Volume
LIPIcs, Volume 66
34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Theoretical Aspects of Computer Science (STACS)
Event
STACS 2017, March 8-11, 2017, Hannover, GermanyEditors
Publication Details
- published at: 2017-03-06
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-028-6
- DBLP: db/conf/stacs/stacs2017
Access Numbers
- Detailed Access Statistics available here
-
Total Document Accesses (updated on a weekly basis):
0PDF Downloads
Documents
No documents found matching your filter selection.
Document
Complete Volume
LIPIcs, Volume 66, STACS'17, Complete Volume
Abstract
LIPIcs, Volume 66, STACS'17, Complete Volume
Cite as
34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@Proceedings{vollmer_et_al:LIPIcs.STACS.2017,
title = {{LIPIcs, Volume 66, STACS'17, Complete Volume}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017},
URN = {urn:nbn:de:0030-drops-70366},
doi = {10.4230/LIPIcs.STACS.2017},
annote = {Keywords: Models of Computation, Nonnumerical Algorithms and Problems, Mathematical Logic, Formal Languages, Combinatorics, Graph Theory}
}
Document
Front Matter
Front Matter, Table of Contents, Foreword, Conference Organization, External Reviewers
Abstract
Front Matter, Table of Contents, Foreword, Conference Organization, External Reviewers
Cite as
34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{vollmer_et_al:LIPIcs.STACS.2017.0,
author = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
title = {{Front Matter, Table of Contents, Foreword, Conference Organization, External Reviewers}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {0:i--0:xvi},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.0},
URN = {urn:nbn:de:0030-drops-70327},
doi = {10.4230/LIPIcs.STACS.2017.0},
annote = {Keywords: Front Matter, Table of Contents, Foreword, Conference Organization, External Reviewers}
}
Document
Tutorial
Computational Aspects of Logics in Team Semantics (Tutorial)
Abstract
Team Semantics is a logical framework for the study of various dependency notions that are important in many areas of science. The starting point of this research is marked by the publication of the monograph Dependence Logic (Jouko Väänänen, 2007) in which first-order dependence logic is developed and studied. Since then team semantics has evolved into a flexible framework in which numerous logics have been studied.
Much of the work in team semantics has so far focused on results concerning either axiomatic characterizations or the expressive power and computational aspects of various logics. This tutorial provides an introduction to team semantics with a focus on results regarding expressivity and computational aspects of the most prominent logics of the area. In particular, we discuss dependence, independence and inclusion logics in first-order, propositional, and modal team semantics. We show that first-order dependence and independence logic are equivalent with existential second-order logic and inclusion logic with greatest fixed point logic. In the propositional and modal settings we characterize the expressive power of these logics by so-called team bisimulations and determine the complexity of their model checking and satisfiability problems.
Cite as
Juha Kontinen. Computational Aspects of Logics in Team Semantics (Tutorial). In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{kontinen:LIPIcs.STACS.2017.1,
author = {Kontinen, Juha},
title = {{Computational Aspects of Logics in Team Semantics}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {1:1--1:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.1},
URN = {urn:nbn:de:0030-drops-70333},
doi = {10.4230/LIPIcs.STACS.2017.1},
annote = {Keywords: team semantics, dependence logic, model checking, satisfiability problem, team bisimulation}
}
Document
Invited Talk
Recompression: New Approach to Word Equations and Context Unification (Invited Talk)
Abstract
Word equations is given by two strings over disjoint alphabets of letters and variables and we ask whether there is a substitution that satisfies this equation. Recently, a new PSPACE solution to this problem was proposed, it is based on compressing simple substrings of the equation and modifying the equation so that such operations are sound. The analysis focuses on the way the equation is stored and changed rather than on the combinatorics of words. This approach greatly simplified many existing proofs and algorithms. In particular, unlike the previous solutions, it generalises to equations over contexts (known for historical reasons as context unification): contexts are terms with one special symbol that represent a missing argument and they can be applied on terms, in which case their argument replaces the special constant.
Cite as
Artur Jez. Recompression: New Approach to Word Equations and Context Unification (Invited Talk). In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 2:1-2:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{jez:LIPIcs.STACS.2017.2,
author = {Jez, Artur},
title = {{Recompression: New Approach to Word Equations and Context Unification}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {2:1--2:3},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.2},
URN = {urn:nbn:de:0030-drops-70280},
doi = {10.4230/LIPIcs.STACS.2017.2},
annote = {Keywords: Word equations, exponent of periodicity, semantic unification, string unification, context unification, compression}
}
Document
Invited Talk
Discrete Logarithms in Small Characteristic Finite Fields: a Survey of Recent Advances (Invited Talk)
Abstract
The discrete logarithm problem is one of the few hard problems on which public-key cryptography can be based. It was introduced in the field by the famous Diffie-Hellman key exchange protocol. Initially, the cryptographic use of the problem was considered in prime fields, but was readily generalized to arbitrary finite fields and, later, to elliptic or higher genus curves.
In this talk, we survey the key technical ideas that can be used to compute discrete logarithms, especially in the case of small characteristic finite fields. These ideas stem from about 40 years of research on the topic. They appeared along the long road that leads from the initial belief that this problem was hard enough for cryptographic purpose to the current state of the art where it can no longer be considered for cryptographic use. Indeed, after the recent developments started in 2012, we now have some very efficient practical algorithms to solve this problem. Unfortunately, these algorithms remain heuristic and one important direction for future research is to lift the remaining heuristic assumptions.
Cite as
Antoine Joux. Discrete Logarithms in Small Characteristic Finite Fields: a Survey of Recent Advances (Invited Talk). In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{joux:LIPIcs.STACS.2017.3,
author = {Joux, Antoine},
title = {{Discrete Logarithms in Small Characteristic Finite Fields: a Survey of Recent Advances}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {3:1--3:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.3},
URN = {urn:nbn:de:0030-drops-70313},
doi = {10.4230/LIPIcs.STACS.2017.3},
annote = {Keywords: Cryptography, Discrete logarithms, Finite fields}
}
Document
Invited Talk
Applications of Algorithmic Metatheorems to Space Complexity and Parallelism (Invited Talk)
Abstract
Algorithmic metatheorems state that if a problem can be described in a certain logic and the inputs are structured in a certain way, then the problem can be solved with a certain amount of resources. As an example, by Courcelle's Theorem all monadic second-order ("in a certain logic") properties of graphs of bounded tree width ("structured in a certain way") can be solved in linear time ("with a certain amount of resources"). Such theorems have become a valuable tool in algorithmics: If a problem happens to have the right structure and can be described in the right logic, they immediately yield a (typically tight) upper bound on the time complexity of the problem. Perhaps even more importantly, several complex algorithms rely on algorithmic metatheorems internally to solve subproblems, which considerably broadens the range of applications of these theorems. The talk is intended as a gentle introduction to the ideas behind algorithmic metatheorems, especially behind some recent results concerning space classes and parallel computation, and tries to give a flavor of the range of
Cite as
Till Tantau. Applications of Algorithmic Metatheorems to Space Complexity and Parallelism (Invited Talk). In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 4:1-4:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{tantau:LIPIcs.STACS.2017.4,
author = {Tantau, Till},
title = {{Applications of Algorithmic Metatheorems to Space Complexity and Parallelism}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {4:1--4:4},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.4},
URN = {urn:nbn:de:0030-drops-70303},
doi = {10.4230/LIPIcs.STACS.2017.4},
annote = {Keywords: Algorithmic metatheorems, Courcelle’s Theorem, tree width, monadic second-order logic, logarithmic space, parallel computations}
}
Document
Split Contraction: The Untold Story
Abstract
The edit operation that contracts edges, which is a fundamental operation in the theory of graph minors, has recently gained substantial scientific attention from the viewpoint of Parameterized Complexity. In this paper, we examine an important family of graphs, namely the family of split graphs, which in the context of edge contractions, is proven to be significantly less obedient than one might expect. Formally, given a graph G and an integer k, the Split Contraction problem asks whether there exists a subset X of edges of G such that G/X is a split graph and X has at most k elements. Here, G/X is the graph obtained from G by contracting edges in X. It was previously claimed that the Split Contraction problem is fixed-parameter tractable. However, we show that, despite its deceptive simplicity, it is W[1]-hard. Our main result establishes the following conditional lower bound: under the Exponential Time Hypothesis, the Split Contraction problem cannot be solved in time 2^(o(l^2)) * poly(n) where l is the vertex cover number of the input graph. We also verify that this lower bound is essentially tight. To the best of our knowledge, this is the first tight lower bound of the form 2^(o(l^2)) * poly(n) for problems parameterized by the vertex cover number of the input graph. In particular, our approach to obtain this lower bound borrows the notion of harmonious coloring from Graph Theory, and might be of independent interest.
Cite as
Akanksha Agrawal, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Split Contraction: The Untold Story. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{agrawal_et_al:LIPIcs.STACS.2017.5,
author = {Agrawal, Akanksha and Lokshtanov, Daniel and Saurabh, Saket and Zehavi, Meirav},
title = {{Split Contraction: The Untold Story}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {5:1--5:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.5},
URN = {urn:nbn:de:0030-drops-70297},
doi = {10.4230/LIPIcs.STACS.2017.5},
annote = {Keywords: Split Graph, Parameterized Complexity, Edge Contraction}
}
Document
The Operator Approach to Entropy Games
Abstract
Entropy games and matrix multiplication games have been recently introduced by Asarin et al. They model the situation in which one player (Despot) wishes to minimize the growth rate of a matrix product, whereas the other player (Tribune) wishes to maximize it. We develop an operator approach to entropy games. This allows us to show that entropy games can be cast as stochastic mean payoff games in which some action spaces are simplices and payments are given by a relative entropy (Kullback-Leibler divergence). In this way, we show that entropy games with a fixed number of states belonging to Despot can be solved in polynomial time. This approach also allows us to solve these games by a policy iteration algorithm, which we compare with the spectral simplex algorithm developed by Protasov.
Cite as
Marianne Akian, Stéphane Gaubert, Julien Grand-Clément, and Jérémie Guillaud. The Operator Approach to Entropy Games. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{akian_et_al:LIPIcs.STACS.2017.6,
author = {Akian, Marianne and Gaubert, St\'{e}phane and Grand-Cl\'{e}ment, Julien and Guillaud, J\'{e}r\'{e}mie},
title = {{The Operator Approach to Entropy Games}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {6:1--6:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.6},
URN = {urn:nbn:de:0030-drops-70260},
doi = {10.4230/LIPIcs.STACS.2017.6},
annote = {Keywords: Stochastic games, Shapley operators, policy iteration, Perron eigenvalues, Risk sensitive control}
}
Document
Parameterized Complexity of Small Weight Automorphisms
Abstract
We show that checking if a given hypergraph has an automorphism that moves exactly k vertices is fixed parameter tractable, using k and additionally either the maximum hyperedge size or the maximum color class size as parameters. In particular, it suffices to use k as parameter if the hyperedge size is at most polylogarithmic in the size of the given hypergraph.
