Volume
LIPIcs, Volume 96
35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Theoretical Aspects of Computer Science (STACS)
Event
STACS 2018, February 28 to March 3, 2018, Caen, FranceEditors
Publication Details
- published at: 2018-02-27
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-062-0
- DBLP: db/conf/stacs/stacs2018
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Document
Complete Volume
LIPIcs, Volume 96, STACS'18, Complete Volume
Abstract
LIPIcs, Volume 96, STACS'18, Complete Volume
Cite as
35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@Proceedings{niedermeier_et_al:LIPIcs.STACS.2018,
title = {{LIPIcs, Volume 96, STACS'18, Complete Volume}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018},
URN = {urn:nbn:de:0030-drops-86179},
doi = {10.4230/LIPIcs.STACS.2018},
annote = {Keywords: Mathematics of computing, Theory of computation}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{niedermeier_et_al:LIPIcs.STACS.2018.0,
author = {Niedermeier, Rolf and Vall\'{e}e, Brigitte},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {0:i--0:xvi},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.0},
URN = {urn:nbn:de:0030-drops-84807},
doi = {10.4230/LIPIcs.STACS.2018.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Recursive Combinatorial Structures: Enumeration, Probabilistic Analysis and Random Generation
Abstract
In a probabilistic context, the main data structures of computer science are viewed as random combinatorial objects.
Analytic Combinatorics, as described in the book by Flajolet and Sedgewick, provides a set of high-level tools for their probabilistic analysis.
Recursive combinatorial definitions lead to generating function equations from which efficient algorithms can be designed for enumeration, random generation and, to some extent, asymptotic analysis. With a focus on random generation, this tutorial first covers the basics of Analytic Combinatorics and then describes the idea of Boltzmann sampling and its realisation.
The tutorial addresses a broad TCS audience and no particular pre-knowledge on analytic combinatorics is expected.
Cite as
Bruno Salvy. Recursive Combinatorial Structures: Enumeration, Probabilistic Analysis and Random Generation. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 1:1-1:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{salvy:LIPIcs.STACS.2018.1,
author = {Salvy, Bruno},
title = {{Recursive Combinatorial Structures: Enumeration, Probabilistic Analysis and Random Generation}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {1:1--1:5},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.1},
URN = {urn:nbn:de:0030-drops-85352},
doi = {10.4230/LIPIcs.STACS.2018.1},
annote = {Keywords: Analytic Combinatorics, Generating Functions, Random Generation}
}
Document
Lower Bound Techniques for QBF Proof Systems
Abstract
How do we prove that a false QBF is inded false? How big a proof is needed? The special case when all quantifiers are existential is the well-studied setting of propositional proof complexity. Expectedly, universal quantifiers change the game significantly. Several proof systems have been designed in the last couple of decades to handle QBFs. Lower bound paradigms from propositional proof complexity cannot always be extended - in most cases feasible interpolation and consequent transfer of circuit lower bounds works, but obtaining lower bounds on size by providing lower bounds on width fails dramatically. A new paradigm with no analogue in the propositional world has emerged in the form of strategy extraction, allowing for transfer of circuit lower bounds, as well as obtaining independent
genuine QBF lower bounds based on a semantic cost measure.
This talk will provide a broad overview of some of these developments.
Cite as
Meena Mahajan. Lower Bound Techniques for QBF Proof Systems. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 2:1-2:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{mahajan:LIPIcs.STACS.2018.2,
author = {Mahajan, Meena},
title = {{Lower Bound Techniques for QBF Proof Systems}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {2:1--2:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.2},
URN = {urn:nbn:de:0030-drops-85362},
doi = {10.4230/LIPIcs.STACS.2018.2},
annote = {Keywords: Proof Complexity, Quantified Boolean formulas, Resolution, Lower Bound Techniques}
}
Document
On the Positive Calculus of Relations with Transitive Closure
Abstract
Binary relations are such a basic object that they appear in many
places in mathematics and computer science. For instance, when
dealing with graphs, program semantics, or termination guarantees,
binary relations are always used at some point.
In this survey, we focus on the relations themselves, and we
consider algebraic and algorithmic questions. On the algebraic side, we want to understand and characterise the laws governing the behaviour of the following standard operations on relations: union, intersection, composition, converse, and reflexive-transitive closure. On the algorithmic side, we look for decision procedures for equality or inequality of relations.
After having formally defined the calculus of relations, we recall
the existing results about two well-studied fragments of particular importance: Kleene algebras and allegories. Unifying those fragments yields a decidable theory whose axiomatisability remains an open problem.
Cite as
Damien Pous. On the Positive Calculus of Relations with Transitive Closure. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{pous:LIPIcs.STACS.2018.3,
author = {Pous, Damien},
title = {{On the Positive Calculus of Relations with Transitive Closure}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {3:1--3:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.3},
URN = {urn:nbn:de:0030-drops-85382},
doi = {10.4230/LIPIcs.STACS.2018.3},
annote = {Keywords: Relation Algebra, Kleene Algebra, Allegories, Automata, Graphs}
}
Document
The Open Shop Scheduling Problem
Abstract
We discuss the computational complexity, the approximability, the algorithmics and the combinatorics of the open shop scheduling problem. We summarize the most important results from the literature and explain their main ideas, we sketch the most beautiful proofs, and we also list a number of open problems.
Cite as
Gerhard J. Woeginger. The Open Shop Scheduling Problem. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{woeginger:LIPIcs.STACS.2018.4,
author = {Woeginger, Gerhard J.},
title = {{The Open Shop Scheduling Problem}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {4:1--4:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.4},
URN = {urn:nbn:de:0030-drops-85375},
doi = {10.4230/LIPIcs.STACS.2018.4},
annote = {Keywords: Algorithms, Complexity, Scheduling, Approximation}
}
Document
Approximating Airports and Railways
Abstract
In this paper we consider the airport and railway problem (AR), which combines capacitated facility location with network design, both in the general metric and the two-dimensional Euclidean space. An instance of the airport and railway problem consists of a set of points in the corresponding metric, together with a non-negative weight for each point, and a parameter k. The points represent cities, the weights denote costs of opening an airport in the corresponding city, and the parameter k is a maximum capacity of an airport. The goal is to construct a minimum cost network of airports and railways connecting all the cities, where railways correspond to edges connecting pairs of points, and the cost of a railway is equal to the distance between the corresponding points. The network is partitioned into components, where each component contains an open airport, and spans at most k cities. For the Euclidean case, any points in the plane can be used as Steiner vertices of the network. We obtain the first bicriteria approximation algorithm for AR for the general metric case, which yields a 4-approximate solution with a resource augmentation of the airport capacity k by a factor of 2. More generally, for any parameter 0 < p <= 1 where pk is an integer we develop a (4/3)(2 + 1/p)-approximation algorithm for metric AR with a resource augmentation by a factor of 1 + p.
Furthermore, we obtain the first constant factor approximation algorithm that does not resort to resource augmentation for AR in the Euclidean plane. Additionally, for the Euclidean setting we provide a quasi-polynomial time approximation scheme for the same problem with a resource augmentation by a factor of 1 + mu on the airport capacity, for any fixed mu > 0.
Cite as
Anna Adamaszek, Antonios Antoniadis, Amit Kumar, and Tobias Mömke. Approximating Airports and Railways. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 5:1-5:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{adamaszek_et_al:LIPIcs.STACS.2018.5,
author = {Adamaszek, Anna and Antoniadis, Antonios and Kumar, Amit and M\"{o}mke, Tobias},
title = {{Approximating Airports and Railways}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {5:1--5:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.5},
URN = {urn:nbn:de:0030-drops-85183},
doi = {10.4230/LIPIcs.STACS.2018.5},
annote = {Keywords: Network Design, Facility Location, Approximation Algorithms, PTAS, Metric, Euclidean}
}
Document
Property Testing for Bounded Degree Databases
Abstract
Aiming at extremely efficient algorithms for big data sets, we introduce property testing of relational databases of bounded degree. Our model generalises the bounded degree model for graphs (Goldreich and Ron, STOC 1997). We prove that in this model, if the databases have bounded tree-width, then every query definable in monadic second-order logic with modulo counting is testable with a constant number of oracle queries and polylogarithmic running time. This is the first logical meta-theorem in property testing of sparse models. Furthermore, we discuss conditions for the existence of uniform and non-uniform testers.
