algorithmica, 64(1):19-37, 2012.
Dániel Marx. Parameterized complexity and approximation algorithms. Comput. J., 51(1):60-78, 2008.
Sebastian Ordyniak, Daniël Paulusma, and Stefan Szeider. Satisfiability of acyclic and almost acyclic CNF formulas. Theor. Comput. Sci., 481:85-99, 2013.
Daniël Paulusma, Friedrich Slivovsky, and Stefan Szeider. Model counting for CNF formulas of bounded modular treewidth. In Natacha Portier and Thomas Wilke, editors, STACS 2013, February 27 - March 2, 2013, Kiel, Germany, volume 20 of LIPIcs, pages 55-66. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2013.
Reinhard Pichler, Stefan Rümmele, Stefan Szeider, and Stefan Woltran. Tractable answer-set programming with weight constraints: bounded treewidth is not enough. TPLP, 14(2):141-164, 2014.
Sigve Hortemo Sæther, Jan Arne Telle, and Martin Vatshelle. Solving maxsat and #sat on structured CNF formulas. In Carsten Sinz and Uwe Egly, editors, SAT 2014 - Vienna, Austria, July 14-17, 2014. Proceedings, volume 8561 of Lecture Notes in Computer Science, pages 16-31. Springer, 2014.
Marko Samer and Stefan Szeider. Constraint satisfaction with bounded treewidth revisited. J. Comput. Syst. Sci., 76(2):103-114, 2010.
Thomas J. Schaefer. The complexity of satisfiability problems. In Richard J. Lipton, Walter A. Burkhard, Walter J. Savitch, Emily P. Friedman, and Alfred V. Aho, editors, Proceedings of the 10th Annual ACM Symposium on Theory of Computing, May 1-3, 1978, San Diego, California, USA, pages 216-226. ACM, 1978.
Friedrich Slivovsky and Stefan Szeider. Model counting for formulas of bounded clique-width. In Leizhen Cai, Siu-Wing Cheng, and Tak Wah Lam, editors, ISAAC 2013, Hong Kong, China, Proceedings, volume 8283 of Lecture Notes in Computer Science, pages 677-687. Springer, 2013.
Stefan Szeider. On fixed-parameter tractable parameterizations of SAT. In Enrico Giunchiglia and Armando Tacchella, editors, Theory and Applications of Satisfiability Testing, 6th International Conference, SAT 2003. Santa Margherita Ligure, Italy, May 5-8, 2003 Selected Revised Papers, volume 2919 of Lecture Notes in Computer Science, pages 188-202. Springer, 2003.
Stefan Szeider. Not so easy problems for tree decomposable graphs. CoRR, abs/1107.1177, 2011.
Stefan Szeider. The parameterized complexity of constraint satisfaction and reasoning. In Hans Tompits, Salvador Abreu, Johannes Oetsch, Jörg Pührer, Dietmar Seipel, Masanobu Umeda, and Armin Wolf, editors, INAP 2011, and WLP 2011, Vienna, Austria, September 28-30, 2011, Revised Selected Papers, volume 7773 of Lecture Notes in Computer Science, pages 27-37. Springer, 2011.
Stefan Szeider. The parameterized complexity of k-flip local search for SAT and MAX SAT. Discrete Optimization, 8(1):139-145, 2011.
Luca Trevisan. Inapproximability of combinatorial optimization problems. Electronic Colloquium on Computational Complexity (ECCC), 2004.
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Enumerating Minimal Connected Dominating Sets in Graphs of Bounded Chordality
Listing, generating or enumerating objects of specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. We study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality. We establish enumeration algorithms as well as lower and upper bounds for the maximum number of minimal connected dominating sets in such graphs. In particular, we present algorithms to enumerate all minimal connected dominating sets of chordal graphs in time O(1.7159^n), of split graphs in time O(1.3803^n), and of AT-free, strongly chordal, and distance-hereditary graphs in time O^*(3^{n/3}), where n is the number of vertices of the input graph. Our algorithms imply corresponding upper bounds for the number of minimal connected dominating sets for these graph classes.
Minimal connected dominating set
exact algorithms
enumeration
307-318
Regular Paper
Petr A.
Golovach
Petr A. Golovach
Pinar
Heggernes
Pinar Heggernes
Dieter
Kratsch
Dieter Kratsch
10.4230/LIPIcs.IPEC.2015.307
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The Graph Motif Problem Parameterized by the Structure of the Input Graph
The Graph Motif problem was introduced in 2006 in the context of biological networks. It consists of deciding whether or not a multiset of colors occurs in a connected subgraph of a vertex-colored graph. Graph Motif has been analyzed from the standpoint of parameterized complexity. The main parameters which came into consideration were the size of the multiset and the number of colors. Though, in the many applications of Graph Motif, the input graph originates from real-life and has structure. Motivated by this prosaic observation, we systematically study its complexity relatively to graph structural parameters. For a wide range of parameters, we give new or improved FPT algorithms, or show that the problem remains intractable. Interestingly, we establish that Graph Motif is W[1]-hard (while in W[P]) for parameter max leaf number, which is, to the best of our knowledge, the first problem to behave this way.