As a building block for our algorithms, we generalize Schweitzer's FPT algorithm [ESA 2011] that, given two graphs on the same vertex set and a parameter k, decides whether there is an isomorphism between the two graphs that moves at most k vertices. We extend this result to hypergraphs, using the maximum hyperedge size as a second parameter.
Another key component of our algorithm is an orbit-shrinking technique that preserves permutations that move few points and that may be of independent interest. Applying it to a suitable subgroup of the automorphism group allows us to switch from bounded hyperedge size to bounded color classes in the exactly-k case.
Cite as
Vikraman Arvind, Johannes Köbler, Sebastian Kuhnert, and Jacobo Torán. Parameterized Complexity of Small Weight Automorphisms. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{arvind_et_al:LIPIcs.STACS.2017.7,
author = {Arvind, Vikraman and K\"{o}bler, Johannes and Kuhnert, Sebastian and Tor\'{a}n, Jacobo},
title = {{Parameterized Complexity of Small Weight Automorphisms}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {7:1--7:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.7},
URN = {urn:nbn:de:0030-drops-70278},
doi = {10.4230/LIPIcs.STACS.2017.7},
annote = {Keywords: Parameterized algorithms, hypergraph isomorphism.}
}
Document
What Can Be Verified Locally?
Abstract
We are considering distributed network computing, in which computing entities are connected by a network modeled as a connected graph. These entities are located at the nodes of the graph, and they exchange information by message-passing along its edges. In this context, we are adopting the classical framework for local distributed decision, in which nodes must collectively decide whether their network configuration satisfies some given boolean predicate, by having each node interacting with the nodes in its vicinity only. A network configuration is accepted if and only if every node individually accepts. It is folklore that not every Turing-decidable network property (e.g., whether the network is planar) can be decided locally whenever the computing entities are Turing machines (TM). On the other hand, it is known that every Turing-decidable network property can be decided locally if nodes are running non-deterministic Turing machines (NTM). However, this holds only if the nodes have the ability to guess the identities of the nodes currently in the network. That is, for different sets of identities assigned to the nodes, the correct guesses of the nodes might be different. If one asks the nodes to use the same guess in the same network configuration even with different identity assignments, i.e., to perform identity-oblivious guesses, then it is known that not every Turing-decidable network property can be decided locally.
In this paper, we show that every Turing-decidable network property can be decided locally if nodes are running alternating Turing machines (ATM), and this holds even if nodes are bounded to perform identity-oblivious guesses. More specifically, we show that, for every network property, there is a local algorithm for ATMs, with at most 2 alternations, that decides that property. To this aim, we define a hierarchy of classes of decision tasks where the lowest level contains tasks solvable with TMs, the first level those solvable with NTMs, and level k contains those tasks solvable with ATMs with k alternations. We characterize the entire hierarchy, and show that it collapses in the second level. In addition, we show separation results between the classes of network properties that are locally decidable with TMs, NTMs, and ATMs. Finally, we establish the existence of completeness results for each of these classes, using novel notions of local reduction.
Cite as
Alkida Balliu, Gianlorenzo D'Angelo, Pierre Fraigniaud, and Dennis Olivetti. What Can Be Verified Locally?. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{balliu_et_al:LIPIcs.STACS.2017.8,
author = {Balliu, Alkida and D'Angelo, Gianlorenzo and Fraigniaud, Pierre and Olivetti, Dennis},
title = {{What Can Be Verified Locally?}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {8:1--8:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.8},
URN = {urn:nbn:de:0030-drops-70253},
doi = {10.4230/LIPIcs.STACS.2017.8},
annote = {Keywords: Distributed Network Computing, Distributed Algorithm, Distributed Decision, Locality}
}
Document
Improved Time-Space Trade-Offs for Computing Voronoi Diagrams
Abstract
Let P be a planar n-point set in general position. For k between 1 and n-1, the Voronoi diagram of order k is obtained by subdividing the plane into regions such that points in the same cell have the same set of nearest k neighbors in P. The (nearest point) Voronoi diagram (NVD) and the farthest point Voronoi diagram (FVD) are the particular cases of k=1 and k=n-1, respectively. It is known that the family of all higher-order Voronoi diagrams of order 1 to K for P can be computed in total time O(n K^2 + n log n) using O(K^2(n-K)) space. Also NVD and FVD can be computed in O(n log n) time using O(n) space.
For s in {1, ..., n}, an s-workspace algorithm has random access to a read-only array with the sites of P in arbitrary order. Additionally, the algorithm may use O(s) words of Theta(log n) bits each for reading and writing intermediate data. The output can be written only once and cannot be accessed afterwards.
We describe a deterministic s-workspace algorithm for computing an NVD and also an FVD for P that runs in O((n^2/s) log s) time. Moreover, we generalize our s-workspace algorithm for computing the family of all higher-order Voronoi diagrams of P up to order K in O(sqrt(s)) in total time O( (n^2 K^6 / s) log^(1+epsilon)(K) (log s / log K)^(O(1)) ) for any fixed epsilon > 0. Previously, for Voronoi diagrams, the only known s-workspace algorithm was to find an NVD for P in expected time O((n^2/s) log s + n log s log^*s). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques.
Cite as
Bahareh Banyassady, Matias Korman, Wolfgang Mulzer, André van Renssen, Marcel Roeloffzen, Paul Seiferth, and Yannik Stein. Improved Time-Space Trade-Offs for Computing Voronoi Diagrams. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{banyassady_et_al:LIPIcs.STACS.2017.9,
author = {Banyassady, Bahareh and Korman, Matias and Mulzer, Wolfgang and van Renssen, Andr\'{e} and Roeloffzen, Marcel and Seiferth, Paul and Stein, Yannik},
title = {{Improved Time-Space Trade-Offs for Computing Voronoi Diagrams}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {9:1--9:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.9},
URN = {urn:nbn:de:0030-drops-70249},
doi = {10.4230/LIPIcs.STACS.2017.9},
annote = {Keywords: memory-constrained model, Voronoi diagram, time-space trade-off}
}
Document
Energy-Efficient Delivery by Heterogeneous Mobile Agents
Abstract
We consider the problem of delivering m messages between specified source-target pairs in an undirected graph, by k mobile agents initially located at distinct nodes of the graph. Each edge has a designated length and each agent consumes energy proportional to the distance it travels in the graph. We are interested in optimizing the total energy consumption for the team of agents. Unlike previous related work, we consider heterogeneous agents with different rates of energy consumption (weights w_i). To solve the delivery problem, agents face three major challenges: Collaboration (how to work together on each message), Planning (which route to take) and Coordination (how to assign agents to messages).
We first show that the delivery problem can be 2-approximated without collaborating and that this is best possible, i.e., we show that the benefit of collaboration is 2 in general. We also show that the benefit of collaboration for a single message is 1 / log 2 which is approximately 1.44. Planning turns out to be NP-hard to approximate even for a single agent, but can be 2-approximated in polynomial time if agents have unit capacities and do not collaborate. We further show that coordination is NP-hard even for agents with unit capacity, but can be efficiently solved exactly if they additionally have uniform weights. Finally, we give a polynomial-time c-approximation for message delivery with unit capacities.
Cite as
Andreas Bärtschi, Jérémie Chalopin, Shantanu Das, Yann Disser, Daniel Graf, Jan Hackfeld, and Paolo Penna. Energy-Efficient Delivery by Heterogeneous Mobile Agents. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{bartschi_et_al:LIPIcs.STACS.2017.10,
author = {B\"{a}rtschi, Andreas and Chalopin, J\'{e}r\'{e}mie and Das, Shantanu and Disser, Yann and Graf, Daniel and Hackfeld, Jan and Penna, Paolo},
title = {{Energy-Efficient Delivery by Heterogeneous Mobile Agents}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {10:1--10:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.10},
URN = {urn:nbn:de:0030-drops-70233},
doi = {10.4230/LIPIcs.STACS.2017.10},
annote = {Keywords: message delivery, mobile agents, energy optimization, approximation algorithms}
}
Document
Towards Tighter Space Bounds for Counting Triangles and Other Substructures in Graph Streams
Abstract
We revisit the much-studied problem of space-efficiently estimating the number of triangles in a graph stream, and extensions of this problem to counting fixed-sized cliques and cycles. For the important special case of counting triangles, we give a 4-pass, (1 +/- epsilon)-approximate, randomized algorithm using O-tilde(epsilon^(-2) m^(3/2) / T) space, where m is the number of edges and T is a promised lower bound on the number of triangles. This matches the space bound of a recent algorithm (McGregor et al., PODS 2016), with an arguably simpler and more general technique. We give an improved multi-pass lower bound of Omega(min{m^(3/2)/T , m/sqrt(T)}), applicable at essentially all densities Omega(n) <= m <= O(n^2). We prove other multi-pass lower bounds in terms of various structural parameters of the input graph. Together, our results resolve a couple of open questions raised in recent work (Braverman et al., ICALP 2013).
Our presentation emphasizes more general frameworks, for both upper and lower bounds. We give a sampling algorithm for counting arbitrary subgraphs and then improve it via combinatorial means in the special cases of counting odd cliques and odd cycles. Our results show that these problems are considerably easier in the cash-register streaming model than in the turnstile model, where previous work had focused. We use Turán graphs and related gadgets to derive lower bounds for counting cliques and cycles, with triangle-counting lower bounds following as a corollary.
Cite as
Suman K. Bera and Amit Chakrabarti. Towards Tighter Space Bounds for Counting Triangles and Other Substructures in Graph Streams. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{bera_et_al:LIPIcs.STACS.2017.11,
author = {Bera, Suman K. and Chakrabarti, Amit},
title = {{Towards Tighter Space Bounds for Counting Triangles and Other Substructures in Graph Streams}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {11:1--11:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.11},
URN = {urn:nbn:de:0030-drops-70222},
doi = {10.4230/LIPIcs.STACS.2017.11},
annote = {Keywords: data streaming, graph algorithms, triangles, subgraph counting, lower bounds}
}
Document
On Polynomial Approximations Over Z/2^kZ*
Abstract
We study approximation of Boolean functions by low-degree polynomials over the ring Z/2^kZ. More precisely, given a Boolean function F:{0,1}^n -> {0,1}, define its k-lift to be F_k:{0,1}^n -> {0,2^(k-1)} by F_k(x) = 2^(k-F(x)) (mod 2^k). We consider the fractional agreement (which we refer to as \gamma_{d,k}(F)) of F_k with degree d polynomials from Z/2^kZ[x_1,..,x_n].
Our results are the following:
* Increasing k can help: We observe that as k increases, gamma_{d,k}(F) cannot decrease. We give two kinds of examples where gamma_{d,k}(F) actually increases. The first is an infinite family of functions F such that gamma_{2d,2}(F) - gamma_{3d-1,1}(F) >= Omega(1). The second is an infinite family of functions F such that gamma_{d,1}(F) <= 1/2+o(1) - as small as possible - but gamma_{d,3}(F) >= 1/2 + Omega(1).
* Increasing k doesn't always help: Adapting a proof of Green [Comput. Complexity, 9(1):16--38, 2000], we show that irrespective of the value of k, the Majority function Maj_n satisfies gamma_{d,k}(Maj_n) <= 1/2+ O(d)/sqrt{n}. In other words, polynomials over Z/2^kZ for large k do not approximate the majority function any better than polynomials over Z/2Z.