Cite as
Isolde Adler and Frederik Harwath. Property Testing for Bounded Degree Databases. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{adler_et_al:LIPIcs.STACS.2018.6,
author = {Adler, Isolde and Harwath, Frederik},
title = {{Property Testing for Bounded Degree Databases}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {6:1--6:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.6},
URN = {urn:nbn:de:0030-drops-85288},
doi = {10.4230/LIPIcs.STACS.2018.6},
annote = {Keywords: logic and databases, property testing, logical meta-theorems, bounded degree model, sublinear algorithms}
}
Document
Erdös-Pósa Property of Obstructions to Interval Graphs
Abstract
The duality between packing and covering problems lies at the heart of fundamental combinatorial proofs, as well as well-known algorithmic methods such as the primal-dual method for approximation and win/win-approach for parameterized analysis. The very essence of this duality is encompassed by a well-known property called the Erdös-Pósa property, which has been extensively studied for over five decades. Informally, we say that a class of graphs F admits the Erdös-Pósa property if there exists f such that for any graph G, either G has vertex-disjoint "copies" of the graphs in F, or there is a set S \subseteq V(G) of f(k) vertices that intersects all copies of the graphs in F. In the context of any graph class G, the most natural question that arises in this regard is as follows - do obstructions to G have the Erdös-Pósa property? Having this view in mind, we focus on the class of interval graphs. Structural properties of interval graphs are intensively studied, also as they lead to the design of polynomial-time algorithms for classic problems that are NP-hard on general graphs. Nevertheless, about one of the most basic properties of such graphs, namely, the Erdös-Pósa property, nothing is known. In this paper, we settle this anomaly: we prove that the family of obstructions to interval graphs - namely, the family of chordless cycles and ATs---admits the Erdös-Pósa property. Our main theorem immediately results in an algorithm to decide whether an input graph G has vertex-disjoint ATs and chordless cycles, or there exists a set of O(k^2 log k) vertices in G that hits all ATs and chordless cycles.
Cite as
Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. Erdös-Pósa Property of Obstructions to Interval Graphs. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{agrawal_et_al:LIPIcs.STACS.2018.7,
author = {Agrawal, Akanksha and Lokshtanov, Daniel and Misra, Pranabendu and Saurabh, Saket and Zehavi, Meirav},
title = {{Erd\"{o}s-P\'{o}sa Property of Obstructions to Interval Graphs}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {7:1--7:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.7},
URN = {urn:nbn:de:0030-drops-84815},
doi = {10.4230/LIPIcs.STACS.2018.7},
annote = {Keywords: Interval Graphs, Obstructions, Erd\"{o}s-P\'{o}sa Property}
}
Document
All Classical Adversary Methods are Equivalent for Total Functions
Abstract
We show that all known classical adversary lower bounds on randomized query complexity are equivalent for total functions, and are equal to the fractional block sensitivity fbs(f). That includes the Kolmogorov complexity bound of Laplante and Magniez and the earlier relational adversary bound of Aaronson. For partial functions, we show unbounded separations between fbs(f) and other adversary bounds, as well as between the relational and Kolmogorov complexity bounds.
We also show that, for partial functions, fractional block sensitivity cannot give lower bounds larger than sqrt(n * bs(f)), where n is the number of variables and bs(f) is the block sensitivity. Then we exhibit a partial function f that matches this upper bound, fbs(f) = Omega(sqrt(n * bs(f))).
Cite as
Andris Ambainis, Martins Kokainis, Krisjanis Prusis, and Jevgenijs Vihrovs. All Classical Adversary Methods are Equivalent for Total Functions. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{ambainis_et_al:LIPIcs.STACS.2018.8,
author = {Ambainis, Andris and Kokainis, Martins and Prusis, Krisjanis and Vihrovs, Jevgenijs},
title = {{All Classical Adversary Methods are Equivalent for Total Functions}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {8:1--8:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.8},
URN = {urn:nbn:de:0030-drops-84953},
doi = {10.4230/LIPIcs.STACS.2018.8},
annote = {Keywords: Randomized Query Complexity, Lower Bounds, Adversary Bounds, Fractional Block Sensitivity}
}
Document
Computing Hitting Set Kernels By AC^0-Circuits
Abstract
Given a hypergraph H = (V,E), what is the smallest subset X of V such that e and X are not disjoint for all e in E? This problem, known as the hitting set problem, is a basic problem in parameterized complexity theory. There are well-known kernelization algorithms for it, which get a hypergraph H and a number k as input and output a hypergraph H' such that (1) H has a hitting set of size k if, and only if, H' has such a hitting set and (2) the size of H' depends only on k and on the maximum cardinality d of edges in H. The algorithms run in polynomial time, but are highly sequential. Recently, it has been shown that one of them can be parallelized to a certain degree: one can compute hitting set kernels in parallel time O(d) - but it was conjectured that this is the best parallel algorithm possible. We refute this conjecture and show how hitting set kernels can be computed in constant parallel time. For our proof, we introduce a new, generalized notion of hypergraph sunflowers and show how iterated applications of the color coding technique can sometimes be collapsed into a single application.
Cite as
Max Bannach and Till Tantau. Computing Hitting Set Kernels By AC^0-Circuits. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{bannach_et_al:LIPIcs.STACS.2018.9,
author = {Bannach, Max and Tantau, Till},
title = {{Computing Hitting Set Kernels By AC^0-Circuits}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {9:1--9:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.9},
URN = {urn:nbn:de:0030-drops-84998},
doi = {10.4230/LIPIcs.STACS.2018.9},
annote = {Keywords: parallel computation, fixed-parameter tractability, kernelization}
}
Document
Parameterized (Approximate) Defective Coloring
Abstract
In Defective Coloring we are given a graph G=(V,E) and two integers chi_d,Delta^* and are asked if we can partition V into chi_d color classes, so that each class induces a graph of maximum degree Delta^*. We investigate the complexity of this generalization of Coloring with respect to several well-studied graph parameters, and show that the problem is W-hard parameterized by treewidth, pathwidth, tree-depth, or feedback vertex set, if chi_d=2. As expected, this hardness can be extended to larger values of chi_d for most of these parameters, with one surprising exception: we show that the problem is FPT parameterized by feedback vertex set for any chi_d != 2, and hence 2-coloring is the only hard case for this parameter. In addition to the above, we give an ETH-based lower bound for treewidth and pathwidth, showing that no algorithm can solve the
problem in n^{o(pw)}, essentially matching the complexity of an algorithm obtained with standard techniques.
We complement these results by considering the problem's approximability and show that, with respect to Delta^*, the problem admits an algorithm which for any epsilon>0 runs in time (tw/epsilon)^{O(tw)} and returns a solution with exactly the desired number of colors that approximates the optimal Delta^* within (1+epsilon). We also give a (tw)^{O(tw)} algorithm which achieves the desired Delta^* exactly while 2-approximating the minimum value of chi_d. We show that this is close to optimal, by establishing that no FPT algorithm can (under standard assumptions) achieve a better than 3/2-approximation to chi_d, even when an extra constant additive error is also allowed.
Cite as
Rémy Belmonte, Michael Lampis, and Valia Mitsou. Parameterized (Approximate) Defective Coloring. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{belmonte_et_al:LIPIcs.STACS.2018.10,
author = {Belmonte, R\'{e}my and Lampis, Michael and Mitsou, Valia},
title = {{Parameterized (Approximate) Defective Coloring}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {10:1--10:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.10},
URN = {urn:nbn:de:0030-drops-85304},
doi = {10.4230/LIPIcs.STACS.2018.10},
annote = {Keywords: Treewidth, Parameterized Complexity, Approximation, Coloring}
}
Document
The Relation between Polynomial Calculus, Sherali-Adams, and Sum-of-Squares Proofs
Abstract
We relate different approaches for proving the unsatisfiability of a system of real polynomial equations over Boolean variables. On the one hand, there are the static proof systems Sherali-Adams and
sum-of-squares (a.k.a. Lasserre), which are based on linear and
semi-definite programming relaxations. On the other hand, we
consider polynomial calculus, which is a dynamic algebraic proof
system that models Gröbner basis computations.