Parameterized Complexity
Structural Parameters
Graph Motif
Computational Biology
319-330
Regular Paper
Édouard
Bonnet
Édouard Bonnet
Florian
Sikora
Florian Sikora
10.4230/LIPIcs.IPEC.2015.319
Eric Alm and Adam P. Arkin. Biological Networks. Current Opinion in Structural Biology, 13(2):193-202, 2003.
Abhimanyu M. Ambalath, Radheshyam Balasundaram, Chintan Rao H., Venkata Koppula, Neeldhara Misra, Geevarghese Philip, and M. S. Ramanujan. On the Kernelization Complexity of Colorful Motifs. In Proc. of the 5th IPEC, volume 6478 of LNCS, pages 14-25. Springer, 2010.
Daniel Berend and Tamir Tassa. Improved bounds on bell numbers and on moments of sums of random variables. Probab. and Math. Statist., 30(2):185-205, 2010.
Nadja Betzler, René van Bevern, Michael R. Fellows, Christian Komusiewicz, and Rolf Niedermeier. Parameterized algorithmics for finding connected motifs in biological networks. IEEE/ACM Trans. Comput. Biology Bioinform., 8(5):1296-1308, 2011.
Andreas Björklund, Petteri Kaski, and Lukasz Kowalik. Probably optimal graph motifs. In Proc. of the 30th International Symposium on Theoretical Aspects of Computer Science (STACS), volume 20 of LIPIcs, 2012.
Andreas Björklund, Petteri Kaski, Lukasz Kowalik, and Juho Lauri. Engineering motif search for large graphs. In Ulrik Brandes and David Eppstein, editors, Proc. of the 17th ALENEX, pages 104-118. SIAM, 2015.
Sebastian Böcker. A golden ratio parameterized algorithm for cluster editing. J. Discrete Algorithms, 16:79-89, 2012.
Sebastian Böcker, Florian Rasche, and Tamara Steijger. Annotating Fragmentation Patterns. In Proc. of the 9th International Workshop Algorithms in Bioinformatics (WABI), volume 5724 of LNCS, pages 13-24. Springer, 2009.
Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM J. Discrete Math., 28(1):277-305, 2014.
Sharon Bruckner, Falk Hüffner, Richard M. Karp, Ron Shamir, and Roded Sharan. Topology-Free Querying of Protein Interaction Networks. Journal of Computational Biology, 17(3):237-252, 2010.
Jianer Chen, Iyad A. Kanj, and Ge Xia. Improved upper bounds for vertex cover. Theoretical Computer Science, 411(40–42):3736 - 3756, 2010.
Marek Cygan, Marcin Pilipczuk, and Michal Pilipczuk. Known algorithms for EDGE CLIQUE COVER are probably optimal. In Proc. of Symposium on Discrete Algorithms, SODA 2013, pages 1044-1053. SIAM, 2013.
Riccardo Dondi, Guillaume Fertin, and Stéphane Vialette. Complexity issues in vertex-colored graph pattern matching. J Discr Algo, 9(1):82-99, 2011.
Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Springer, 2013.
Michael R. Fellows, Guillaume Fertin, Danny Hermelin, and Stéphane Vialette. Upper and lower bounds for finding connected motifs in vertex-colored graphs. J. Comput. Syst. Sci., 77(4):799-811, 2011.
Michael R. Fellows, Daniel Lokshtanov, Neeldhara Misra, Matthias Mnich, Frances A. Rosamond, and Saket Saurabh. The complexity ecology of parameters: An illustration using bounded max leaf number. Theory Comput. Syst., 45(4):822-848, 2009.
Robert Ganian. Twin-cover: Beyond vertex cover in parameterized algorithmics. In Proc. of the 6th International Symposium Parameterized and Exact Computation IPEC 2011, volume 7112 of LNCS, pages 259-271. Springer, 2011.
Robert Ganian. Using neighborhood diversity to solve hard problems. CoRR, abs/1201.3091, 2012.
Jens Gramm, Jiong Guo, Falk Hüffner, and Rolf Niedermeier. Data reduction and exact algorithms for clique cover. ACM Journal of Experimental Algorithmics, 13, 2008.
Sylvain Guillemot and Florian Sikora. Finding and counting vertex-colored subtrees. Algorithmica, 65(4):828-844, 2013.
Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001.
Christian Komusiewicz and Rolf Niedermeier. New races in parameterized algorithmics. In Proc. of Mathematical Foundations of Computer Science MFCS 2012, volume 7464 of LNCS, pages 19-30. Springer, 2012.
Ioannis Koutis. Constrained multilinear detection for faster functional motif discovery. Inf. Process. Lett., 112(22):889-892, 2012.
Vincent Lacroix, Cristina G. Fernandes, and Marie-France Sagot. Motif search in graphs: application to metabolic networks. IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB), 3(4):360-368, 2006.