We observe that the model we study subsumes the model of non-classical polynomials, in the sense that proving bounds in our model implies bounds on the agreement of non-classical polynomials with Boolean functions. In particular, our results answer questions raised by Bhowmick and Lovett [In Proc. 30th Computational Complexity Conf., pages 72-87, 2015] that ask whether non-classical polynomials approximate Boolean functions better than classical polynomials of the same degree.
Cite as
Abhishek Bhrushundi, Prahladh Harsha, and Srikanth Srinivasan. On Polynomial Approximations Over Z/2^kZ*. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 12:1-12:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{bhrushundi_et_al:LIPIcs.STACS.2017.12,
author = {Bhrushundi, Abhishek and Harsha, Prahladh and Srinivasan, Srikanth},
title = {{On Polynomial Approximations Over Z/2^kZ*}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {12:1--12:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.12},
URN = {urn:nbn:de:0030-drops-70212},
doi = {10.4230/LIPIcs.STACS.2017.12},
annote = {Keywords: Polynomials over rings, Approximation by polynomials, Boolean functions, Non-classical polynomials}
}
Document
Existential-R-Complete Decision Problems about Symmetric Nash Equilibria in Symmetric Multi-Player Games
Abstract
We study the complexity of decision problems about symmetric Nash equilibria for symmetric multi-player games. These decision problems concern the existence of a symmetric Nash equilibrium with certain natural properties. We show that a handful of such decision problems are Existential-R-complete; that is, they are exactly as hard as deciding the Existential Theory of the Reals.
Cite as
Vittorio Bilò and Marios Mavronicolas. Existential-R-Complete Decision Problems about Symmetric Nash Equilibria in Symmetric Multi-Player Games. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{bilo_et_al:LIPIcs.STACS.2017.13,
author = {Bil\`{o}, Vittorio and Mavronicolas, Marios},
title = {{Existential-R-Complete Decision Problems about Symmetric Nash Equilibria in Symmetric Multi-Player Games}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {13:1--13:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.13},
URN = {urn:nbn:de:0030-drops-70200},
doi = {10.4230/LIPIcs.STACS.2017.13},
annote = {Keywords: Nash equilibrium, complexity of equilibria, ExistentialR-completeness}
}
Document
On Büchi One-Counter Automata
Abstract
Equivalence of deterministic pushdown automata is a famous problem in theoretical computer science whose decidability has been shown by Sénizergues. Our first result shows that decidability no longer holds when moving from finite words to infinite words. This solves an open problem that has recently been raised by Löding. In fact, we show that already the equivalence problem for deterministic Büchi one-counter automata is undecidable. Hence, the decidability border is rather tight when taking into account a recent result by Löding and Repke that equivalence of deterministic weak parity pushdown automata (a subclass of deterministic Büchi pushdown automata) is decidable.
Another known result on finite words is that the universality problem for vector addition systems is decidable. We show undecidability when moving to infinite words. In fact, we prove that already the universality problem for nondeterministic Büchi one-counter nets (or equivalently vector addition systems with one unbounded dimension) is undecidable.
Cite as
Stanislav Böhm, Stefan Göller, Simon Halfon, and Piotr Hofman. On Büchi One-Counter Automata. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{bohm_et_al:LIPIcs.STACS.2017.14,
author = {B\"{o}hm, Stanislav and G\"{o}ller, Stefan and Halfon, Simon and Hofman, Piotr},
title = {{On B\"{u}chi One-Counter Automata}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {14:1--14:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.14},
URN = {urn:nbn:de:0030-drops-70194},
doi = {10.4230/LIPIcs.STACS.2017.14},
annote = {Keywords: infinite words, deterministic pushdown automata}
}
Document
Optimizing Tree Decompositions in MSO
Abstract
The classic algorithm of Bodlaender and Kloks solves the following problem in linear fixed-parameter time: given a tree decomposition of a graph of (possibly suboptimal) width k, compute an optimum-width tree decomposition of the graph. In this work, we prove that this problem can also be solved in MSO in the following sense: for every positive integer k, there is an MSO transduction from tree decompositions of width k to tree decompositions of optimum width. Together with our recent results, this implies that for every k there exists an MSO transduction which inputs a graph of treewidth k, and nondeterministically outputs its tree decomposition of optimum width.
Cite as
Mikolaj Bojanczyk and Michal Pilipczuk. Optimizing Tree Decompositions in MSO. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{bojanczyk_et_al:LIPIcs.STACS.2017.15,
author = {Bojanczyk, Mikolaj and Pilipczuk, Michal},
title = {{Optimizing Tree Decompositions in MSO}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {15:1--15:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.15},
URN = {urn:nbn:de:0030-drops-70173},
doi = {10.4230/LIPIcs.STACS.2017.15},
annote = {Keywords: tree decomposition, treewidth, transduction, monadic second-order logic}
}
Document
Complexity of Token Swapping and its Variants
Abstract
In the Token Swapping problem we are given a graph with a token placed on each vertex. Each token has exactly one destination vertex, and we try to move all the tokens to their destinations, using the minimum number of swaps, i.e., operations of exchanging the tokens on two adjacent vertices. As the main result of this paper, we show that Token Swapping is W[1]-hard parameterized by the length k of a shortest sequence of swaps. In fact, we prove that, for any computable function f, it cannot be solved in time f(k)*n^(o(k / log k)) where n is the number of vertices of the input graph, unless the ETH fails. This lower bound almost matches the trivial n^O(k)-time algorithm.
We also consider two generalizations of the Token Swapping, namely Colored Token Swapping (where the tokens have colors and tokens of the same color are indistinguishable), and Subset Token Swapping (where each token has a set of possible destinations). To complement the hardness result, we prove that even the most general variant, Subset Token Swapping, is FPT in nowhere-dense graph classes.
Finally, we consider the complexities of all three problems in very restricted classes of graphs: graphs of bounded treewidth and diameter, stars, cliques, and paths, trying to identify the borderlines between polynomial and NP-hard cases.
Cite as
Édouard Bonnet, Tillmann Miltzow, and Pawel Rzazewski. Complexity of Token Swapping and its Variants. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{bonnet_et_al:LIPIcs.STACS.2017.16,
author = {Bonnet, \'{E}douard and Miltzow, Tillmann and Rzazewski, Pawel},
title = {{Complexity of Token Swapping and its Variants}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {16:1--16:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.16},
URN = {urn:nbn:de:0030-drops-70185},
doi = {10.4230/LIPIcs.STACS.2017.16},
annote = {Keywords: token swapping, parameterized complexity, NP-hardness, W\lbrack1\rbrack-hardness}
}
Document
Monte Carlo Computability
Abstract
We introduce Monte Carlo computability as a probabilistic concept of computability on infinite objects and prove that Monte Carlo computable functions are closed under composition. We then mutually separate the following classes of functions from each other: the class of multi-valued functions that are non-deterministically computable, that of Las Vegas computable functions, and that of Monte Carlo computable functions. We give natural examples of computational problems witnessing these separations. As a specific problem which is Monte Carlo computable but neither Las Vegas computable nor non-deterministically computable, we study the problem of sorting infinite sequences that was recently introduced by Neumann and Pauly. Their results allow us to draw conclusions about the relation between algebraic models and Monte Carlo computability.
Cite as
Vasco Brattka, Rupert Hölzl, and Rutger Kuyper. Monte Carlo Computability. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{brattka_et_al:LIPIcs.STACS.2017.17,
author = {Brattka, Vasco and H\"{o}lzl, Rupert and Kuyper, Rutger},
title = {{Monte Carlo Computability}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {17:1--17:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.17},
URN = {urn:nbn:de:0030-drops-70164},
doi = {10.4230/LIPIcs.STACS.2017.17},
annote = {Keywords: Weihrauch degrees, Weak Weak Konig's Lemma, Monte Carlo computability, algorithmic randomness, sorting}
}
Document
The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex
Abstract
We consider the problem of finding a subcomplex K' of a simplicial complex K such that K' is homeomorphic to the 2-dimensional sphere, S^2. We study two variants of this problem. The first asks if there exists such a K' with at most k triangles, and we show that this variant is W[1]-hard and, assuming ETH, admits no O(n^(o(sqrt(k)))) time algorithm. We also give an algorithm that is tight with regards to this lower bound. The second problem is the dual of the first, and asks if K' can be found by removing at most k triangles from K. This variant has an immediate O(3^k poly(|K|)) time algorithm, and we show that it admits a polynomial kernelization to O(k^2) triangles, as well as a polynomial compression to a weighted version with bit-size O(k log k).
Cite as
Benjamin Burton, Sergio Cabello, Stefan Kratsch, and William Pettersson. The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{burton_et_al:LIPIcs.STACS.2017.18,
author = {Burton, Benjamin and Cabello, Sergio and Kratsch, Stefan and Pettersson, William},
title = {{The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {18:1--18:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.18},
URN = {urn:nbn:de:0030-drops-70156},
doi = {10.4230/LIPIcs.STACS.2017.18},
annote = {Keywords: computational topology, parameterized complexity, simplicial complex}
}
Document
On Long Words Avoiding Zimin Patterns
Abstract
A pattern is encountered in a word if some infix of the word is the image of the pattern under some non-erasing morphism. A pattern p is unavoidable if, over every finite alphabet, every sufficiently long word encounters p. A theorem by Zimin and independently by Bean, Ehrenfeucht and McNulty states that a pattern over n distinct variables is unavoidable if, and only if, p itself is encountered in the n-th Zimin pattern. Given an alphabet size k, we study the minimal length f(n,k) such that every word of length f(n,k) encounters the n-th Zimin pattern. It is known that f is upper-bounded by a tower of exponentials. Our main result states that f(n,k) is lower-bounded by a tower of n-3 exponentials, even for k=2. To the best of our knowledge, this improves upon a previously best-known doubly-exponential lower bound. As a further result, we prove a doubly-exponential upper bound for encountering Zimin patterns in the abelian sense.
Cite as
Arnaud Carayol and Stefan Göller. On Long Words Avoiding Zimin Patterns. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 19:1-19:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{carayol_et_al:LIPIcs.STACS.2017.19,
author = {Carayol, Arnaud and G\"{o}ller, Stefan},
title = {{On Long Words Avoiding Zimin Patterns}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {19:1--19:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.19},
URN = {urn:nbn:de:0030-drops-70140},
doi = {10.4230/LIPIcs.STACS.2017.19},
annote = {Keywords: Unavoidable patterns, combinatorics on words, lower bounds}
}
Document
Extended Learning Graphs for Triangle Finding
Abstract
We present new quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and sparse instances. For dense graphs on n vertices, we get a query complexity of O(n^(5/4)) without any of the extra logarithmic factors present in the previous algorithm of Le Gall [FOCS'14]. For sparse graphs with m >= n^(5/4) edges, we get a query complexity of O(n^(11/12) m^(1/6) sqrt(log n)), which is better than the one obtained by Le Gall and Nakajima [ISAAC'15] when m >= n^(3/2). We also obtain an algorithm with query complexity O(n^(5/6) (m log n)^(1/6) + d_2 sqrt(n)) where d_2 is the variance of the degree distribution.