Our first result is that sum-of-squares simulates polynomial
calculus: any polynomial calculus refutation of degree d can be
transformed into a sum-of-squares refutation of degree 2d and only
polynomial increase in size.
In contrast, our second result shows that this is not the case for Sherali-Adams: there are systems of polynomial equations that have polynomial calculus refutations of degree 3 and polynomial size, but require Sherali-Adams refutations of large degree and exponential size.
A corollary of our first result is that the proof systems
Positivstellensatz and Positivstellensatz Calculus, which have been separated over non-Boolean polynomials, simulate
each other in the presence of Boolean axioms.
Cite as
Christoph Berkholz. The Relation between Polynomial Calculus, Sherali-Adams, and Sum-of-Squares Proofs. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{berkholz:LIPIcs.STACS.2018.11,
author = {Berkholz, Christoph},
title = {{The Relation between Polynomial Calculus, Sherali-Adams, and Sum-of-Squares Proofs}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {11:1--11:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.11},
URN = {urn:nbn:de:0030-drops-85279},
doi = {10.4230/LIPIcs.STACS.2018.11},
annote = {Keywords: Proof Complexity, Polynomial Calculus, Sum-of-Squares, Sherali-Adams}
}
Document
Genuine Lower Bounds for QBF Expansion
Abstract
We propose the first general technique for proving genuine lower bounds in expansion-based QBF proof systems. We present the technique in a framework centred on natural properties of winning strategies in the 'evaluation game' interpretation of QBF semantics. As applications, we prove an exponential proof-size lower bound for a whole class of formula families, and demonstrate the power of our approach over existing methods by providing alternative short proofs of two known hardness results. We also use our technique to deduce a result with manifest practical import: in the absence of propositional hardness, formulas separating the two major QBF expansion systems must have unbounded quantifier alternations.
Cite as
Olaf Beyersdorff and Joshua Blinkhorn. Genuine Lower Bounds for QBF Expansion. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{beyersdorff_et_al:LIPIcs.STACS.2018.12,
author = {Beyersdorff, Olaf and Blinkhorn, Joshua},
title = {{Genuine Lower Bounds for QBF Expansion}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {12:1--12:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.12},
URN = {urn:nbn:de:0030-drops-85174},
doi = {10.4230/LIPIcs.STACS.2018.12},
annote = {Keywords: QBF, proof complexity, lower-bound techniques, resolution}
}
Document
Efficient Oracles and Routing Schemes for Replacement Paths
Abstract
Real life graphs and networks are prone to failure of nodes (vertices) and links (edges). In particular, for a pair of nodes s and t and a failing edge e in an n-vertex unweighted graph G=(V(G),E(G)), the replacement path pi_{G-e}(s,t) is a shortest s-t path that avoids e. In this paper we present several efficient constructions that, for every (s,t) \in S x T, where S, T \subseteq V(G), and every e \in E(G), maintain the collection of all pi_{G-e}(s,t), either implicitly (i.e., through compact data structures a.k.a. distance sensitivity oracles (DSO)), or explicitly (i.e., through sparse subgraphs a.k.a. fault-tolerant preservers (FTP)).
More precisely, we provide the following results:
(1) DSO:
For every S,T \subseteq V(G), we construct a DSO for maintaining S x T distances under single edge (or vertex) faults. This DSO has size tilde{O}(n\sqrt{|S||T|}) and query time of
O(\sqrt{|S||T|}). At the expense of having quasi-polynomial query time,
the size of the oracle can be improved to tilde{O}(n|S|+|T|\sqrt{|S|n}), which is optimal for |T| = Omega(sqrt{n|S|}). When |T| = Omega(n^frac{3}{4} |S|^frac{1}{4}), the construction can be further refined in order to get a polynomial query time. We also consider the approximate additive setting, and show a family of DSOs that exhibits a tradeoff between the additive stretch and the size of the oracle. Finally, for the meaningful single-source case, the above result is complemented by a lower bound conditioned on the Set-Intersection conjecture. This lower bound establishes a separation between the oracle and the subgraph settings.
(2) FTP:
We show the construction of a path-reporting DSO of size tilde{O}(n^{4/3}(|S||T|)^{1/3}) reporting pi_{G-e}(s,t) in O(|pi_{G-e}(s,t)|+(n|S||T|)^{1/3}) time. Such a DSO can be transformed into a FTP having the same size, and moreover it can be elaborated in order to make it optimal (up to a poly-logarithmic factor) both in space and query time for the special case in which T=V(G). Our FTP improves over previous constructions when |T|=O(sqrt{|S|n}) (up to inverse poly-logarithmic factors).
(3) Routing and Labeling Schemes:
For the well-studied single-source setting, we present a novel routing scheme, that allows to route messages on pi_{G-e}(s,t) by using edge labels and routing tables of size tilde{O}(\sqrt{n}), and a header message of poly-logarithmic size. We also present a labeling scheme for the setting which is optimal in space up to constant factors.
Cite as
Davide Bilò, Keerti Choudhary, Luciano Gualà, Stefano Leucci, Merav Parter, and Guido Proietti. Efficient Oracles and Routing Schemes for Replacement Paths. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{bilo_et_al:LIPIcs.STACS.2018.13,
author = {Bil\`{o}, Davide and Choudhary, Keerti and Gual\`{a}, Luciano and Leucci, Stefano and Parter, Merav and Proietti, Guido},
title = {{Efficient Oracles and Routing Schemes for Replacement Paths}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {13:1--13:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.13},
URN = {urn:nbn:de:0030-drops-85249},
doi = {10.4230/LIPIcs.STACS.2018.13},
annote = {Keywords: Fault tolerant, Shortest path, Oracle, Routing}
}
Document
On the Tree Conjecture for the Network Creation Game
Abstract
Selfish Network Creation focuses on modeling real world networks from a game-theoretic point of view. One of the classic models by Fabrikant et al.[PODC'03] is the network creation game, where agents correspond to nodes in a network which buy incident edges for the price of alpha per edge to minimize their total distance to all other nodes. The model is well-studied but still has intriguing open problems. The most famous conjectures state that the price of anarchy is constant for all alpha and that for alpha >= n all equilibrium networks are trees.
We introduce a novel technique for analyzing stable networks for high edge-price alpha and employ it to improve on the best known bounds for both conjectures. In particular we show that for alpha > 4n-13 all equilibrium networks must be trees, which implies a constant price of anarchy for this range of alpha. Moreover, we also improve the constant upper bound on the price of anarchy for equilibrium trees.
Cite as
Davide Bilò and Pascal Lenzner. On the Tree Conjecture for the Network Creation Game. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{bilo_et_al:LIPIcs.STACS.2018.14,
author = {Bil\`{o}, Davide and Lenzner, Pascal},
title = {{On the Tree Conjecture for the Network Creation Game}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {14:1--14:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.14},
URN = {urn:nbn:de:0030-drops-85092},
doi = {10.4230/LIPIcs.STACS.2018.14},
annote = {Keywords: Algorithmic Game Theory, Network Creation Game, Price of Anarchy, Quality of Nash Equilibria}
}
Document
On Low for Speed Oracles
Abstract
Relativizing computations of Turing machines to an oracle is a central concept in the theory of computation, both in complexity theory and in computability theory(!). Inspired by lowness notions from computability theory, Allender introduced the concept of "low for speed" oracles. An oracle A is low for speed if relativizing to A has essentially no effect on computational complexity, meaning that if a decidable language can be decided in time f(n) with access to oracle A, then it can be decided in time poly(f(n)) without any oracle. The existence of non-computable such A's was later proven by Bayer and Slaman, who even constructed a computably enumerable one, and exhibited a number of properties of these oracles as well as interesting connections with computability theory. In this paper, we pursue this line of research, answering the questions left by Bayer and Slaman and give further evidence that the structure of the class of low for speed oracles is a very rich one.