Michael Lampis. Algorithmic meta-theorems for restrictions of treewidth. Algorithmica, 64(1):19-37, 2012.
Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Slightly superexponential parameterized problems. In Proc. of SODA 2011, pages 760-776, 2011.
Carsten Lund and Mihalis Yannakakis. On the hardness of approximating minimization problems. J. ACM, 41(5):960-981, 1994.
Rolf Niedermeier. Invitation to Fixed Parameter Algorithms. Lecture Series in Mathematics and Its Applications. Oxford University Press, 2006.
Ron Y. Pinter, Hadas Shachnai, and Meirav Zehavi. Deterministic parameterized algorithms for the graph motif problem. In Proc. of Mathematical Foundations of Computer Science MFCS 2014, volume 8635 of LNCS, pages 589-600. Springer, 2014.
Ron Y. Pinter and Meirav Zehavi. Algorithms for topology-free and alignment network queries. J. Discrete Algorithms, 27:29-53, 2014.
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Kernels for Structural Parameterizations of Vertex Cover - Case of Small Degree Modulators
Vertex Cover is one of the most well studied problems in the realm of parameterized algorithms and admits a kernel with O(l^2) edges and 2*l vertices. Here, l denotes the size of a vertex cover we are seeking for. A natural question is whether Vertex Cover admits a polynomial kernel (or a parameterized algorithm) with respect to a parameter k, that is, provably smaller than the size of the vertex cover. Jansen and Bodlaender [STACS 2011, TOCS 2013] raised this question and gave a kernel for Vertex Cover of size O(f^3), where f is the size of a feedback vertex set of the input graph. We continue this line of work and study Vertex Cover with respect to a parameter that is always smaller than the solution size and incomparable to the size of the feedback vertex set of the input graph. Our parameter is the number of vertices whose removal results in a graph of maximum degree two. While vertex cover with this parameterization can easily be shown to be fixed-parameter tractable (FPT), we show that it has a polynomial sized kernel.
The input to our problem consists of an undirected graph G, S \subseteq V(G) such that |S| = k and G[V(G)\S] has maximum degree at most 2 and a positive integer l. Given (G,S,l), in polynomial time we output an instance (G',S',l') such that |V(G')|<= O(k^5), |E(G')|<= O(k^6) and G has a vertex cover of size at most l if and only if G' has a vertex cover of size at most l'.
When G[V(G)\S] has maximum degree at most 1, we improve the known kernel bound from O(k^3) vertices to O(k^2) vertices (and O(k^3) edges). In general, if G[V(G)\S] is simply a collection of cliques of size at most d, then we transform the graph in polynomial time to an equivalent hypergraph with O(k^d) vertices and show that, for d >= 3, a kernel with O(k^{d-epsilon}) vertices is unlikely to exist for any epsilon >0 unless NP is a subset of coNO/poly.
Parameterized Complexity
Kernelization
expansion lemma
vertex cover
structural parameterization
331-342
Regular Paper
Diptapriyo
Majumdar
Diptapriyo Majumdar
Venkatesh
Raman
Venkatesh Raman
Saket
Saurabh
Saket Saurabh
10.4230/LIPIcs.IPEC.2015.331
Faisal N. Abu-Khzam. A kernelization algorithm for d-hitting set. J. Comput. Syst. Sci., 76(7):524-531, 2010.
Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM J. Discrete Math., 28(1):277-305, 2014.
Hans L. Bodlaender, Stéphan Thomassé, and Anders Yeo. Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci., 412(35):4570-4578, 2011.
Leizhen Cai. Parameterized complexity of vertex colouring. Discrete Applied Mathematics, 127(3):415-429, 2003.
Jianer Chen, Iyad A. Kanj, and Ge Xia. Improved upper bounds for vertex cover. Theor. Comput. Sci., 411(40-42):3736-3756, 2010.
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.
Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM, 61(4):23:1-23:27, 2014.
Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013.
Michael R. Fellows, Bart M. P. Jansen, and Frances A. Rosamond. Towards fully multivariate algorithmics: Parameter ecology and the deconstruction of computational complexity. Eur. J. Comb., 34(3):541-566, 2013.
Michael R. Fellows, Daniel Lokshtanov, Neeldhara Misra, Matthias Mnich, Frances A. Rosamond, and Saket Saurabh. The complexity ecology of parameters: An illustration using bounded max leaf number. Theory Comput. Syst., 45(4):822-848, 2009.
Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, and Saket Saurabh. Hitting forbidden minors: Approximation and kernelization. In STACS, pages 189-200, 2011.
Martin Grötschel and George L. Nemhauser. A polynomial algorithm for the max-cut problem on graphs without long odd cycles. Math. Programming, 29(1):28-40, 1984.
Gregory Gutin, Eun Jung Kim, Stefan Szeider, and Anders Yeo. A probabilistic approach to problems parameterized above or below tight bounds. J. Comput. Syst. Sci., 77(2):422-429, 2011.