Our algorithms are designed and analyzed in a new model of learning graphs that we call extended learning graphs. In addition, we present a framework in order to easily combine and analyze them. As a consequence we get much simpler algorithms and analyses than previous algorithms of Le Gall based on the MNRS quantum walk framework [SICOMP'11].
Cite as
Titouan Carette, Mathieu Laurière, and Frédéric Magniez. Extended Learning Graphs for Triangle Finding. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{carette_et_al:LIPIcs.STACS.2017.20,
author = {Carette, Titouan and Lauri\`{e}re, Mathieu and Magniez, Fr\'{e}d\'{e}ric},
title = {{Extended Learning Graphs for Triangle Finding}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {20:1--20:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.20},
URN = {urn:nbn:de:0030-drops-70132},
doi = {10.4230/LIPIcs.STACS.2017.20},
annote = {Keywords: Quantum query complexity, learning graphs, triangle finding}
}
Document
Lower Bounds for Elimination via Weak Regularity
Abstract
We consider the problem of elimination in communication complexity, that was first raised by Ambainis et al. and later studied by Beimel et al. for its connection to the famous direct sum question. In this problem, let f: {0,1}^2n -> {0,1} be any boolean function. Alice and Bob get k inputs x_1, ..., x_k and y_1, ..., y_k respectively, with x_i,y_i in {0,1}^n. They want to output a k-bit vector v, such that there exists one index i for which v_i is not equal f(x_i,y_i). We prove a general result lower bounding the randomized communication complexity of the elimination problem for f using its discrepancy. Consequently, we obtain strong lower bounds for the functions Inner-Product and Greater-Than, that work for exponentially larger values of k than the best previous bounds.
To prove our result, we use a pseudo-random notion called regularity that was first used by Raz and Wigderson. We show that functions with small discrepancy are regular. We also observe that a weaker notion, that we call weak-regularity, already implies hardness of elimination. Finally, we give a different proof, borrowing ideas from Viola, to show that Greater-Than is weakly regular.
Cite as
Arkadev Chattopadhyay, Pavel Dvorák, Michal Koucký, Bruno Loff, and Sagnik Mukhopadhyay. Lower Bounds for Elimination via Weak Regularity. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{chattopadhyay_et_al:LIPIcs.STACS.2017.21,
author = {Chattopadhyay, Arkadev and Dvor\'{a}k, Pavel and Kouck\'{y}, Michal and Loff, Bruno and Mukhopadhyay, Sagnik},
title = {{Lower Bounds for Elimination via Weak Regularity}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {21:1--21:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.21},
URN = {urn:nbn:de:0030-drops-70128},
doi = {10.4230/LIPIcs.STACS.2017.21},
annote = {Keywords: communication complexity, elimination, discrepancy, regularity, greater-than}
}
Document
Parameterized and Approximation Results for Scheduling with a Low Rank Processing Time Matrix
Abstract
We study approximation and parameterized algorithms for R||C_max, focusing on the problem when the rank of the matrix formed by job processing times is small. Bhaskara et al. initiated the study of approximation algorithms with respect to the rank, showing that R||C_max admits a QPTAS (Quasi-polynomial time approximation scheme) when the rank is 2, and becomes APX-hard when the rank is 4.
We continue this line of research. We prove that R||C_max is APX-hard even if the rank is 3, resolving an open problem. We then show that R||C_max is FPT parameterized by the rank and the largest job processing time p_max. This generalizes the parameterized results on P||C_max and R||C_max with few different types of machines. We also provide nearly tight lower bounds under Exponential Time Hypothesis which suggests that the running time of the FPT algorithm is unlikely to be improved significantly.
Cite as
Lin Chen, Dániel Marx, Deshi Ye, and Guochuan Zhang. Parameterized and Approximation Results for Scheduling with a Low Rank Processing Time Matrix. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{chen_et_al:LIPIcs.STACS.2017.22,
author = {Chen, Lin and Marx, D\'{a}niel and Ye, Deshi and Zhang, Guochuan},
title = {{Parameterized and Approximation Results for Scheduling with a Low Rank Processing Time Matrix}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {22:1--22:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.22},
URN = {urn:nbn:de:0030-drops-70110},
doi = {10.4230/LIPIcs.STACS.2017.22},
annote = {Keywords: APX-hardness, Parameterized algorithm, Scheduling, Exponential Time Hypothesis}
}
Document
Fractional Coverings, Greedy Coverings, and Rectifier Networks
Abstract
A rectifier network is a directed acyclic graph with distinguished sources and sinks; it is said to compute a Boolean matrix M that has a 1 in the entry (i,j) iff there is a path from the j-th source to the i-th sink. The smallest number of edges in a rectifier network that computes M is a classic complexity measure on matrices, which has been studied for more than half a century.
We explore two techniques that have hitherto found little to no applications in this theory. They build upon a basic fact that depth-2 rectifier networks are essentially weighted coverings of Boolean matrices with rectangles. Using fractional and greedy coverings (defined in the standard way), we obtain new results in this area.
First, we show that all fractional coverings of the so-called full triangular matrix have cost at least n log n. This provides (a fortiori) a new proof of the tight lower bound on its depth-2 complexity (the exact value has been known since 1965, but previous proofs are based on different arguments). Second, we show that the greedy heuristic is instrumental in tightening the upper bound on the depth-2 complexity of the Kneser-Sierpinski (disjointness) matrix. The previous upper bound is O(n^{1.28}), and we improve it to O(n^{1.17}), while the best known lower bound is Omega(n^{1.16}). Third, using fractional coverings, we obtain a form of direct product theorem that gives a lower bound on unbounded-depth complexity of Kronecker (tensor) products of matrices. In this case, the greedy heuristic shows (by an argument due to Lovász) that our result is only a logarithmic factor away from the "full" direct product theorem. Our second and third results constitute progress on open problem 7.3 and resolve, up to a logarithmic factor, open problem 7.5 from a recent book by Jukna and Sergeev (in Foundations and Trends in Theoretical Computer Science (2013)).
Cite as
Dmitry Chistikov, Szabolcs Iván, Anna Lubiw, and Jeffrey Shallit. Fractional Coverings, Greedy Coverings, and Rectifier Networks. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{chistikov_et_al:LIPIcs.STACS.2017.23,
author = {Chistikov, Dmitry and Iv\'{a}n, Szabolcs and Lubiw, Anna and Shallit, Jeffrey},
title = {{Fractional Coverings, Greedy Coverings, and Rectifier Networks}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {23:1--23:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.23},
URN = {urn:nbn:de:0030-drops-70107},
doi = {10.4230/LIPIcs.STACS.2017.23},
annote = {Keywords: rectifier network, OR-circuit, biclique covering, fractional covering, greedy covering}
}
Document
Separability of Reachability Sets of Vector Addition Systems
Abstract
Given two families of sets F and G, the F-separability problem for G asks whether for two given sets U, V in G there exists a set S in F, such that U is included in S and V is disjoint with S. We consider two families of sets F: modular sets S which are subsets of N^d, defined as unions of equivalence classes modulo some natural number n in N, and unary sets, which extend modular sets by requiring equality below a threshold n, and equivalence modulo n above n. Our main result is decidability of modular- and unary-separability for the class G of reachability sets of Vector Addition Systems, Petri Nets, Vector Addition Systems with States, and for sections thereof.
Cite as
Lorenzo Clemente, Wojciech Czerwinski, Slawomir Lasota, and Charles Paperman. Separability of Reachability Sets of Vector Addition Systems. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{clemente_et_al:LIPIcs.STACS.2017.24,
author = {Clemente, Lorenzo and Czerwinski, Wojciech and Lasota, Slawomir and Paperman, Charles},
title = {{Separability of Reachability Sets of Vector Addition Systems}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {24:1--24:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.24},
URN = {urn:nbn:de:0030-drops-70091},
doi = {10.4230/LIPIcs.STACS.2017.24},
annote = {Keywords: separability, Petri nets, modular sets, unary sets, decidability}
}
Document
Counting Edge-Injective Homomorphisms and Matchings on Restricted Graph Classes
Abstract
We consider the parameterized problem of counting all matchings with exactly k edges in a given input graph G. This problem is #W[1]-hard (Curticapean, ICALP 2013), so it is unlikely to admit f(k)poly(n) time algorithms. We show that #W[1]-hardness persists even when the input graph G comes from restricted graph classes, such as line graphs and bipartite graphs of arbitrary constant girth and maximum degree two on one side.
To prove the result for line graphs, we observe that k-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of k paths of length two into (arbitrary) host graphs. Here, a homomorphism from H to G is edge-injective if it maps any two distinct edges of H to distinct edges in G. We show that edge-injective homomorphisms from a pattern graph H can be counted in polynomial time if H has bounded vertex-cover number after removing isolated edges. For hereditary classes H of pattern graphs, we obtain a full complexity dichotomy theorem by proving that counting edge-injective homomorphisms, restricted to patterns from H, is #W[1]-hard if no such bound exists.
Our proofs rely on an edge-colored variant of Holant problems and a delicate interpolation argument; both may be of independent interest.
Cite as
Radu Curticapean, Holger Dell, and Marc Roth. Counting Edge-Injective Homomorphisms and Matchings on Restricted Graph Classes. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{curticapean_et_al:LIPIcs.STACS.2017.25,
author = {Curticapean, Radu and Dell, Holger and Roth, Marc},
title = {{Counting Edge-Injective Homomorphisms and Matchings on Restricted Graph Classes}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {25:1--25:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.25},
URN = {urn:nbn:de:0030-drops-70080},
doi = {10.4230/LIPIcs.STACS.2017.25},
annote = {Keywords: matchings, homomorphisms, line graphs, counting complexity, parameterized complexity}
}
Document
Robust and Adaptive Search
Abstract
Binary search finds a given element in a sorted array with an optimal number of log n queries. However, binary search fails even when the array is only slightly disordered or access to its elements is subject to errors. We study the worst-case query complexity of search algorithms that are robust to imprecise queries and that adapt to perturbations of the order of the elements. We give (almost) tight results for various parameters that quantify query errors and that measure array disorder. In particular, we exhibit settings where query complexities of log n + ck, (1+epsilon) log n + ck, and sqrt(cnk)+o(nk) are best-possible for parameter value k, any epsilon > 0, and constant c.
Cite as
Yann Disser and Stefan Kratsch. Robust and Adaptive Search. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{disser_et_al:LIPIcs.STACS.2017.26,
author = {Disser, Yann and Kratsch, Stefan},
title = {{Robust and Adaptive Search}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {26:1--26:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.26},
URN = {urn:nbn:de:0030-drops-70077},
doi = {10.4230/LIPIcs.STACS.2017.26},
annote = {Keywords: searching, robustness, adaptive algorithms, memory faults, array disorder}
}
Document
Graphic TSP in Cubic Graphs
Abstract
We present a polynomial-time 9/7-approximation algorithm for the graphic TSP for cubic graphs, which improves the previously best approximation factor of 1.3 for 2-connected cubic graphs and drops the requirement of 2-connectivity at the same time. To design our algorithm, we prove that every simple 2-connected cubic n-vertex graph contains a spanning closed walk of length at most 9n/7-1, and that such a walk can be found in polynomial time.