Cite as
Laurent Bienvenu and Rodney Downey. On Low for Speed Oracles. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{bienvenu_et_al:LIPIcs.STACS.2018.15,
author = {Bienvenu, Laurent and Downey, Rodney},
title = {{On Low for Speed Oracles}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {15:1--15:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.15},
URN = {urn:nbn:de:0030-drops-85226},
doi = {10.4230/LIPIcs.STACS.2018.15},
annote = {Keywords: Lowness for speed, Oracle computations, Turing degrees}
}
Document
Large Flocks of Small Birds: on the Minimal Size of Population Protocols
Abstract
Population protocols are a well established model of distributed computation by mobile finite-state agents with very limited storage. A classical result establishes that population protocols compute exactly predicates definable in Presburger arithmetic. We initiate the study of the minimal amount of memory required to compute a given predicate as a function of its size. We present results on the predicates x >= n for n \in N, and more generally on the predicates corresponding to systems of linear inequalities. We show that they can be computed by protocols with O(log n) states (or, more generally, logarithmic in the coefficients of the predicate), and that, surprisingly, some families of predicates can be computed by protocols with O(log log n) states. We give essentially matching lower bounds for the class of 1-aware protocols.
Cite as
Michael Blondin, Javier Esparza, and Stefan Jaax. Large Flocks of Small Birds: on the Minimal Size of Population Protocols. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{blondin_et_al:LIPIcs.STACS.2018.16,
author = {Blondin, Michael and Esparza, Javier and Jaax, Stefan},
title = {{Large Flocks of Small Birds: on the Minimal Size of Population Protocols}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {16:1--16:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.16},
URN = {urn:nbn:de:0030-drops-85116},
doi = {10.4230/LIPIcs.STACS.2018.16},
annote = {Keywords: Population protocols, Presburger arithmetic}
}
Document
Communicating Finite-State Machines and Two-Variable Logic
Abstract
Communicating finite-state machines are a fundamental, well-studied model of finite-state processes that communicate via unbounded first-in first-out channels. We show that they are expressively equivalent to existential MSO logic with two first-order variables and the order relation.
Cite as
Benedikt Bollig, Marie Fortin, and Paul Gastin. Communicating Finite-State Machines and Two-Variable Logic. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{bollig_et_al:LIPIcs.STACS.2018.17,
author = {Bollig, Benedikt and Fortin, Marie and Gastin, Paul},
title = {{Communicating Finite-State Machines and Two-Variable Logic}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {17:1--17:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.17},
URN = {urn:nbn:de:0030-drops-85298},
doi = {10.4230/LIPIcs.STACS.2018.17},
annote = {Keywords: communicating finite-state machines, MSO logic, message sequence charts}
}
Document
On Approximating the Stationary Distribution of Time-reversible Markov Chains
Abstract
Approximating the stationary probability of a state in a Markov chain through Markov chain Monte Carlo techniques is, in general, inefficient. Standard random walk approaches require tilde{O}(tau/pi(v)) operations to approximate the probability pi(v) of a state v in a chain with mixing time tau, and even the best available techniques still have complexity tilde{O}(tau^1.5 / pi(v)^0.5); and since these complexities depend inversely on pi(v), they can grow beyond any bound in the size of the chain or in its mixing time.
In this paper we show that, for time-reversible Markov chains, there exists a simple randomized approximation algorithm that breaks this "small-pi(v) barrier".
Cite as
Marco Bressan, Enoch Peserico, and Luca Pretto. On Approximating the Stationary Distribution of Time-reversible Markov Chains. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{bressan_et_al:LIPIcs.STACS.2018.18,
author = {Bressan, Marco and Peserico, Enoch and Pretto, Luca},
title = {{On Approximating the Stationary Distribution of Time-reversible Markov Chains}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {18:1--18:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.18},
URN = {urn:nbn:de:0030-drops-84949},
doi = {10.4230/LIPIcs.STACS.2018.18},
annote = {Keywords: Markov chains, MCMC sampling, large graph algorithms, randomized algorithms, sublinear algorithms}
}
Document
On Singleton Arc Consistency for CSPs Defined by Monotone Patterns
Abstract
Singleton arc consistency is an important type of local consistency which has been recently shown to solve all constraint satisfaction problems (CSPs) over constraint languages of bounded width. We aim to characterise all classes of CSPs defined by a forbidden pattern that are solved by singleton arc consistency and closed under removing constraints. We identify five new patterns whose absence ensures solvability by singleton arc consistency, four of which are provably maximal and three of which generalise 2-SAT. Combined with simple counter-examples for other patterns, we make significant progress towards a complete classification.
Cite as
Clément Carbonnel, David A. Cohen, Martin C. Cooper, and Stanislav Zivny. On Singleton Arc Consistency for CSPs Defined by Monotone Patterns. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{carbonnel_et_al:LIPIcs.STACS.2018.19,
author = {Carbonnel, Cl\'{e}ment and Cohen, David A. and Cooper, Martin C. and Zivny, Stanislav},
title = {{On Singleton Arc Consistency for CSPs Defined by Monotone Patterns}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {19:1--19:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.19},
URN = {urn:nbn:de:0030-drops-84876},
doi = {10.4230/LIPIcs.STACS.2018.19},
annote = {Keywords: constraint satisfaction problems, forbidden patterns, singleton arc consistency}
}
Document
The Firing Squad Problem Revisited
Abstract
In the classical firing squad problem, an unknown number of nodes represented by identical finite state machines is arranged on a line and in each time unit each node may change its state according to its neighbors' states. Initially all nodes are passive, except one specific node located at an end of the line, which issues a fire command. This command needs to be propagated to all other nodes, so that eventually all nodes simultaneously enter some designated ``firing" state.
A natural extension of the firing squad problem, introduced in this paper, allows each node to postpone its participation in the squad for an arbitrary time, possibly forever, and firing is allowed only after all nodes decided to participate. This variant is highly relevant in the context of decentralized distributed computing, where processes have to coordinate for initiating various tasks simultaneously.
The main goal of this paper is to study the above variant of the firing squad problem under the assumptions that the nodes are infinite state machines, and that the inter-node communication links can be changed arbitrarily in each time unit, i.e., are defined by a dynamic graph. In this setting, we study the following fundamental question: what connectivity requirements enable a solution to the firing squad problem?
Our main result is an exact characterization of the dynamic graphs for which the firing squad problem can be solved. When restricted to static directed graphs, this characterization implies that the problem can be solved if and only if the graph is strongly connected. We also discuss how information on the number of nodes or on the diameter of the network, and the use of randomization, can improve the solutions to the problem.
Cite as
Bernadette Charron-Bost and Shlomo Moran. The Firing Squad Problem Revisited. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{charronbost_et_al:LIPIcs.STACS.2018.20,
author = {Charron-Bost, Bernadette and Moran, Shlomo},
title = {{The Firing Squad Problem Revisited}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {20:1--20:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.20},
URN = {urn:nbn:de:0030-drops-85195},
doi = {10.4230/LIPIcs.STACS.2018.20},
annote = {Keywords: Synchronization, Detection, Simultaneity, Dynamic Networks}
}
Document
Small-depth Multilinear Formula Lower Bounds for Iterated Matrix Multiplication, with Applications
Abstract
The complexity of Iterated Matrix Multiplication is a central theme in Computational Complexity theory, as the problem is closely related to the problem of separating various complexity classes within P. In this paper, we study the algebraic formula complexity of multiplying d many 2x2 matrices, denoted IMM_d, and show that the well-known divide-and-conquer algorithm cannot be significantly improved at any depth, as long as the formulas are multilinear.
Formally, for each depth Delta <= log d, we show that any product-depth Delta multilinear formula for IMM_d must have size exp(Omega(Delta d^{1/Delta})). It also follows from this that any multilinear circuit of product-depth Delta for the same polynomial of the above form must have a size of exp(Omega(d^{1/Delta})). In particular, any polynomial-sized multilinear formula for IMM_d must have depth Omega(log d), and any polynomial-sized multilinear circuit for IMM_d must have depth Omega(log d/log log d). Both these bounds are tight up to constant factors.
Our lower bound has the following consequences for multilinear formula complexity.
Depth-reduction: A well-known result of Brent (JACM 1974) implies that any formula of size s can be converted to one of size s^{O(1)} and depth O(log s); further, this reduction continues to hold for multilinear formulas. On the other hand, our lower bound implies that any depth-reduction in the multilinear setting cannot reduce the depth to o(log s) without a superpolynomial blow-up in size.