Gregory Gutin and Anders Yeo. Constraint satisfaction problems parameterized above or below tight bounds: A survey. In The Multivariate Algorithmic Revolution and Beyond - Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday, volume 7370 of Lecture Notes in Computer Science, pages 257-286. Springer, 2012.
Wen Lian Hsu, Yoshiro Ikura, and George L. Nemhauser. A polynomial algorithm for maximum weighted vertex packings on graphs without long odd cycles. Math. Programming, 20(2):225-232, 1981.
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.
Stefan Kratsch and Magnus Wahlström. Representative sets and irrelevant vertices: New tools for kernelization. In FOCS, pages 450-459, 2012.
Daniel Lokshtanov, N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. Faster parameterized algorithms using linear programming. ACM Transactions on Algorithms, 11(2):15:1-15:31, 2014.
George L. Nemhauser and Leslie E. Trotter Jr. Vertex packings: Structural properties and algorithms. Math. Program., 8(1):232-248, 1975.
Michael Okun and Amnon Barak. A new approach for approximating node deletion problems. Inf. Process. Lett., 88(5):231-236, 2003.
Fahad Panolan and Ashutosh Rai. On the kernelization complexity of problems on graphs without long odd cycles. In COCOON 2012, volume 7434 of LNCS, pages 445-457. Springer, 2012.
Elena Prieto. Systematic kernelization in FPT algorithm design. PhD thesis, The University of Newcastle, Australia, 2005.
Michael Sipser. Introduction to the theory of computation. PWS Publishing Company, 1997.
Stéphan Thomassé. A 4k² kernel for feedback vertex set. ACM Transactions on Algorithms, 6(2), 2010.
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Parameterized Complexity of Critical Node Cuts
We consider the following graph cut problem called Critical Node Cut (CNC): Given a graph G on n vertices, and two positive integers k and x, determine whether G has a set of k vertices whose removal leaves G with at most x connected pairs of vertices. We analyze this problem in the framework of parameterized complexity. That is, we are interested in whether or not this problem is solvable in f(kappa) * n^{O(1)} time (i.e., whether or not it is fixed-parameter tractable), for various natural parameters kappa. We consider four such parameters:
- The size k of the required cut.
- The upper bound x on the number of remaining connected pairs.
- The lower bound y on the number of connected pairs to be removed.
- The treewidth w of G.
We determine whether or not CNC is fixed-parameter tractable for each of these parameters. We determine this also for all possible aggregations of these four parameters, apart from w+k. Moreover, we also determine whether or not CNC admits a polynomial kernel for all these parameterizations. That is, whether or not there is an algorithm that reduces each instance of CNC in polynomial time to an equivalent instance of size kappa^{O(1)}, where kappa is the given parameter.
graph cut problem
NP-hard problem
treewidth
343-354
Regular Paper
Danny
Hermelin
Danny Hermelin
Moshe
Kaspi
Moshe Kaspi
Christian
Komusiewicz
Christian Komusiewicz
Barak
Navon
Barak Navon
10.4230/LIPIcs.IPEC.2015.343
Bernardetta Addis, Marco Di Summa, and Andrea Grosso. Removing critical nodes from a graph: complexity results and polynomial algorithms for the case of bounded treewidth. Optimization online (www.optimization-online.org), 2011.
Stefan Arnborg, Jens Lagergren, and Detlef Seese. Easy problems for tree-decomposable graphs. Journal of Algorithms, 12(2):308-340, 1991.
Ashwin Arulselvan, Clayton W Commander, Lily Elefteriadou, and Panos M Pardalos. Detecting critical nodes in sparse graphs. Computers & Operations Research, 36(7):2193-2200, 2009.
Hans L Bodlaender. A tourist guide through treewidth. Acta Cybernetica, 11(1-2):1, 1994.
Hans L Bodlaender, Rodney G Downey, Michael R Fellows, and Danny Hermelin. On problems without polynomial kernels. Journal of Computer and System Sciences, 75(8):423-434, 2009.
Vladimir Boginski and Clayton W Commander. Identifying critical nodes in protein-protein interaction networks. Clustering challenges in biological networks, pages 153-167, 2009.
Nicolas Bousquet, Jean Daligault, and Stéphan Thomassé. Multicut is FPT. In Proceedings of the 43rd Annual Symposium on Theory of Computing (STOC), pages 459-468. ACM, 2011.
Karl Bringmann, Danny Hermelin, Matthias Mnich, and Erik Jan van Leeuwen. Parameterized complexity dichotomy for steiner multicut. In Proceedings of the 32nd Annual Symposium on Theoretical Aspects of Computer Science (STACS), pages 157-170, 2015.
Yixin Cao, Jianer Chen, and Jia-Hao Fan. An O(1.84^k) parameterized algorithm for the multiterminal cut problem. In Proceedings of the 19th Annual Symposium on Fundamentals of Computation Theory (FCT), pages 84-94, 2013.