Cite as
Zdenek Dvorák, Daniel Král, and Bojan Mohar. Graphic TSP in Cubic Graphs. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{dvorak_et_al:LIPIcs.STACS.2017.27,
author = {Dvor\'{a}k, Zdenek and Kr\'{a}l, Daniel and Mohar, Bojan},
title = {{Graphic TSP in Cubic Graphs}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {27:1--27:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.27},
URN = {urn:nbn:de:0030-drops-70068},
doi = {10.4230/LIPIcs.STACS.2017.27},
annote = {Keywords: Graphic TSP, approximation algorithms, cubic graphs}
}
Document
Independent Sets near the Lower Bound in Bounded Degree Graphs
Abstract
By Brook's Theorem, every n-vertex graph of maximum degree at most Delta >= 3 and clique number at most Delta is Delta-colorable, and thus it has an independent set of size at least n/Delta. We give an approximate characterization of graphs with independence number close to this bound, and use it to show that the problem of deciding whether such a graph has an independent set of size at least n/Delta+k has a kernel of size O(k).
Cite as
Zdenek Dvorák and Bernard Lidický. Independent Sets near the Lower Bound in Bounded Degree Graphs. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{dvorak_et_al:LIPIcs.STACS.2017.28,
author = {Dvor\'{a}k, Zdenek and Lidick\'{y}, Bernard},
title = {{Independent Sets near the Lower Bound in Bounded Degree Graphs}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {28:1--28:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.28},
URN = {urn:nbn:de:0030-drops-70042},
doi = {10.4230/LIPIcs.STACS.2017.28},
annote = {Keywords: independent set, bounded degree, Delta-colorable, fixed parameter tractability}
}
Document
Semialgebraic Invariant Synthesis for the Kannan-Lipton Orbit Problem
Abstract
The Orbit Problem consists of determining, given a linear transformation A on d-dimensional rationals Q^d, together with vectors x and y, whether the orbit of x under repeated applications of A can ever reach y. This problem was famously shown to be decidable by Kannan and Lipton in the 1980s.
In this paper, we are concerned with the problem of synthesising suitable invariants P which are subsets of R^d, i.e., sets that are stable under A and contain x and not y, thereby providing compact and versatile certificates of non-reachability. We show that whether a given instance of the Orbit Problem admits a semialgebraic invariant is decidable, and moreover in positive instances we provide an algorithm to synthesise suitable invariants of polynomial size.
It is worth noting that the existence of semilinear invariants, on the other hand, is (to the best of our knowledge) not known to be decidable.
Cite as
Nathanaël Fijalkow, Pierre Ohlmann, Joël Ouaknine, Amaury Pouly, and James Worrell. Semialgebraic Invariant Synthesis for the Kannan-Lipton Orbit Problem. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 29:1-29:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{fijalkow_et_al:LIPIcs.STACS.2017.29,
author = {Fijalkow, Nathana\"{e}l and Ohlmann, Pierre and Ouaknine, Jo\"{e}l and Pouly, Amaury and Worrell, James},
title = {{Semialgebraic Invariant Synthesis for the Kannan-Lipton Orbit Problem}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {29:1--29:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.29},
URN = {urn:nbn:de:0030-drops-70059},
doi = {10.4230/LIPIcs.STACS.2017.29},
annote = {Keywords: Verification,algebraic computation,Skolem Problem,Orbit Problem,invariants}
}
Document
The First-Order Logic of Hyperproperties
Abstract
We investigate the logical foundations of hyperproperties. Hyperproperties generalize trace properties, which are sets of traces, to sets of sets of traces. The most prominent application of hyperproperties is information flow security: information flow policies characterize the secrecy and integrity of a system by comparing two or more execution traces, for example by comparing the observations made by an external observer on execution traces that result from different values of a secret variable.
In this paper, we establish the first connection between temporal logics for hyperproperties and first-order logic. Kamp's seminal theorem (in the formulation due to Gabbay et al.) states that linear-time temporal logic (LTL) is expressively equivalent to first-order logic over the natural numbers with order. We introduce first-order logic over sets of traces and prove that HyperLTL, the extension of LTL to hyperproperties, is strictly subsumed by this logic. We furthermore exhibit a fragment that is expressively equivalent to HyperLTL, thereby establishing Kamp's theorem for hyperproperties.
Cite as
Bernd Finkbeiner and Martin Zimmermann. The First-Order Logic of Hyperproperties. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{finkbeiner_et_al:LIPIcs.STACS.2017.30,
author = {Finkbeiner, Bernd and Zimmermann, Martin},
title = {{The First-Order Logic of Hyperproperties}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {30:1--30:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.30},
URN = {urn:nbn:de:0030-drops-70031},
doi = {10.4230/LIPIcs.STACS.2017.30},
annote = {Keywords: Hyperproperties, Linear Temporal Logic, First-order Logic}
}
Document
Improving and Extending the Testing of Distributions for Shape-Restricted Properties
Abstract
Distribution testing deals with what information can be deduced about an unknown distribution over {1,...,n}, where the algorithm is only allowed to obtain a relatively small number of independent samples from the distribution. In the extended conditional sampling model, the algorithm is also allowed to obtain samples from the restriction of the original distribution on subsets of {1,...,n}.
In 2015, Canonne, Diakonikolas, Gouleakis and Rubinfeld unified several previous results, and showed that for any property of distributions satisfying a "decomposability" criterion, there exists an algorithm (in the basic model) that can distinguish with high probability distributions satisfying the property from distributions that are far from it in variation distance.
We present here a more efficient yet simpler algorithm for the basic model, as well as very efficient algorithms for the conditional model, which until now was not investigated under the umbrella of decomposable properties. Additionally, we provide an algorithm for the conditional model that handles a much larger class of properties.
Our core mechanism is a way of efficiently producing an interval-partition of {1,...,n} that satisfies a "fine-grain" quality. We show that with such a partition at hand we can directly move forward with testing individual intervals, instead of first searching for the "correct" partition of {1,...,n}.
Cite as
Eldar Fischer, Oded Lachish, and Yadu Vasudev. Improving and Extending the Testing of Distributions for Shape-Restricted Properties. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{fischer_et_al:LIPIcs.STACS.2017.31,
author = {Fischer, Eldar and Lachish, Oded and Vasudev, Yadu},
title = {{Improving and Extending the Testing of Distributions for Shape-Restricted Properties}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {31:1--31:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.31},
URN = {urn:nbn:de:0030-drops-70024},
doi = {10.4230/LIPIcs.STACS.2017.31},
annote = {Keywords: conditional sampling, distribution testing, property testing, statistics}
}
Document
Matrix Rigidity from the Viewpoint of Parameterized Complexity
Abstract
The rigidity of a matrix A for a target rank r over a field F is the minimum Hamming distance between A and a matrix of rank at most r. Rigidity is a classical concept in Computational Complexity Theory: constructions of rigid matrices are known to imply lower bounds of significant importance relating to arithmetic circuits. Yet, from the viewpoint of Parameterized Complexity, the study of central properties of matrices in general, and of the rigidity of a matrix in particular, has been neglected. In this paper, we conduct a comprehensive study of different aspects of the computation of the rigidity of general matrices in the framework of Parameterized Complexity. Naturally, given parameters r and k, the Matrix Rigidity problem asks whether the rigidity of A for the target rank r is at most k. We show that in case F equals the reals or F is any finite field, this problem is fixed-parameter tractable with respect to k+r. To this end, we present a dimension reduction procedure, which may be a valuable primitive in future studies of problems of this nature. We also employ central tools in Real Algebraic Geometry, which are not well known in Parameterized Complexity, as a black box. In particular, we view the output of our dimension reduction procedure as an algebraic variety. Our main results are complemented by a W[1]-hardness result and a subexponential-time parameterized algorithm for a special case of Matrix Rigidity, highlighting the different flavors of this problem.
Cite as
Fedor V. Fomin, Daniel Lokshtanov, S. M. Meesum, Saket Saurabh, and Meirav Zehavi. Matrix Rigidity from the Viewpoint of Parameterized Complexity. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{fomin_et_al:LIPIcs.STACS.2017.32,
author = {Fomin, Fedor V. and Lokshtanov, Daniel and Meesum, S. M. and Saurabh, Saket and Zehavi, Meirav},
title = {{Matrix Rigidity from the Viewpoint of Parameterized Complexity}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {32:1--32:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.32},
URN = {urn:nbn:de:0030-drops-70019},
doi = {10.4230/LIPIcs.STACS.2017.32},
annote = {Keywords: Matrix Rigidity, Parameterized Complexity, Linear Algebra}
}
Document
Deterministic Regular Expressions with Back-References
Abstract
Most modern libraries for regular expression matching allow back-references (i.e. repetition operators) that substantially increase expressive power, but also lead to intractability. In order to find a better balance between expressiveness and tractability, we combine these with the notion of determinism for regular expressions used in XML DTDs and XML Schema. This includes the definition of a suitable automaton model, and a generalization of the Glushkov construction.
Cite as
Dominik D. Freydenberger and Markus L. Schmid. Deterministic Regular Expressions with Back-References. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{freydenberger_et_al:LIPIcs.STACS.2017.33,
author = {Freydenberger, Dominik D. and Schmid, Markus L.},
title = {{Deterministic Regular Expressions with Back-References}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {33:1--33:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.33},
URN = {urn:nbn:de:0030-drops-70004},
doi = {10.4230/LIPIcs.STACS.2017.33},
annote = {Keywords: Deterministic Regular Expression, Regex, Glushkov Automaton}
}
Document
On the Decomposition of Finite-Valued Streaming String Transducers
Abstract
We prove the following decomposition theorem: every 1-register streaming string transducer that associates a uniformly bounded number of outputs with each input can be effectively decomposed as a finite union of functional 1-register streaming string transducers. This theorem relies on a combinatorial result by Kortelainen concerning word equations with iterated factors. Our result implies the decidability of the equivalence problem for the considered class of transducers. This can be seen as a first step towards proving a more general decomposition theorem for streaming string transducers with multiple registers.
Cite as
Paul Gallot, Anca Muscholl, Gabriele Puppis, and Sylvain Salvati. On the Decomposition of Finite-Valued Streaming String Transducers. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{gallot_et_al:LIPIcs.STACS.2017.34,
author = {Gallot, Paul and Muscholl, Anca and Puppis, Gabriele and Salvati, Sylvain},
title = {{On the Decomposition of Finite-Valued Streaming String Transducers}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {34:1--34:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.34},
URN = {urn:nbn:de:0030-drops-69997},
doi = {10.4230/LIPIcs.STACS.2017.34},
annote = {Keywords: Streaming Transducers, finite valuedness, equivalence}
}
Document
Circuit Evaluation for Finite Semirings
Abstract
The circuit evaluation problem for finite semirings is considered, where semirings are not assumed to have an additive or multiplicative identity. The following dichotomy is shown: If a finite semiring R (i) has a solvable multiplicative semigroup and (ii) does not contain a subsemiring with an additive identity 0 and a multiplicative identity 1 != 0, then its circuit evaluation problem is in the complexity class DET (which is contained in NC^2). In all other cases, the circuit evaluation problem is P-complete.