Separations from general formulas: Shpilka and Yehudayoff (FnTTCS 2010) asked whether general formulas can be more efficient than multilinear formulas for computing multilinear polynomials. Our result, along with a non-trivial upper bound for IMM_d implied by a result of Gupta, Kamath, Kayal and Saptharishi (SICOMP 2016), shows that for any size s and product-depth Delta = o(log s), general formulas of size s and product-depth Delta cannot be converted to multilinear formulas of size s^{O(1)} and product-depth Delta, when the underlying field has characteristic zero.
Cite as
Suryajith Chillara, Nutan Limaye, and Srikanth Srinivasan. Small-depth Multilinear Formula Lower Bounds for Iterated Matrix Multiplication, with Applications. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{chillara_et_al:LIPIcs.STACS.2018.21,
author = {Chillara, Suryajith and Limaye, Nutan and Srinivasan, Srikanth},
title = {{Small-depth Multilinear Formula Lower Bounds for Iterated Matrix Multiplication, with Applications}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {21:1--21:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.21},
URN = {urn:nbn:de:0030-drops-85235},
doi = {10.4230/LIPIcs.STACS.2018.21},
annote = {Keywords: Algebraic circuit complexity, Multilinear formulas, Lower Bounds}
}
Document
Upper and Lower Bounds for Dynamic Data Structures on Strings
Abstract
We consider a range of simply stated dynamic data structure problems on strings. An update changes one symbol in the input and a query asks us to compute some function of the pattern of length m and a substring of a longer text. We give both conditional and unconditional lower bounds for variants of exact matching with wildcards, inner product, and Hamming distance computation via a sequence of reductions. As an example, we show that there does not exist an O(m^{1/2-epsilon}) time algorithm for a large range of these problems unless the online Boolean matrix-vector multiplication conjecture is false. We also provide nearly matching upper bounds for most of the problems we consider.
Cite as
Raphaël Clifford, Allan Grønlund, Kasper Green Larsen, and Tatiana Starikovskaya. Upper and Lower Bounds for Dynamic Data Structures on Strings. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{clifford_et_al:LIPIcs.STACS.2018.22,
author = {Clifford, Rapha\"{e}l and Gr{\o}nlund, Allan and Larsen, Kasper Green and Starikovskaya, Tatiana},
title = {{Upper and Lower Bounds for Dynamic Data Structures on Strings}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {22:1--22:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.22},
URN = {urn:nbn:de:0030-drops-85088},
doi = {10.4230/LIPIcs.STACS.2018.22},
annote = {Keywords: exact pattern matching with wildcards, hamming distance, inner product, conditional lower bounds}
}
Document
Lower Bounds for Combinatorial Algorithms for Boolean Matrix Multiplication
Abstract
In this paper we propose models of combinatorial algorithms for the Boolean
Matrix Multiplication (BMM), and prove lower bounds on computing BMM in these models.
First, we give a relatively relaxed combinatorial model which is an extension of the model by Angluin (1976),
and we prove that the time required by any algorithm
for the BMM is at least Omega(n^3 / 2^{O( sqrt{ log n })}). Subsequently, we propose a more general model capable of simulating the
"Four Russian Algorithm". We prove a lower bound of Omega(n^{7/3} / 2^{O(sqrt{ log n })}) for the BMM under this model.
We use a special class of graphs, called (r,t)-graphs, originally discovered by Rusza and Szemeredi (1978),
along with randomization, to construct matrices that are hard instances for our combinatorial models.
Cite as
Debarati Das, Michal Koucký, and Michael Saks. Lower Bounds for Combinatorial Algorithms for Boolean Matrix Multiplication. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{das_et_al:LIPIcs.STACS.2018.23,
author = {Das, Debarati and Kouck\'{y}, Michal and Saks, Michael},
title = {{Lower Bounds for Combinatorial Algorithms for Boolean Matrix Multiplication}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {23:1--23:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.23},
URN = {urn:nbn:de:0030-drops-85050},
doi = {10.4230/LIPIcs.STACS.2018.23},
annote = {Keywords: Lower bounds, Combinatorial algorithm, Boolean matrix multiplication}
}
Document
Solving the Rubik's Cube Optimally is NP-complete
Abstract
In this paper, we prove that optimally solving an n x n x n Rubik's Cube is NP-complete by reducing from the Hamiltonian Cycle problem in square grid graphs. This improves the previous result that optimally solving an n x n x n Rubik's Cube with missing stickers is NP-complete. We prove this result first for the simpler case of the Rubik's Square--an n x n x 1 generalization of the Rubik's Cube--and then proceed with a similar but more complicated proof for the Rubik's Cube case. Our results hold both when the goal is make the sides monochromatic and when the goal is to put each sticker into a specific location.
Cite as
Erik D. Demaine, Sarah Eisenstat, and Mikhail Rudoy. Solving the Rubik's Cube Optimally is NP-complete. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{demaine_et_al:LIPIcs.STACS.2018.24,
author = {Demaine, Erik D. and Eisenstat, Sarah and Rudoy, Mikhail},
title = {{Solving the Rubik's Cube Optimally is NP-complete}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {24:1--24:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.24},
URN = {urn:nbn:de:0030-drops-85336},
doi = {10.4230/LIPIcs.STACS.2018.24},
annote = {Keywords: combinatorial puzzles, NP-hardness, group theory, Hamiltonicity}
}
Document
Approximation Algorithms for Scheduling with Resource and Precedence Constraints
Abstract
We study non-preemptive scheduling problems on identical parallel machines and uniformly related machines under both resource constraints and general precedence constraints between jobs. Our first result is an O(logn)-approximation algorithm for the objective of minimizing the makespan on parallel identical machines under resource and general precedence constraints. We then use this result as a subroutine to obtain an O(logn)-approximation algorithm for the
more general objective of minimizing the total weighted completion time on parallel identical machines under both constraints. Finally, we present an O(logm logn)-approximation algorithm for scheduling under these constraints on uniformly related machines. We show that these results can all be generalized to include the case where each job has a release time. This is the first upper bound on the approximability of this class of scheduling problems where both resource and general precedence constraints must be satisfied simultaneously.
Cite as
Gökalp Demirci, Henry Hoffmann, and David H. K. Kim. Approximation Algorithms for Scheduling with Resource and Precedence Constraints. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{demirci_et_al:LIPIcs.STACS.2018.25,
author = {Demirci, G\"{o}kalp and Hoffmann, Henry and Kim, David H. K.},
title = {{Approximation Algorithms for Scheduling with Resource and Precedence Constraints}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {25:1--25:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.25},
URN = {urn:nbn:de:0030-drops-85319},
doi = {10.4230/LIPIcs.STACS.2018.25},
annote = {Keywords: scheduling, resource, precedence, weighted completion time}
}
Document
Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices
Abstract
We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on one hand Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (Steiner vertices) in the optimum solution. In contrast to this we give an efficient parameterized approximation scheme (EPAS), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (PSAKS) for the considered parameter.
We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither of these an EPAS is likely to exist for the studied parameter: for Steiner Forest an easy observation shows that the problem is APX-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that computing a constant approximation for this parameter is W[1]-hard. Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree. Also we prove that there is an EPAS and a PSAKS for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter.
Cite as
Pavel Dvorák, Andreas Emil Feldmann, Dušan Knop, Tomáš Masarík, Tomáš Toufar, and Pavel Veselý. Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{dvorak_et_al:LIPIcs.STACS.2018.26,
author = {Dvor\'{a}k, Pavel and Feldmann, Andreas Emil and Knop, Du\v{s}an and Masar{\'\i}k, Tom\'{a}\v{s} and Toufar, Tom\'{a}\v{s} and Vesel\'{y}, Pavel},
title = {{Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {26:1--26:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.26},
URN = {urn:nbn:de:0030-drops-85158},
doi = {10.4230/LIPIcs.STACS.2018.26},
annote = {Keywords: Steiner Tree, Steiner Forest, Approximation Algorithms, Parameterized Algorithms, Lossy Kernelization}
}
Document
Finding List Homomorphisms from Bounded-treewidth Graphs to Reflexive Graphs: a Complete Complexity Characterization
Abstract
In the list homomorphism problem, the input consists of two graphs G and H, together with a list L(v) \subseteq V(H) for every vertex v \in V(G). The task is to find a homomorphism phi:V(G) -> V(H) respecting the lists, that is, we have that phi(v) \in L(v) for every v \in V(H) and if u and v are adjacent in G, then phi(u) and phi(v) are adjacent in H. If H is a fixed graph, then the problem is denoted LHom(H). We consider the reflexive version of the problem, where we assume that every vertex
in H has a self-loop. If is known that reflexive LHom(H) is polynomial-time solvable if H is an interval graph and it is NP-complete otherwise [Feder and Hell, JCTB 1998].