Jianer Chen, Yang Liu, and Songjian Lu. An improved parameterized algorithm for the minimum node multiway cut problem. Algorithmica, 55(1):1-13, 2009.
Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk, and Jakub Onufry Wojtaszczyk. On multiway cut parameterized above lower bounds. ACM Transactions on Computation Theory, 5(1):3, 2013.
Marco Di Summa, Andrea Grosso, and Marco Locatelli. Complexity of the critical node problem over trees. Computers & Operations Research, 38(12):1766-1774, 2011.
Rodney G Downey and Michael R Fellows. Parameterized Complexity. Springer-Verlag, 1999.
Pål Grønås Drange, Markus Sortland Dregi, and Pim van't Hof. On the computational complexity of vertex integrity and component order connectivity. In Proceedings of the 25th Annual International Symposium on Algorithms and Computation (ISAAC), pages 285-297, 2014.
Andrew Drucker. New limits to classical and quantum instance compression. In Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2012.
Jessica Enright and Kitty Meeks. Deleting edges to restrict the size of an epidemic. CoRR, abs/1504.05773, 2015.
Michael R Fellows, Danny Hermelin, Frances A Rosamond, and Stéphane Vialette. On the parameterized complexity of multiple-interval graph problems. Theoretical Computer Science, 410(1):53-61, 2009.
Michael R Fellows and Sam Stueckle. The immersion order, forbidden subgraphs and the complexity of network integrity. Journal of Combinatorial Mathematics and Combinatorial Computing, 6(1):23-32, 1989.
Michael R Garey and David S Johnson. Computers and intractability; a guide to the theory of NP-completeness. San Francisco, LA: Freeman, 1979.
Sylvain Guillemot. FPT algorithms for path-transversal and cycle-transversal problems. Discrete Optimization, 8(1):61-71, 2011.
Karen E Joyce, Paul J Laurienti, Jonathan H Burdette, and Satoru Hayasaka. A new measure of centrality for brain networks. PLoS One, 5(8):e12200, 2010.
Stefan Kratsch, Marcin Pilipczuk, Michal Pilipczuk, and Magnus Wahlström. Fixed-parameter tractability of multicut in directed acyclic graphs. SIAM Journal on Discrete Mathematics, 29(1):122-144, 2015.
Vito Latora and Massimo Marchiori. How the science of complex networks can help developing strategies against terrorism. Chaos, solitons & fractals, 20(1):69-75, 2004.
Dániel Marx. Parameterized graph separation problems. Theoretical Computer Science, 351(3):394-406, 2006.
Dániel Marx, Barry O'Sullivan, and Igor Razgon. Finding small separators in linear time via treewidth reduction. ACM Transaction on Algorithms, 9(4):1-30, 2013.
Dániel Marx and Igor Razgon. Fixed-parameter tractability of multicut parameterized by the size of the cutset. In Proceedings of the 43rd Annual Symposium on Theory of Computing (STOC), pages 469-478, 2011.
Cullen M Taniguchi, Brice Emanuelli, and C Ronald Kahn. Critical nodes in signalling pathways: insights into insulin action. Nature Reviews Molecular Cell Biology, 7(2):85-96, 2006.
Mario Ventresca. Global search algorithms using a combinatorial unranking-based problem representation for the critical node detection problem. Computers & Operations Research, 39(11):2763-2775, 2012.
Mario Ventresca and Dionne Aleman. A derandomized approximation algorithm for the critical node detection problem. Computers & Operations Research, 43:261-270, 2014.
Mingyu Xiao. Simple and improved parameterized algorithms for multiterminal cuts. Theory of Computing Systems, 46(4):723-736, 2010.
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Parameterized Complexity of Sparse Linear Complementarity Problems
In this paper, we study the parameterized complexity of the linear complementarity problem (LCP), which is one of the most fundamental mathematical optimization problems. The parameters we focus on are the sparsities of the input and the output of the LCP: the maximum numbers of nonzero entries per row/column in the coefficient matrix and the number of nonzero entries in a solution. Our main result is to present a fixed-parameter algorithm for the LCP with all the parameters. We also show that if we drop any of the three parameters, then the LCP is fixed-parameter intractable.
In addition, we discuss the nonexistence of a polynomial kernel for the LCP.
linear complementarity problem
sparsity
parameterized complexity
355-364
Regular Paper
Hanna
Sumita
Hanna Sumita
Naonori
Kakimura
Naonori Kakimura
Kazuhisa
Makino
Kazuhisa Makino
10.4230/LIPIcs.IPEC.2015.355
V. Arvind, J. Köbler, S. Kuhnert, and J.Torán. Solving linear equations parameterized by Hamming weight. In Proceedings of the 9th International Symposium on Parameterized and Exact Computation, pages 39-50, 2014.
H. Björklund, O. Svensson, and S. Vorobyov. Linear complementarity algorithms for mean payoff games. Technical Report 2005-05, DIMACS: Center for Discrete Mathematics and Theoretical Computer Science, 2005.