Cite as
Moses Ganardi, Danny Hucke, Daniel König, and Markus Lohrey. Circuit Evaluation for Finite Semirings. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 35:1-35:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{ganardi_et_al:LIPIcs.STACS.2017.35,
author = {Ganardi, Moses and Hucke, Danny and K\"{o}nig, Daniel and Lohrey, Markus},
title = {{Circuit Evaluation for Finite Semirings}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {35:1--35:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.35},
URN = {urn:nbn:de:0030-drops-69978},
doi = {10.4230/LIPIcs.STACS.2017.35},
annote = {Keywords: circuit value problem, finite semirings, circuit complexity}
}
Document
Combining Treewidth and Backdoors for CSP
Abstract
We show that CSP is fixed-parameter tractable when parameterized by the treewidth of a backdoor into any tractable CSP problem over a finite constraint language. This result combines the two prominent approaches for achieving tractability for CSP: (i) structural restrictions on the interaction between the variables and the constraints and (ii) language restrictions on the relations that can be used inside the constraints. Apart from defining the notion of backdoor-treewidth and showing how backdoors of small treewidth can be used to efficiently solve CSP, our main technical contribution is a fixed-parameter algorithm that finds a backdoor of small treewidth.
Cite as
Robert Ganian, M. S. Ramanujan, and Stefan Szeider. Combining Treewidth and Backdoors for CSP. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 36:1-36:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{ganian_et_al:LIPIcs.STACS.2017.36,
author = {Ganian, Robert and Ramanujan, M. S. and Szeider, Stefan},
title = {{Combining Treewidth and Backdoors for CSP}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {36:1--36:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.36},
URN = {urn:nbn:de:0030-drops-69986},
doi = {10.4230/LIPIcs.STACS.2017.36},
annote = {Keywords: Algorithms and data structures, Fixed Parameter Tractability, Constraint Satisfaction}
}
Document
On the Complexity of Partial Derivatives
Abstract
The method of partial derivatives is one of the most successful lower bound methods for arithmetic circuits. It uses as a complexity measure the dimension of the span of the partial derivatives of a polynomial. In this paper, we consider this complexity measure as a computational problem: for an input polynomial given as the sum of its nonzero monomials, what is the complexity of computing the dimension of its space of partial derivatives?
We show that this problem is #P-hard and we ask whether it belongs to #P. We analyze the "trace method", recently used in combinatorics and in algebraic complexity to lower bound the rank of certain matrices. We show that this method provides a polynomial-time computable lower bound on the dimension of the span of partial derivatives, and from this method we derive closed-form lower bounds. We leave as an open problem the existence of an approximation algorithm with reasonable performance guarantees.
Cite as
Ignacio Garcia-Marco, Pascal Koiran, Timothée Pecatte, and Stéphan Thomassé. On the Complexity of Partial Derivatives. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 37:1-37:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{garciamarco_et_al:LIPIcs.STACS.2017.37,
author = {Garcia-Marco, Ignacio and Koiran, Pascal and Pecatte, Timoth\'{e}e and Thomass\'{e}, St\'{e}phan},
title = {{On the Complexity of Partial Derivatives}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {37:1--37:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.37},
URN = {urn:nbn:de:0030-drops-69964},
doi = {10.4230/LIPIcs.STACS.2017.37},
annote = {Keywords: counting complexity, simplicial complex, lower bounds, arithmetic circuits}
}
Document
Set Membership with Non-Adaptive Bit Probes
Abstract
We consider the non-adaptive bit-probe complexity of the set membership problem, where a set S of size at most n from a universe of size m is to be represented as a short bit vector in order to answer membership queries of the form "Is x in S?" by non-adaptively probing the bit vector at t places. Let s_N(m,n,t) be the minimum number of bits of storage needed for such a scheme. In this work, we show existence of non-adaptive and adaptive schemes for a range of t that improves an upper bound of Buhrman, Miltersen, Radhakrishnan and Srinivasan (2002) on s_N(m,n,t). For three non-adaptive probes, we improve the previous best lower bound on s_N(m,n,3) by Alon and Feige (2009).
Cite as
Mohit Garg and Jaikumar Radhakrishnan. Set Membership with Non-Adaptive Bit Probes. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 38:1-38:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{garg_et_al:LIPIcs.STACS.2017.38,
author = {Garg, Mohit and Radhakrishnan, Jaikumar},
title = {{Set Membership with Non-Adaptive Bit Probes}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {38:1--38:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.38},
URN = {urn:nbn:de:0030-drops-69952},
doi = {10.4230/LIPIcs.STACS.2017.38},
annote = {Keywords: Data Structures, Bit-probe model, Compression, Bloom filters, Expansion}
}
Document
Pro-Aperiodic Monoids via Saturated Models
Abstract
We apply Stone duality and model theory to study the structure theory of free pro-aperiodic monoids. Stone duality implies that elements of the free pro-aperiodic monoid may be viewed as elementary equivalence classes of pseudofinite words. Model theory provides us with saturated words in each such class, i.e., words in which all possible factorizations are realized. We give several applications of this new approach, including a solution to the word problem for omega-terms that avoids using McCammond's normal forms, as well as new proofs and extensions of other structural results concerning free pro-aperiodic monoids.
Cite as
Samuel J. van Gool and Benjamin Steinberg. Pro-Aperiodic Monoids via Saturated Models. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{vangool_et_al:LIPIcs.STACS.2017.39,
author = {van Gool, Samuel J. and Steinberg, Benjamin},
title = {{Pro-Aperiodic Monoids via Saturated Models}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {39:1--39:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.39},
URN = {urn:nbn:de:0030-drops-69945},
doi = {10.4230/LIPIcs.STACS.2017.39},
annote = {Keywords: aperiodic monoids, profinite monoids, Stone duality, saturated models}
}
Document
Trimming and Gluing Gray Codes
Abstract
We consider the algorithmic problem of generating each subset of [n]:={1,2,...,n} whose size is in some interval [k,l], 0 <= k <= l <= n, exactly once (cyclically) by repeatedly adding or removing a single element, or by exchanging a single element. For k=0 and l=n this is the classical problem of generating all 2^n subsets of [n] by element additions/removals, and for k=l this is the classical problem of generating all n over k subsets of [n] by element exchanges. We prove the existence of such cyclic minimum-change enumerations for a large range of values n, k, and l, improving upon and generalizing several previous results. For all these existential results we provide optimal algorithms to compute the corresponding Gray codes in constant time O(1) per generated set and space O(n). Rephrased in terms of graph theory, our results establish the existence of (almost) Hamilton cycles in the subgraph of the n-dimensional cube Q_n induced by all levels [k,l]. We reduce all remaining open cases to a generalized version of the middle levels conjecture, which asserts that the subgraph of Q_(2k+1) induced by all levels [k-c,k+1+c], c in {0, 1, ... , k}, has a Hamilton cycle. We also prove an approximate version of this conjecture, showing that this graph has a cycle that visits a (1-o(1))-fraction of all vertices.
Cite as
Petr Gregor and Torsten Mütze. Trimming and Gluing Gray Codes. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{gregor_et_al:LIPIcs.STACS.2017.40,
author = {Gregor, Petr and M\"{u}tze, Torsten},
title = {{Trimming and Gluing Gray Codes}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {40:1--40:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.40},
URN = {urn:nbn:de:0030-drops-69930},
doi = {10.4230/LIPIcs.STACS.2017.40},
annote = {Keywords: Gray code, subset, combination, loopless algorithm}
}
Document
Mixing of Permutations by Biased Transposition
Abstract
Markov chains defined on the set of permutations of n elements have been studied widely by mathematicians and theoretical computer scientists. We consider chains in which a position i<n is chosen uniformly at random, and then sigma(i) and sigma(i+1) are swapped with probability depending on sigma(i) and sigma(i+1). Our objective is to identify some conditions that assure rapid mixing.
One case of particular interest is what we call the "gladiator chain," in which each number g is assigned a "strength" s_g and when g and g' are swapped, g comes out on top with probability s_g / ( s_g + s_g' ). The stationary probability of this chain is the same as that of the slow-mixing "move ahead one" chain for self-organizing lists, but an open conjecture of Jim Fill's implies that all gladiator chains mix rapidly. Here we obtain some positive partial results by considering cases where the gladiators fall into only a few strength classes.
Cite as
Shahrzad Haddadan and Peter Winkler. Mixing of Permutations by Biased Transposition. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 41:1-41:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{haddadan_et_al:LIPIcs.STACS.2017.41,
author = {Haddadan, Shahrzad and Winkler, Peter},
title = {{Mixing of Permutations by Biased Transposition}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {41:1--41:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.41},
URN = {urn:nbn:de:0030-drops-69928},
doi = {10.4230/LIPIcs.STACS.2017.41},
annote = {Keywords: Markov chains, permutations, self organizing lists, mixing time}
}
Document
Efficient Quantum Walk on the Grid with Multiple Marked Elements
Abstract
We give a quantum algorithm for finding a marked element on the grid when there are multiple marked elements. Our algorithm uses quadratically fewer steps than a random walk on the grid, ignoring logarithmic factors. This is the first known quantum walk that finds a marked element in a number of steps less than the square-root of the extended hitting time. We also give a new tighter upper bound on the extended hitting time of a marked subset, expressed in terms of the hitting times of its members.
Cite as
Peter Hoyer and Mojtaba Komeili. Efficient Quantum Walk on the Grid with Multiple Marked Elements. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 42:1-42:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{hoyer_et_al:LIPIcs.STACS.2017.42,
author = {Hoyer, Peter and Komeili, Mojtaba},
title = {{Efficient Quantum Walk on the Grid with Multiple Marked Elements}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {42:1--42:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.42},
URN = {urn:nbn:de:0030-drops-69902},
doi = {10.4230/LIPIcs.STACS.2017.42},
annote = {Keywords: Quantum walks, random walks, query complexity, spatial search}
}
Document
On OBDD-Based Algorithms and Proof Systems That Dynamically Change Order of Variables
Abstract
In 2004 Atserias, Kolaitis and Vardi proposed OBDD-based propositional proof systems that prove unsatisfiability of a CNF formula by deduction of identically false OBDD from OBDDs representing clauses of the initial formula. All OBDDs in such proofs have the same order of variables. We initiate the study of OBDD based proof systems that additionally contain a rule that allows to change the order in OBDDs. At first we consider a proof system OBDD(and, reordering) that uses the conjunction (join) rule and the rule that allows to change the order. We exponentially separate this proof system from OBDD(and)-proof system that uses only the conjunction rule. We prove two exponential lower bounds on the size of OBDD(and, reordering)-refutations of Tseitin formulas and the pigeonhole principle. The first lower bound was previously unknown even for OBDD(and)-proofs and the second one extends the result of Tveretina et al. from OBDD(and) to OBDD(and, reordering).