We explore the complexity of the problem parameterized by the treewidth tw(G) of the input graph G. If a tree decomposition of G of width tw(G) is given in the input, then the problem can be solved in time |V(H)|^{tw(G)} n^{O(1)} by naive dynamic programming. Our main result completely reveals when and by exactly how much this naive algorithm can be improved. We introduce a simple combinatorial invariant i^*(H), which is based on the existence of decompositions and incomparable sets, and show that this number should appear as the base of the exponent in the best possible running time. Specifically, we prove for every fixed non-interval graph H that
* If a tree decomposition of width tw(G) is given in the input, then the problem can be solved in time i^*(H)^{tw(G)} n^{O(1)}.
* Assuming the Strong Exponential-Time Hypothesis (SETH), the probem cannot be solved in time (i^*(H)-epsilon)^{tw(G)} n^{O(1)} for any epsilon>0.
Thus by matching upper and lower bounds, our result exactly characterizes for every fixed H the complexity of reflexive LHom(H) parameterized by treewidth.
Cite as
László Egri, Dániel Marx, and Pawel Rzazewski. Finding List Homomorphisms from Bounded-treewidth Graphs to Reflexive Graphs: a Complete Complexity Characterization. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{egri_et_al:LIPIcs.STACS.2018.27,
author = {Egri, L\'{a}szl\'{o} and Marx, D\'{a}niel and Rzazewski, Pawel},
title = {{Finding List Homomorphisms from Bounded-treewidth Graphs to Reflexive Graphs: a Complete Complexity Characterization}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {27:1--27:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.27},
URN = {urn:nbn:de:0030-drops-84867},
doi = {10.4230/LIPIcs.STACS.2018.27},
annote = {Keywords: graph homomorphism, list homomorphism, reflexive graph, treewidth}
}
Document
Small Resolution Proofs for QBF using Dependency Treewidth
Abstract
In spite of the close connection between the evaluation of quantified Boolean formulas (QBF) and propositional satisfiability (SAT), tools and techniques which exploit structural properties of SAT instances are known to fail for QBF. This is especially true for the structural parameter treewidth, which has allowed the design of successful algorithms for SAT but cannot be straightforwardly applied to QBF since it does not take into account the interdependencies between quantified variables.
In this work we introduce and develop dependency treewidth, a new structural parameter based on treewidth which allows the efficient solution of QBF instances. Dependency treewidth pushes the frontiers of tractability for QBF by overcoming the limitations of previously introduced variants of treewidth for QBF. We augment our results by developing algorithms for computing the decompositions that are required to use the parameter.
Cite as
Eduard Eiben, Robert Ganian, and Sebastian Ordyniak. Small Resolution Proofs for QBF using Dependency Treewidth. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{eiben_et_al:LIPIcs.STACS.2018.28,
author = {Eiben, Eduard and Ganian, Robert and Ordyniak, Sebastian},
title = {{Small Resolution Proofs for QBF using Dependency Treewidth}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {28:1--28:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.28},
URN = {urn:nbn:de:0030-drops-85135},
doi = {10.4230/LIPIcs.STACS.2018.28},
annote = {Keywords: QBF, treewidth, fixed parameter tractability, dependency schemes}
}
Document
Lossy Kernels for Connected Dominating Set on Sparse Graphs
Abstract
For alpha > 1, an alpha-approximate (bi-)kernel for a problem Q is a polynomial-time algorithm that takes as input an instance (I, k) of Q and outputs an instance (I',k') (of a problem Q') of size bounded by a function of k such that, for every c >= 1, a c-approximate solution for the new instance can be turned into a (c alpha)-approximate solution of the original instance in polynomial time. This framework of lossy kernelization was recently introduced by Lokshtanov et al. We study Connected Dominating Set (and its distance-r variant) parameterized by solution size on sparse graph classes like biclique-free graphs, classes of bounded expansion, and nowhere dense classes. We prove that for every alpha > 1, Connected Dominating Set admits a polynomial-size alpha-approximate (bi-)kernel on all the aforementioned classes. Our results are in sharp contrast to the kernelization complexity of Connected Dominating Set, which is known to not admit a polynomial kernel even on 2-degenerate graphs and graphs of bounded expansion, unless NP \subseteq coNP/poly. We complement our results by the following conditional lower bound. We show that if a class C is somewhere dense and closed under taking subgraphs, then for some value of r \in N there cannot exist an alpha-approximate bi-kernel for the (Connected) Distance-r Dominating Set problem on C for any alpha > 1 (assuming the Gap Exponential Time Hypothesis).
Cite as
Eduard Eiben, Mithilesh Kumar, Amer E. Mouawad, Fahad Panolan, and Sebastian Siebertz. Lossy Kernels for Connected Dominating Set on Sparse Graphs. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{eiben_et_al:LIPIcs.STACS.2018.29,
author = {Eiben, Eduard and Kumar, Mithilesh and Mouawad, Amer E. and Panolan, Fahad and Siebertz, Sebastian},
title = {{Lossy Kernels for Connected Dominating Set on Sparse Graphs}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {29:1--29:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.29},
URN = {urn:nbn:de:0030-drops-85027},
doi = {10.4230/LIPIcs.STACS.2018.29},
annote = {Keywords: Lossy Kernelization, Connected Dominating Set, Sparse Graph Classes}
}
Document
The Intersection Problem for Finite Monoids
Abstract
We investigate the intersection problem for finite monoids, which asks for a given set of regular languages, represented by recognizing morphisms to finite monoids from a variety V, whether there exists a word contained in their intersection. Our main result is that the problem is PSPACE-complete if V is contained in DS and NP-complete if V is non-trivial and contained in DO. Our NP-algorithm for the case that V is contained in DO uses novel methods, based on compression techniques and combinatorial properties of DO. We also show that the problem is log-space reducible to the intersection problem for deterministic finite automata (DFA) and that a variant of the problem is log-space reducible to the membership problem for transformation monoids. In light of these reductions, our hardness results can be seen as a generalization of both a classical result by Kozen and a theorem by Beaudry, McKenzie and Thérien.
Cite as
Lukas Fleischer and Manfred Kufleitner. The Intersection Problem for Finite Monoids. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{fleischer_et_al:LIPIcs.STACS.2018.30,
author = {Fleischer, Lukas and Kufleitner, Manfred},
title = {{The Intersection Problem for Finite Monoids}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {30:1--30:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.30},
URN = {urn:nbn:de:0030-drops-85079},
doi = {10.4230/LIPIcs.STACS.2018.30},
annote = {Keywords: intersection problem, finite monoid, recognizing morphism, complexity}
}
Document
Automata Theory on Sliding Windows
Abstract
In a recent paper we analyzed the space complexity of streaming algorithms whose goal is to decide membership of a sliding window to a fixed language. For the class of regular languages we proved a space trichotomy theorem: for every regular language the optimal space bound is either constant, logarithmic or linear. In this paper we continue this line of research: We present natural characterizations for the constant and logarithmic space classes and establish tight relationships to the concept of language growth. We also analyze the space complexity with respect to automata size and prove almost matching lower and upper bounds. Finally, we consider the decision problem whether a language given by a DFA/NFA admits a sliding window algorithm using logarithmic/constant space.