H. L. Bodlaender, R. G. Downey, M. R. Fellows, and D. Hermelin. On problems without polynomial kernels. Journal of Computer and System Sciences, 75:423-434, 2009.
X. Chen, X. Deng, and S. Teng. Sparse games are hard. In Proceedings of the 2nd International Workshop on Internet and Network Economics, pages 262-273, 2006.
X. Chen, X. Deng, and S. Teng. Settling the complexity of computing two-player Nash equilibria. Journal of the ACM, 56:14:1-14:57, 2009.
S. J. Chung. NP-completeness of the linear complementarity problem. Journal of Optimization Theory and Applications, 60:393-399, 1989.
B. Codenotti, M. Leoncini, and G. Resta. Efficient computation of Nash equilibria for very sparse win-lose bimatrix games. In Proceedings of the 14th Annual European Symposium on Algorithms, pages 232-243, 2006.
E. Cohen and N. Megiddo. Improved algorithms for linear inequalities with two variables per inequality. SIAM Journal on Computing, 23:1313-1347, 1994.
R. W. Cottle. The principal pivoting method of quadratic programming. In Mathematics of Decision Sciences, Part 1, pages 142-162. American Mathematical Society, Providence R. I., 1968.
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R. W. Cottle, J. S. Pang, and R. E. Stone. The Linear Complementarity Problem. Academic Press, Boston, 1992.
P. Damaschke. Sparse solutions of sparse linear systems: fixed-parameter tractability and an application of complex group testing. Theoretical Computer Science, 511:137-146, 2013.
C. Daskalakis and C. H. Papadimitriou. On oblivious PTAS’s for Nash equilibrium. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, pages 75-84, 2009.
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R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer, New York, 1999.
V. Estivill-Castro and M. Parsa. Computing Nash equilibria gets harder: new results show hardness even for parameterized complexity. In Proceedings of the 15th Australasian Symposium on Computing, pages 83-90, 2009.
J. Flum and M. Grohe. Parameterized Complexity Theory. Springer, Berlin, 2006.
I. Gilboa and E. Zemel. Nash and correlated equilibria: some complexity considerations. Games and Economic Behavior, 1:80-93, 1989.
D. Hermelin, C. Huang, S. Kratsch, and M. Wahlström. Parameterized two-player Nash equilibrium. Algorithmica, 65:1-15, 2013.
D. S. Hochbaum and J. Naor. Simple and fast algorithms for linear and integer programs with two variables per inequality. SIAM Journal on Computing, 23:1179-1192, 1994.
S. Kratsch. On polynomial kernels for integer linear programs: covering, packing and feasibility. In Proceedings of the 21st Annual European Symposium, pages 647-658, 2013.
S. Kratsch. On polynomial kernels for sparse integer linear programs. In Proceedings of the 30th Symposium on Theoretical Aspects of Computer Science, pages 80-91, 2013.
S. Kratsch and V. A. Quyen. On kernels for covering and packing ILPs with small coefficients. In Proceedings of the 9th International Symposium on Parameterized and Exact Computation, pages 307-318, 2014.
C. E. Lemke. Bimatrix equilibrium points and mathematical programming. Management Science, 11:681-689, 1965.
K. G. Murty. Linear Complementarity, Linear and Nonlinear Programming. Internet Edition, 1997.
T. J. Schaefer. The complexity of satisfiability problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing, pages 216-226, 1978.
H. Sumita, N. Kakimura, and K. Makino. The linear complementarity problems with a few variables per constraint. Mathematics of Operations Research, to appear.
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Parameterized Lower Bound and Improved Kernel for Diamond-free Edge Deletion
A diamond is a graph obtained by removing an edge from a complete graph on four vertices. A graph is diamond-free if it does not contain an induced diamond. The Diamond-free Edge Deletion problem asks to find whether there exist at most k edges in the input graph whose deletion results in a diamond-free graph. The problem was proved to be NP-complete and a polynomial kernel of O(k^4) vertices was found by Fellows et. al. (Discrete Optimization, 2011).
In this paper, we give an improved kernel of O(k^3) vertices for Diamond-free Edge Deletion. We give an alternative proof of the NP-completeness of the problem and observe that it cannot be solved in time 2^{o(k)} * n^{O(1)}, unless the Exponential Time Hypothesis fails.
edge deletion problems
polynomial kernelization
365-376
Regular Paper
R. B.
Sandeep
R. B. Sandeep
Naveen
Sivadasan
Naveen Sivadasan
10.4230/LIPIcs.IPEC.2015.365
N. R. Aravind, R. B. Sandeep, and Naveen Sivadasan. On polynomial kernelization of ℋ-free edge deletion. In Parameterized and Exact Computation, pages 28-38. Springer International Publishing, 2014.
Nikhil Bansal, Avrim Blum, and Shuchi Chawla. Correlation clustering. Machine Learning, 56(1-3):89-113, 2004.
Hans L. Bodlaender and Babette van Antwerpen-de Fluiter. On intervalizing k-colored graphs for DNA physical mapping. Discrete Applied Mathematics, 71(1-3):55-77, 1996.