In 2004 Pan and Vardi proposed an approach to the propositional satisfiability problem based on OBDDs and symbolic quantifier elimination (we denote algorithms based on this approach as OBDD(and, exists)-algorithms. We notice that there exists an OBDD(and, exists)-algorithm that solves satisfiable and unsatisfiable Tseitin formulas in polynomial time. In contrast, we show that there exist formulas representing systems of linear equations over F_2 that are hard for OBDD(and, exists, reordering)-algorithms. Our hard instances are satisfiable formulas representing systems of linear equations over F_2 that correspond to some checksum matrices of error correcting codes.
Cite as
Dmitry Itsykson, Alexander Knop, Andrey Romashchenko, and Dmitry Sokolov. On OBDD-Based Algorithms and Proof Systems That Dynamically Change Order of Variables. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 43:1-43:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{itsykson_et_al:LIPIcs.STACS.2017.43,
author = {Itsykson, Dmitry and Knop, Alexander and Romashchenko, Andrey and Sokolov, Dmitry},
title = {{On OBDD-Based Algorithms and Proof Systems That Dynamically Change Order of Variables}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {43:1--43:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.43},
URN = {urn:nbn:de:0030-drops-69914},
doi = {10.4230/LIPIcs.STACS.2017.43},
annote = {Keywords: Proof complexity, OBDD, error-correcting codes, Tseitin formulas, expanders}
}
Document
Multiple Random Walks on Paths and Grids
Abstract
We derive several new results on multiple random walks on "low dimensional" graphs.
First, inspired by an example of a weighted random walk on a path of three vertices given by Efremenko and Reingold, we prove the following dichotomy: as the path length n tends to infinity, we have a super-linear speed-up w.r.t. the cover time if and only if the number of walks k is equal to 2. An important ingredient of our proofs is the use of a continuous-time analogue of multiple random walks, which might be of independent interest. Finally, we also present the first tight bounds on the speed-up of the cover time for any d-dimensional grid with d >= 2 being an arbitrary constant, and reveal a sharp transition between linear and logarithmic speed-up.
Cite as
Andrej Ivaskovic, Adrian Kosowski, Dominik Pajak, and Thomas Sauerwald. Multiple Random Walks on Paths and Grids. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{ivaskovic_et_al:LIPIcs.STACS.2017.44,
author = {Ivaskovic, Andrej and Kosowski, Adrian and Pajak, Dominik and Sauerwald, Thomas},
title = {{Multiple Random Walks on Paths and Grids}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {44:1--44:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.44},
URN = {urn:nbn:de:0030-drops-69897},
doi = {10.4230/LIPIcs.STACS.2017.44},
annote = {Keywords: random walks, randomized algorithms, parallel computing}
}
Document
On the Size of Lempel-Ziv and Lyndon Factorizations
Abstract
Lyndon factorization and Lempel-Ziv (LZ) factorization are both important tools for analysing the structure and complexity of strings, but their combinatorial structure is very different. In this paper, we establish the first direct connection between the two by showing that while the Lyndon factorization can be bigger than the non-overlapping LZ factorization (which we demonstrate by describing a new, non-trivial family of strings) it is always less than twice the size.
Cite as
Juha Kärkkäinen, Dominik Kempa, Yuto Nakashima, Simon J. Puglisi, and Arseny M. Shur. On the Size of Lempel-Ziv and Lyndon Factorizations. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 45:1-45:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{karkkainen_et_al:LIPIcs.STACS.2017.45,
author = {K\"{a}rkk\"{a}inen, Juha and Kempa, Dominik and Nakashima, Yuto and Puglisi, Simon J. and Shur, Arseny M.},
title = {{On the Size of Lempel-Ziv and Lyndon Factorizations}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {45:1--45:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.45},
URN = {urn:nbn:de:0030-drops-69878},
doi = {10.4230/LIPIcs.STACS.2017.45},
annote = {Keywords: Lempel-Ziv factorization, Lempel-Ziv parsing, LZ, Lyndon word, Lyndon factorization, Standard factorization}
}
Document
Voting and Bribing in Single-Exponential Time
Abstract
We introduce a general problem about bribery in voting systems. In the R-Multi-Bribery problem, the goal is to bribe a set of voters at minimum cost such that a desired candidate wins the manipulated election under the voting rule R. Voters assign prices for withdrawing their vote, for swapping the positions of two consecutive candidates in their preference order, and for perturbing their approval count for a candidate.
As our main result, we show that R-Multi-Bribery is fixed-parameter tractable parameterized by the number of candidates for many natural voting rules R, including Kemeny rule, all scoring protocols, maximin rule, Bucklin rule, fallback rule, SP-AV, and any C1 rule. In particular, our result resolves the parameterized of R-Swap Bribery for all those voting rules, thereby solving a long-standing open problem and "Challenge #2" of the 9 Challenges in computational social choice by Bredereck et al.
Further, our algorithm runs in single-exponential time for arbitrary cost; it thus improves the earlier double-exponential time algorithm by Dorn and Schlotter that is restricted to the unit-cost case for all scoring protocols, the maximin rule, and Bucklin rule.
Cite as
Dusan Knop, Martin Koutecký, and Matthias Mnich. Voting and Bribing in Single-Exponential Time. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{knop_et_al:LIPIcs.STACS.2017.46,
author = {Knop, Dusan and Kouteck\'{y}, Martin and Mnich, Matthias},
title = {{Voting and Bribing in Single-Exponential Time}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {46:1--46:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.46},
URN = {urn:nbn:de:0030-drops-69885},
doi = {10.4230/LIPIcs.STACS.2017.46},
annote = {Keywords: Parameterized algorithm, swap bribery, n-fold integer programming}
}
Document
A Complexity Dichotomy for Poset Constraint Satisfaction
Abstract
We determine the complexity of all constraint satisfaction problems over partial orders, in particular we show that every such problem is NP-complete or can be solved in polynomial time. This result generalises the complexity dichotomy for temporal constraint satisfaction problems by Bodirsky and Kára. We apply the so called universal-algebraic approach together with tools from model theory and Ramsey theory to prove our result. In the course of this analysis we also establish a structural dichotomy regarding the model theoretic properties of the reducts of the random partial order.
Cite as
Michael Kompatscher and Trung Van Pham. A Complexity Dichotomy for Poset Constraint Satisfaction. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 47:1-47:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{kompatscher_et_al:LIPIcs.STACS.2017.47,
author = {Kompatscher, Michael and Pham, Trung Van},
title = {{A Complexity Dichotomy for Poset Constraint Satisfaction}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {47:1--47:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.47},
URN = {urn:nbn:de:0030-drops-69850},
doi = {10.4230/LIPIcs.STACS.2017.47},
annote = {Keywords: Constraint Satisfaction, Random Partial Order, Computational Complexity, Universal Algebra, Ramsey Theory}
}
Document
Structural Properties and Constant Factor-Approximation of Strong Distance-r Dominating Sets in Sparse Directed Graphs
Abstract
Bounded expansion and nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez, form a large variety of classes of uniformly sparse graphs which includes the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs. Since their initial definition it was shown that these graph classes can be defined in many equivalent ways: by generalised colouring numbers, neighbourhood complexity, sparse neighbourhood covers, a game known as the splitter game, and many more.
We study the corresponding concepts for directed graphs. We show that the densities of bounded depth directed minors and bounded depth topological minors relate in a similar way as in the undirected case. We provide a characterisation of bounded expansion classes by a directed version of the generalised colouring numbers. As an application we show how to construct sparse directed neighbourhood covers and how to approximate directed distance-r dominating sets on classes of bounded expansion. On the other hand, we show that linear neighbourhood complexity does not characterise directed classes of bounded expansion.
Cite as
Stephan Kreutzer, Roman Rabinovich, Sebastian Siebertz, and Grischa Weberstädt. Structural Properties and Constant Factor-Approximation of Strong Distance-r Dominating Sets in Sparse Directed Graphs. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 48:1-48:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{kreutzer_et_al:LIPIcs.STACS.2017.48,
author = {Kreutzer, Stephan and Rabinovich, Roman and Siebertz, Sebastian and Weberst\"{a}dt, Grischa},
title = {{Structural Properties and Constant Factor-Approximation of Strong Distance-r Dominating Sets in Sparse Directed Graphs}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {48:1--48:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.48},
URN = {urn:nbn:de:0030-drops-69868},
doi = {10.4230/LIPIcs.STACS.2017.48},
annote = {Keywords: Directed Graph Structure Theory, Bounded Expansion, Generalised Colouring Numbers, Splitter Game, Approximation Algorithms, Dominating Set}
}
Document
Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates
Abstract
We study the following computational problem: for which values of k, the majority of n bits MAJ_n can be computed with a depth two formula whose each gate computes a majority function of at most k bits? The corresponding computational model is denoted by MAJ_k o MAJ_k. We observe that the minimum value of k for which there exists a MAJ_k o MAJ_k circuit that has high correlation with the majority of n bits is equal to Theta(sqrt(n)). We then show that for a randomized MAJ_k o MAJ_k circuit computing the majority of n input bits with high probability for every input, the minimum value of k is equal to n^(2/3+o(1)). We show a worst case lower bound: if a MAJ_k o MAJ_k circuit computes the majority of n bits correctly on all inputs, then k <= n^(13/19+o(1)). This lower bound exceeds the optimal value for randomized circuits and thus is unreachable for pure randomized techniques. For depth 3 circuits we show that a circuit with k= O(n^(2/3)) can compute MAJ_n correctly on all inputs.
Cite as
Alexander S. Kulikov and Vladimir V. Podolskii. Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 49:1-49:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{kulikov_et_al:LIPIcs.STACS.2017.49,
author = {Kulikov, Alexander S. and Podolskii, Vladimir V.},
title = {{Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {49:1--49:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.49},
URN = {urn:nbn:de:0030-drops-69832},
doi = {10.4230/LIPIcs.STACS.2017.49},
annote = {Keywords: circuit complexity, computational complexity, threshold, majority, lower bound, upper bound}
}
Document
Minkowski Games
Abstract
We introduce and study Minkowski games. In these games, two players take turns to choose positions in R^d based on some rules. Variants include boundedness games, where one player wants to keep the positions bounded (while the other wants to escape to infinity), and safety games, where one player wants to stay within a given set (while the other wants to leave it).
We provide some general characterizations of which player can win such games, and explore the computational complexity of the associated decision problems. A natural representation of boundedness games yields coNP-completeness, whereas the safety games are undecidable.