Cite as
Moses Ganardi, Danny Hucke, Daniel König, Markus Lohrey, and Konstantinos Mamouras. Automata Theory on Sliding Windows. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{ganardi_et_al:LIPIcs.STACS.2018.31,
author = {Ganardi, Moses and Hucke, Danny and K\"{o}nig, Daniel and Lohrey, Markus and Mamouras, Konstantinos},
title = {{Automata Theory on Sliding Windows}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {31:1--31:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.31},
URN = {urn:nbn:de:0030-drops-84851},
doi = {10.4230/LIPIcs.STACS.2018.31},
annote = {Keywords: regular languages, sliding window algorithms}
}
Document
Knapsack Problems for Wreath Products
Abstract
In recent years, knapsack problems for (in general non-commutative) groups have attracted attention. In this paper, the knapsack problem for wreath products is studied. It turns out that decidability of knapsack is not preserved under wreath product. On the other hand, the class of knapsack-semilinear groups, where solutions sets of knapsack equations are effectively semilinear, is closed under wreath product. As a consequence, we obtain the decidability of knapsack for free solvable groups. Finally, it is shown that for every non-trivial abelian group G, knapsack (as well as the related subset sum problem)
for the wreath product G \wr Z is NP-complete.
Cite as
Moses Ganardi, Daniel König, Markus Lohrey, and Georg Zetzsche. Knapsack Problems for Wreath Products. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{ganardi_et_al:LIPIcs.STACS.2018.32,
author = {Ganardi, Moses and K\"{o}nig, Daniel and Lohrey, Markus and Zetzsche, Georg},
title = {{Knapsack Problems for Wreath Products}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {32:1--32:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.32},
URN = {urn:nbn:de:0030-drops-85201},
doi = {10.4230/LIPIcs.STACS.2018.32},
annote = {Keywords: knapsack, wreath products, decision problems in group theory}
}
Document
On Structural Parameterizations of the Bounded-Degree Vertex Deletion Problem
Abstract
We study the parameterized complexity of the Bounded-Degree Vertex Deletion problem (BDD), where the aim is to find a maximum induced subgraph whose maximum degree is below a given degree bound. Our focus lies on parameters that measure the structural properties of the input instance. We first show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treedepth, and even the size of a minimum vertex deletion set into graphs of pathwidth and treedepth at most three. We thereby resolve the main open question stated in Betzler, Bredereck, Niedermeier and Uhlmann (2012) concerning the complexity of BDD parameterized by the feedback vertex set number. On the positive side, we obtain fixed-parameter algorithms for the problem with respect to the decompositional parameter treecut width and a novel problem-specific parameter called the core fracture number.
Cite as
Robert Ganian, Fabian Klute, and Sebastian Ordyniak. On Structural Parameterizations of the Bounded-Degree Vertex Deletion Problem. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{ganian_et_al:LIPIcs.STACS.2018.33,
author = {Ganian, Robert and Klute, Fabian and Ordyniak, Sebastian},
title = {{On Structural Parameterizations of the Bounded-Degree Vertex Deletion Problem}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {33:1--33:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.33},
URN = {urn:nbn:de:0030-drops-85140},
doi = {10.4230/LIPIcs.STACS.2018.33},
annote = {Keywords: bounded-degree vertex deletion, feedback vertex set, parameterized algorithms, treecut width}
}
Document
Dependences in Strategy Logic
Abstract
Strategy Logic (SL) is a very expressive temporal logic for specifying and verifying properties of multi-agent systems: in SL, one can quantify over strategies, assign them to agents, and express LTL properties of the resulting plays. Such a powerful framework has two drawbacks: First, model checking SL has non-elementary complexity; second, the exact semantics of SL is rather intricate, and may not correspond to what is expected. In this paper, we focus on strategy dependences in SL, by tracking how existentially-quantified strategies in a formula may (or may not) depend on other strategies selected in the formula, revisiting the approach of [Mogavero et al., Reasoning about strategies: On the model-checking problem, 2014]. We explain why elementary dependences, as defined by Mogavero et al., do not exactly capture the intended concept of behavioral strategies. We address this discrepancy by introducing timeline dependences, and exhibit a large fragment of SL for which model checking can be performed in 2-EXPTIME under this new semantics.
Cite as
Patrick Gardy, Patricia Bouyer, and Nicolas Markey. Dependences in Strategy Logic. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{gardy_et_al:LIPIcs.STACS.2018.34,
author = {Gardy, Patrick and Bouyer, Patricia and Markey, Nicolas},
title = {{Dependences in Strategy Logic}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {34:1--34:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.34},
URN = {urn:nbn:de:0030-drops-85320},
doi = {10.4230/LIPIcs.STACS.2018.34},
annote = {Keywords: strategic reasoning, strategy logic, dependences, behavioural strategies.}
}
Document
Colouring Square-Free Graphs without Long Induced Paths
Abstract
The Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a given integer k such that no two adjacent vertices are coloured alike. The complexity of Colouring is fully understood for graph classes characterized by one forbidden induced subgraph H. Despite a huge body of existing work, there are still major complexity gaps if two induced subgraphs H_1 and H_2 are forbidden. We let H_1 be the s-vertex cycle C_s and H_2 be the t-vertex path P_t. We show that Colouring is polynomial-time solvable for s=4 and t<=6, which unifies several known results for Colouring on (H_1,H_2)-free graphs. Our algorithm is based on a novel decomposition theorem for (C_4,P_6)-free graphs without clique cutsets into homogeneous pairs of sets and a new framework for bounding the clique-width of a graph by the clique-width of its subgraphs induced by homogeneous pairs of sets. To apply this framework, we also need to use divide-and-conquer to bound the clique-width of subgraphs induced by homogeneous pairs of sets. To complement our positive result we also prove that Colouring is NP-complete for s=4 and t>=9, which is the first hardness result on Colouring for (C_4,P_t)-free graphs.
Cite as
Serge Gaspers, Shenwei Huang, and Daniel Paulusma. Colouring Square-Free Graphs without Long Induced Paths. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{gaspers_et_al:LIPIcs.STACS.2018.35,
author = {Gaspers, Serge and Huang, Shenwei and Paulusma, Daniel},
title = {{Colouring Square-Free Graphs without Long Induced Paths}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {35:1--35:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.35},
URN = {urn:nbn:de:0030-drops-84922},
doi = {10.4230/LIPIcs.STACS.2018.35},
annote = {Keywords: graph colouring, hereditary graph class, clique-width, cycle, path}
}
Document
Optimal Dislocation with Persistent Errors in Subquadratic Time
Abstract
We study the problem of sorting N elements in presence of persistent errors in comparisons: In this classical model, each comparison between two elements is wrong independently with some probability p, but repeating the same comparison gives always the same result. The best known algorithms for this problem have running time O(N^2) and achieve an optimal maximum dislocation of O(log N) for constant error probability. Note that no algorithm can achieve dislocation o(log N), regardless of its running time.
In this work we present the first subquadratic time algorithm with optimal maximum dislocation: Our algorithm runs in tilde{O}(N^{3/2}) time and guarantees O(log N) maximum dislocation with high probability. Though the first version of our algorithm is randomized, it can be derandomized by extracting the necessary random bits from the results of the comparisons (errors).
Cite as
Barbara Geissmann, Stefano Leucci, Chih-Hung Liu, and Paolo Penna. Optimal Dislocation with Persistent Errors in Subquadratic Time. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 36:1-36:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{geissmann_et_al:LIPIcs.STACS.2018.36,
author = {Geissmann, Barbara and Leucci, Stefano and Liu, Chih-Hung and Penna, Paolo},
title = {{Optimal Dislocation with Persistent Errors in Subquadratic Time}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {36:1--36:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.36},
URN = {urn:nbn:de:0030-drops-85266},
doi = {10.4230/LIPIcs.STACS.2018.36},
annote = {Keywords: sorting, recurrent comparison errors, maximum dislocation}
}
Document
An Improved Bound for Random Binary Search Trees with Concurrent Insertions
Abstract
Recently, Aspnes and Ruppert (DISC 2016) defined the following simple random experiment to determine the impact of concurrency on the performance of binary search trees: n randomly permuted keys arrive one at a time. When a new key arrives, it is first placed into a buffer of size c. Whenever the buffer is full, or when all keys have arrived, an adversary chooses one key from the buffer and inserts it into the binary search tree.