Leizhen Cai. Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett., 58(4):171-176, 1996.
Leizhen Cai and Yufei Cai. Incompressibility of H-free edge modification problems. Algorithmica, 71(3):731-757, 2015.
Yufei Cai. Polynomial kernelisation of H-free edge modification problems. Mphil thesis, Department of Computer Science and Engineering, The Chinese University of Hong Kong, Hong Kong SAR, China, 2012.
Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk, Erik Jan van Leeuwen, and Marcin Wrochna. Polynomial kernelization for removing induced claws and diamonds. In WG, 2015.
Pål Grønås Drange and Michał Pilipczuk. A polynomial kernel for trivially perfect editing. In ESA, 2015.
Ehab S El-Mallah and Charles J Colbourn. The complexity of some edge deletion problems. Circuits and Systems, IEEE Transactions on, 35(3):354-362, 1988.
A Farrugia. Clique-helly graphs and hereditary clique-helly graphs, a mini-survey. Algoritmic graph theory (CS 762)-2002-Project, Dept. of Comb., University of Waterloo, 2002.
Michael R. Fellows, Jiong Guo, Christian Komusiewicz, Rolf Niedermeier, and Johannes Uhlmann. Graph-based data clustering with overlaps. Discrete Optimization, 8(1):2-17, 2011.
M. R. Garey, David S. Johnson, and Larry J. Stockmeyer. Some simplified NP-complete graph problems. Theor. Comput. Sci., 1(3):237-267, 1976.
Paul W. Goldberg, Martin Charles Golumbic, Haim Kaplan, and Ron Shamir. Four strikes against physical mapping of DNA. Journal of Computational Biology, 2(1):139-152, 1995.
Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001.
Stefan Kratsch and Magnus Wahlström. Two edge modification problems without polynomial kernels. Discrete Optimization, 10(3):193-199, 2013.
Bojan Mohar. Face covers and the genus problem for apex graphs. J. Comb. Theory, Ser. B, 82(1):102-117, 2001.
Stephan Olariu. Paw-fee graphs. Inf. Process. Lett., 28(1):53-54, 1988.
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On Kernelization and Approximation for the Vector Connectivity Problem
In the Vector Connectivity problem we are given an undirected graph G=(V,E), a demand function phi: V => {0,...,d}, and an integer k. The question is whether there exists a set S of at most k vertices such that every vertex v in V\S has at least phi(v) vertex-disjoint paths to S; this abstractly captures questions about placing servers in a network, or warehouses on a map, relative to demands. The problem is NP-hard already for instances with d=4 (Cicalese et al., Theor. Comput. Sci. 2015), admits a log-factor approximation (Boros et al., Networks 2014), and is fixed-parameter tractable in terms of k (Lokshtanov, unpublished 2014).
We prove several results regarding kernelization and approximation for Vector Connectivity and the variant Vector d-Connectivity where the upper bound d on demands is a constant. For Vector d-Connectivity we give a factor d-approximation algorithm and construct a vertex-linear kernelization, i.e., an efficient reduction to an equivalent instance with f(d)k=O(k) vertices. For Vector Connectivity we get a factor opt-approximation and we show that it has no kernelization to size polynomial in k+d unless NP \subseteq coNP/poly, making f(d)\poly(k) optimal for Vector d-Connectivity. Finally, we provide a write-up for fixed-parameter tractability of Vector Connectivity(k) by giving a different algorithm based on matroid intersection.
parameterized complexity
kernelization
approximation
377-388
Regular Paper
Stefan
Kratsch
Stefan Kratsch
Manuel
Sorge
Manuel Sorge
10.4230/LIPIcs.IPEC.2015.377
Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM J. Discrete Math., 28(1):277-305, 2014.
Endre Boros, Pinar Heggernes, Pim van 't Hof, and Martin Milanič. Vector connectivity in graphs. Networks, 63(4):277-285, 2014.
Ferdinando Cicalese, Martin Milanič, and Romeo Rizzi. On the complexity of the vector connectivity problem. Theor. Comput. Sci., 591:60-71, 2015.
Michael Dom, Daniel Lokshtanov, and Saket Saurabh. Kernelization lower bounds through colors and IDs. ACM T. Alg., 11(2):13:1-13:20, 2014.
Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Linear kernels for (connected) dominating set on graphs with excluded topological subgraphs. In Proc. 30th STACS, pages 92-103. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2013.
Danny Hermelin, Stefan Kratsch, Karolina Soltys, Magnus Wahlström, and Xi Wu. A completeness theory for polynomial (Turing) kernelization. Algorithmica, 71(3):702-730, 2015.
Stasys Jukna. Extremal combinatorics - with applications in computer science. Texts in theoretical computer science. Springer, 2001.
Dàniel Marx. A parameterized view on matroid optimization problems. Theor. Comput. Sci., 410(44):4471-4479, 2009.