Cite as
Stéphane Le Roux, Arno Pauly, and Jean-François Raskin. Minkowski Games. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 50:1-50:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{leroux_et_al:LIPIcs.STACS.2017.50,
author = {Le Roux, St\'{e}phane and Pauly, Arno and Raskin, Jean-Fran\c{c}ois},
title = {{Minkowski Games}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {50:1--50:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.50},
URN = {urn:nbn:de:0030-drops-69849},
doi = {10.4230/LIPIcs.STACS.2017.50},
annote = {Keywords: Control in R^d, determinacy, polytopic/arbitrary, coNP-complete, undecidable}
}
Document
On the Sensitivity Complexity of k-Uniform Hypergraph Properties
Abstract
In this paper we investigate the sensitivity complexity of hypergraph properties. We present a k-uniform hypergraph property with sensitivity complexity O(n^{ceil(k/3)}) for any k >= 3, where n is the number of vertices. Moreover, we can do better when k = 1 (mod 3) by presenting a k-uniform hypergraph property with sensitivity O(n^{ceil(k/3)-1/2}). This result disproves a conjecture of Babai, which conjectures that the sensitivity complexity of k-uniform hypergraph properties is at least Omega(n^{k/2}). We also investigate the sensitivity complexity of other weakly symmetric functions and show that for many classes of transitive-invariant Boolean functions the minimum achievable sensitivity complexity can be O(N^{1/3}), where N is the number of variables. Finally, we give a lower bound for sensitivity of k-uniform hypergraph properties, which implies the sensitivity conjecture of k-uniform hypergraph properties for any constant k.
Cite as
Qian Li and Xiaoming Sun. On the Sensitivity Complexity of k-Uniform Hypergraph Properties. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 51:1-51:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{li_et_al:LIPIcs.STACS.2017.51,
author = {Li, Qian and Sun, Xiaoming},
title = {{On the Sensitivity Complexity of k-Uniform Hypergraph Properties}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {51:1--51:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.51},
URN = {urn:nbn:de:0030-drops-69825},
doi = {10.4230/LIPIcs.STACS.2017.51},
annote = {Keywords: Sensitivity Complexity, k-uniform Hypergraph Properties, Boolean Function, Turan's question}
}
Document
The Complexity of Knapsack in Graph Groups
Abstract
Myasnikov et al. have introduced the knapsack problem for arbitrary finitely generated groups. In LohreyZ16 the authors proved that for each graph group, the knapsack problem can be solved in NP. Here, we determine the exact complexity of the problem for every graph group. While the problem is TC^0-complete for complete graphs, it is LogCFL-complete for each (non-complete) transitive forest. For every remaining graph, the problem is NP-complete.
Cite as
Markus Lohrey and Georg Zetzsche. The Complexity of Knapsack in Graph Groups. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{lohrey_et_al:LIPIcs.STACS.2017.52,
author = {Lohrey, Markus and Zetzsche, Georg},
title = {{The Complexity of Knapsack in Graph Groups}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {52:1--52:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.52},
URN = {urn:nbn:de:0030-drops-69810},
doi = {10.4230/LIPIcs.STACS.2017.52},
annote = {Keywords: knapsack, subset sum, graph groups, decision problems in group theory}
}
Document
Algorithmic Information, Plane Kakeya Sets, and Conditional Dimension
Abstract
We formulate the conditional Kolmogorov complexity of x given y at precision r, where x and y are points in Euclidean spaces and r is a natural number. We demonstrate the utility of this notion in two ways.
1. We prove a point-to-set principle that enables one to use the (relativized, constructive) dimension of a single point in a set E in a Euclidean space to establish a lower bound on the (classical) Hausdorff dimension of E. We then use this principle, together with conditional Kolmogorov complexity in Euclidean spaces, to give a new proof of the known, two-dimensional case of the Kakeya conjecture. This theorem of geometric measure theory, proved by Davies in 1971, says that every plane set containing a unit line segment in every direction has Hausdorff dimension 2.
2. We use conditional Kolmogorov complexity in Euclidean spaces to develop the lower and upper conditional dimensions dim(x|y) and Dim(x|y) of x given y, where x and y are points in Euclidean spaces. Intuitively these are the lower and upper asymptotic algorithmic information densities of x conditioned on the information in y. We prove that these conditional dimensions are robust and that they have the correct information-theoretic relationships with the well-studied dimensions dim(x) and Dim(x) and the mutual dimensions mdim(x:y) and Mdim(x:y).
Cite as
Jack H. Lutz and Neil Lutz. Algorithmic Information, Plane Kakeya Sets, and Conditional Dimension. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 53:1-53:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{lutz_et_al:LIPIcs.STACS.2017.53,
author = {Lutz, Jack H. and Lutz, Neil},
title = {{Algorithmic Information, Plane Kakeya Sets, and Conditional Dimension}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {53:1--53:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.53},
URN = {urn:nbn:de:0030-drops-69806},
doi = {10.4230/LIPIcs.STACS.2017.53},
annote = {Keywords: algorithmic randomness, conditional dimension, geometric measure theory, Kakeya sets, Kolmogorov complexity}
}
Document
On the Synchronisation Problem over Cellular Automata
Abstract
Cellular automata are a discrete, synchronous, and uniform dynamical system that give rise to a wide range of dynamical behaviours. In this paper, we investigate whether this system can achieve synchronisation. We study the cases of classical bi-infinite configurations, periodic configurations, and periodic configurations of prime period. In the two former cases, we prove that only a "degenerated" form of synchronisation - there exists a fix-point - is possible. In the latter case, we give an explicit construction of a cellular automaton for which any periodic configuration of prime period eventually converges to cycle of two uniform configurations. Our construction is based upon sophisticated tools: aperiodic NW-deterministic tilings and partitioned intervals.
Cite as
Gaétan Richard. On the Synchronisation Problem over Cellular Automata. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 54:1-54:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{richard:LIPIcs.STACS.2017.54,
author = {Richard, Ga\'{e}tan},
title = {{On the Synchronisation Problem over Cellular Automata}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {54:1--54:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.54},
URN = {urn:nbn:de:0030-drops-69780},
doi = {10.4230/LIPIcs.STACS.2017.54},
annote = {Keywords: cellular automata, dynamical systems, aperiodic tiling, synchronisation}
}
Document
Word Equations Where a Power Equals a Product of Powers
Abstract
We solve a long-standing open problem on word equations by proving that if the words x_0, ..., x_n satisfy the equation x_0^k = x_1^k ... x_n^k for three positive values of k, then the words commute. One of our methods is to assign numerical values for the letters, and then study the sums of the letters of words and their prefixes. We also give a geometric interpretation of our methods.
Cite as
Aleksi Saarela. Word Equations Where a Power Equals a Product of Powers. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 55:1-55:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{saarela:LIPIcs.STACS.2017.55,
author = {Saarela, Aleksi},
title = {{Word Equations Where a Power Equals a Product of Powers}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {55:1--55:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.55},
URN = {urn:nbn:de:0030-drops-69793},
doi = {10.4230/LIPIcs.STACS.2017.55},
annote = {Keywords: Combinatorics on words, Word equations}
}
Document
Improved Distance Queries and Cycle Counting by Frobenius Normal Form
Abstract
Consider an unweighted, directed graph G with the diameter D. In this paper, we introduce the framework for counting cycles and walks of given length in matrix multiplication time O-tilde(n^omega). The framework is based on the fast decomposition into Frobenius normal form and the Hankel matrix-vector multiplication. It allows us to solve the following problems efficiently.
* All Nodes Shortest Cycles - for every node return the length of the shortest cycle containing it. We give an O-tilde(n^omega) algorithm that improves the previous O-tilde(n^((omega + 3)/2)) algorithm for unweighted digraphs.
* We show how to compute all D sets of vertices lying on cycles of length c in {1, ..., D} in randomized time O-tilde(n^omega). It improves upon an algorithm by Cygan where algorithm that computes a single set is presented.
* We present a functional improvement of distance queries for directed, unweighted graphs.
* All Pairs All Walks - we show almost optimal O-tilde(n^3) time algorithm for all walks counting problem. We improve upon the naive O(D n^omega) time algorithm.
Cite as
Piotr Sankowski and Karol Wegrzycki. Improved Distance Queries and Cycle Counting by Frobenius Normal Form. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{sankowski_et_al:LIPIcs.STACS.2017.56,
author = {Sankowski, Piotr and Wegrzycki, Karol},
title = {{Improved Distance Queries and Cycle Counting by Frobenius Normal Form}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {56:1--56:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.56},
URN = {urn:nbn:de:0030-drops-69773},
doi = {10.4230/LIPIcs.STACS.2017.56},
annote = {Keywords: Frobenius Normal Form, Graph Algorithms, All Nodes Shortest Cycles}
}
Document
Lower Bounds on Key Derivation for Square-Friendly Applications
Abstract
Security of cryptographic applications is typically defined by security games. The adversary, within certain resources, cannot win with probability much better than 0 (for unpredictability applications, like one-way functions) or much better than 1/2 (indistinguishability applications for instance encryption schemes). In so called squared-friendly applications the winning probability of the adversary, for different values of the application secret randomness, is not only close to 0 or 1/2 on average, but also concentrated in the sense that its second central moment is small. The class of squared-friendly applications, which contains all unpredictability applications and many indistinguishability applications, is particularly important for key derivation. Barak et al. observed that for square-friendly applications one can beat the "RT-bound", extracting secure keys with significantly smaller entropy loss. In turn Dodis and Yu showed that in squared-friendly applications one can directly use a "weak" key, which has only high entropy, as a secure key.
In this paper we give sharp lower bounds on square security assuming security for "weak" keys. We show that any application which is either (a) secure with weak keys or (b) allows for entropy savings for keys derived by universal hashing, must be square-friendly. Quantitatively, our lower bounds match the positive results of Dodis and Yu and Barak et al. (TCC'13, CRYPTO'11) Hence, they can be understood as a general characterization of squared-friendly applications.
While the positive results on squared-friendly applications where derived by one clever application of the Cauchy-Schwarz Inequality, for tight lower bounds we need more machinery. In our approach we use convex optimization techniques and some theory of circular matrices.
Cite as
Maciej Skorski. Lower Bounds on Key Derivation for Square-Friendly Applications. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 57:1-57:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{skorski:LIPIcs.STACS.2017.57,
author = {Skorski, Maciej},
title = {{Lower Bounds on Key Derivation for Square-Friendly Applications}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {57:1--57:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.57},
URN = {urn:nbn:de:0030-drops-69761},
doi = {10.4230/LIPIcs.STACS.2017.57},
annote = {Keywords: key derivation, square-friendly applications, lower bounds}
}
Document
List Approximation for Increasing Kolmogorov Complexity
Abstract
It is impossible to effectively modify a string in order to increase its Kolmogorov complexity. But is it possible to construct a few strings, not longer than the input string, so that most of them have larger complexity? We show that the answer is yes. We present an algorithm that on input a string x of length n returns a list with O(n^2) many strings, all of length n, such that 99% of them are more complex than x, provided the complexity of x is less than n. We obtain similar results for other parameters, including a polynomial-time construction.
Cite as
Marius Zimand. List Approximation for Increasing Kolmogorov Complexity. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 58:1-58:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
Copy BibTex To Clipboard
@InProceedings{zimand:LIPIcs.STACS.2017.58,
author = {Zimand, Marius},
title = {{List Approximation for Increasing Kolmogorov Complexity}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {58:1--58:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-028-6},
ISSN = {1868-8969},
year = {2017},
volume = {66},
editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.58},
URN = {urn:nbn:de:0030-drops-70347},
doi = {10.4230/LIPIcs.STACS.2017.58},
annote = {Keywords: Kolmogorov complexity, list approximation, randomness extractor}
}