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B-Chromatic Number: Beyond NP-Hardness
The b-chromatic number of a graph G, chi_b(G), is the largest integer k such that G has a k-vertex coloring with the property that each color class has a vertex which is adjacent to at least one vertex in each of the other color classes. In the B-Chromatic Number problem, the objective is to decide whether chi_b(G) >= k. Testing whether chi_b(G)=Delta(G)+1, where Delta(G) is the maximum degree of a graph, itself is NP-complete even for connected bipartite graphs (Kratochvil, Tuza and Voigt, WG 2002). In this paper we study B-Chromatic Number in the realm of parameterized complexity and exact exponential time algorithms. We show that B-Chromatic Number is W[1]-hard when parameterized by k, resolving the open question posed by Havet and Sampaio (Algorithmica 2013). When k=Delta(G)+1, we design an algorithm for B-Chromatic Number running in time 2^{O(k^2 * log(k))}*n^{O(1)}. Finally, we show that B-Chromatic Number for an n-vertex graph can be solved in time O(3^n * n^{4} * log(n)).
b-chromatic number
exact algorithm
parameterized complexity
389-401
Regular Paper
Fahad
Panolan
Fahad Panolan
Geevarghese
Philip
Geevarghese Philip
Saket
Saurabh
Saket Saurabh
10.4230/LIPIcs.IPEC.2015.389
Kenneth Appel and Wolfgang Haken. The solution of the four-color-map problem. Sci Am, 237(4):108-121, October 1977.
Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set partitioning via inclusion-exclusion. SIAM J. Computing, 39(2):546-563, 2009.
Manfred Cochefert. Algorithmes Exacts et Exponentiels pour les Problèmes NP-difficiles sur les Graphes et Hypergraphes. PhD thesis, Université de Lorraine, December 2014.
Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013.
Brice Effantin, Nicolas Gastineau, and Olivier Togni. A characterization of b-chromatic and partial grundy numbers by induced subgraphs. CoRR, abs/1505.07780, 2015.
Frantisek Galcík and Ján Katrenic. A note on approximating the b-chromatic number. Discrete Applied Mathematics, 161(7-8):1137-1140, 2013.
Frédéric Havet, Cláudia Linhares Sales, and Leonardo Sampaio. b-coloring of tight graphs. Discrete Applied Mathematics, 160(18):2709-2715, 2012.
Frédéric Havet and Leonardo Sampaio. On the grundy and b-chromatic numbers of a graph. Algorithmica, 65(4):885-899, 2013.
Robert W. Irving and David Manlove. The b-chromatic number of a graph. Discrete Applied Mathematics, 91(1-3):127-141, 1999.
Richard M. Karp. Reducibility among combinatorial problems. In Complexity of computer computations, pages 85-103. Plenum Press, New York, 1972.
Jan Kratochvíl, Zsolt Tuza, and Margit Voigt. On the b-chromatic number of graphs. In Ludek Kucera, editor, Graph-Theoretic Concepts in Computer Science, 28th International Workshop, WG 2002, Cesky Krumlov, Czech Republic, June 13-15, 2002, Revised Papers, volume 2573 of Lecture Notes in Computer Science, pages 310-320. Springer, 2002.
E. L. Lawler. A note on the complexity of the chromatic number problem. Information Processing Lett., 5(3):66-67, 1976.
Arnold Schönhage and Volker Strassen. Schnelle Multiplikation großer Zahlen. Computing, 7(3-4):281-292, 1971.
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Fast Biclustering by Dual Parameterization
We study two clustering problems, Starforest Editing, the problem of adding and deleting edges to obtain a disjoint union of stars, and the generalization Bicluster Editing. We show that, in addition to being NP-hard, none of the problems can be solved in subexponential time unless the exponential time hypothesis fails.
Misra, Panolan, and Saurabh (MFCS 2013) argue that introducing a bound on the number of connected components in the solution should not make the problem easier: In particular, they argue that the subexponential time algorithm for editing to a fixed number of clusters (p-Cluster Editing) by Fomin et al. (J. Comput. Syst. Sci., 80(7) 2014) is an exception rather than the rule. Here, p is a secondary parameter, bounding the number of components in the solution.
However, upon bounding the number of stars or bicliques in the solution, we obtain algorithms which run in time O(2^{3*sqrt(pk)} + n + m) for p-Starforest Editing and O(2^{O(p * sqrt(k) * log(pk))} + n + m) for p-Bicluster Editing. We obtain a similar result for the more general case of t-Partite p-Cluster Editing. This is subexponential in k for a fixed number of clusters, since p is then considered a constant.
Our results even out the number of multivariate subexponential time algorithms and give reasons to believe that this area warrants further study.
graph editing
subexponential algorithms
parameterized complexity
402-413
Regular Paper
Pål Grønås
Drange
Pål Grønås Drange
Felix
Reidl
Felix Reidl
Fernando
Sánchez Villaamil
Fernando Sánchez Villaamil
Somnath
Sikdar
Somnath Sikdar
10.4230/LIPIcs.IPEC.2015.402
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