The ability of the adversary to choose the next key to insert among c buffered keys, models a distributed system, where up to c processes try to insert keys concurrently. Aspnes and Ruppert showed that the expected average depth of nodes in the resulting tree is O(log(n) + c) for a comparison-based adversary, which can only take the relative order of arrived keys into account. We generalize and strengthen this result. In particular, we allow an adversary that knows the actual values of all keys that have arrived, and show that the resulting expected average node depth is D_{avg}(n) + O(c), where D_{avg}(n) = 2ln(n) - Theta(1) is the expected average node depth of a random tree obtained in the standard unbuffered version of this experiment. Extending the bound by Aspnes and Ruppert to this stronger adversary model answers one of their open questions.
Cite as
George Giakkoupis and Philipp Woelfel. An Improved Bound for Random Binary Search Trees with Concurrent Insertions. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 37:1-37:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{giakkoupis_et_al:LIPIcs.STACS.2018.37,
author = {Giakkoupis, George and Woelfel, Philipp},
title = {{An Improved Bound for Random Binary Search Trees with Concurrent Insertions}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {37:1--37:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.37},
URN = {urn:nbn:de:0030-drops-85108},
doi = {10.4230/LIPIcs.STACS.2018.37},
annote = {Keywords: random binary search tree, buffer, average depth, concurrent data structures}
}
Document
String Periods in the Order-Preserving Model
Abstract
The order-preserving model (op-model, in short) was introduced quite recently but has already attracted significant attention because of its applications in data analysis. We introduce several types of periods in this setting (op-periods). Then we give algorithms to compute these periods in time O(n), O(n log log n), O(n log^2 log n/log log log n), O(n log n) depending on the type of periodicity. In the most general variant the number of different periods can be as big as Omega(n^2), and a compact representation is needed. Our algorithms require novel combinatorial insight into the properties of such periods.
Cite as
Garance Gourdel, Tomasz Kociumaka, Jakub Radoszewski, Wojciech Rytter, Arseny Shur, and Tomasz Walen. String Periods in the Order-Preserving Model. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{gourdel_et_al:LIPIcs.STACS.2018.38,
author = {Gourdel, Garance and Kociumaka, Tomasz and Radoszewski, Jakub and Rytter, Wojciech and Shur, Arseny and Walen, Tomasz},
title = {{String Periods in the Order-Preserving Model}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {38:1--38:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.38},
URN = {urn:nbn:de:0030-drops-85064},
doi = {10.4230/LIPIcs.STACS.2018.38},
annote = {Keywords: order-preserving pattern matching, period, efficient algorithm}
}
Document
Beyond JWP: A Tractable Class of Binary VCSPs via M-Convex Intersection
Abstract
A binary VCSP is a general framework for the minimization problem of a function represented as the sum of unary and binary cost functions.An important line of VCSP research is to investigate what functions can be solved in polynomial time.
Cooper-Zivny classified the tractability of binary VCSP instances according to the concept of "triangle,"
and showed that the only interesting tractable case is the one induced by the joint winner property (JWP).
Recently, Iwamasa-Murota-Zivny made a link between VCSP and discrete convex analysis, showing that a function satisfying the JWP can be transformed into a function represented as the sum of two M-convex functions, which can be minimized in polynomial time via an M-convex intersection algorithm if the value oracle of each M-convex function is given.
In this paper,
we give an algorithmic answer to a natural question: What binary finite-valued CSP instances can be solved in polynomial time via an M-convex intersection algorithm?
We solve this problem by devising a polynomial-time algorithm for obtaining a concrete form of the representation in the representable case.
Our result presents a larger tractable class of binary finite-valued CSPs, which properly contains the JWP class.
Cite as
Hiroshi Hirai, Yuni Iwamasa, Kazuo Murota, and Stanislav Zivny. Beyond JWP: A Tractable Class of Binary VCSPs via M-Convex Intersection. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{hirai_et_al:LIPIcs.STACS.2018.39,
author = {Hirai, Hiroshi and Iwamasa, Yuni and Murota, Kazuo and Zivny, Stanislav},
title = {{Beyond JWP: A Tractable Class of Binary VCSPs via M-Convex Intersection}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {39:1--39:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.39},
URN = {urn:nbn:de:0030-drops-85042},
doi = {10.4230/LIPIcs.STACS.2018.39},
annote = {Keywords: valued constraint satisfaction problems, discrete convex analysis, M-convexity}
}
Document
Nonuniform Reductions and NP-Completeness
Abstract
Nonuniformity is a central concept in computational complexity with powerful connections to circuit complexity and randomness. Nonuniform reductions have been used to study the isomorphism conjecture for NP and completeness for larger complexity classes. We study the power of nonuniform reductions for NP0completeness, obtaining both separations and upper bounds for nonuniform completeness vs uniform complessness in NP.
Under various hypotheses, we obtain the following separations:
1. There is a set complete for NP under nonuniform many-one reductions, but not under uniform many-one reductions. This is true even with a single bit of nonuniform advice.
2. There is a set complete for NP under nonuniform many-one reductions with polynomial-size advice, but not under uniform Turing reductions. That is, polynomial nonuniformity is stronger than a polynomial number of queries.
3. For any fixed polynomial p(n), there is a set complete for NP under uniform 2-truth-table reductions, but not under nonuniform many-one reductions that use p(n) advice. That is, giving a uniform reduction a second query makes it more powerful than a nonuniform reduction with fixed polynomial advice.
4. There is a set complete for NP under nonuniform many-one reductions with polynomial ad- vice, but not under nonuniform many-one reductions with logarithmic advice. This hierarchy theorem also holds for other reducibilities, such as truth-table and Turing.
We also consider uniform upper bounds on nonuniform completeness. Hirahara (2015) showed that unconditionally every set that is complete for NP under nonuniform truth-table reductions that use logarithmic advice is also uniformly Turing-complete. We show that under a derandomization hypothesis, the same statement for truth-table reductions and truth-table completeness also holds.
Cite as
John M. Hitchcock and Hadi Shafei. Nonuniform Reductions and NP-Completeness. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 40:1-40:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{hitchcock_et_al:LIPIcs.STACS.2018.40,
author = {Hitchcock, John M. and Shafei, Hadi},
title = {{Nonuniform Reductions and NP-Completeness}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {40:1--40:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.40},
URN = {urn:nbn:de:0030-drops-85217},
doi = {10.4230/LIPIcs.STACS.2018.40},
annote = {Keywords: computational complexity, NP-completeness, reducibility, nonuniform complexity}
}
Document
On the Power of Tree-Depth for Fully Polynomial FPT Algorithms
Abstract
There are many classical problems in P whose time complexities have not been improved over the past decades.
Recent studies of "Hardness in P" have revealed that, for several of such problems, the current fastest algorithm is the best possible under some complexity assumptions.
To bypass this difficulty, the concept of "FPT inside P" has been introduced.
For a problem with the current best time complexity O(n^c), the goal is to design an algorithm running in k^{O(1)}n^{c'} time for a parameter k and a constant c'<c.
In this paper, we investigate the complexity of graph problems in P parameterized by tree-depth, a graph parameter related to tree-width.
We show that a simple divide-and-conquer method can solve many graph problems, including
Weighted Matching, Negative Cycle Detection, Minimum Weight Cycle, Replacement Paths, and 2-hop Cover,
in O(td m) time or O(td (m+nlog n)) time, where td is the tree-depth of the input graph.
Because any graph of tree-width tw has tree-depth at most (tw+1)log_2 n, our algorithms also run in O(tw mlog n) time or O(tw (m+nlog n)log n) time.
These results match or improve the previous best algorithms parameterized by tree-width.
Especially, we solve an open problem of fully polynomial FPT algorithm for Weighted Matching parameterized by tree-width posed by Fomin et al. (SODA 2017).
Cite as
Yoichi Iwata, Tomoaki Ogasawara, and Naoto Ohsaka. On the Power of Tree-Depth for Fully Polynomial FPT Algorithms. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{iwata_et_al:LIPIcs.STACS.2018.41,
author = {Iwata, Yoichi and Ogasawara, Tomoaki and Ohsaka, Naoto},
title = {{On the Power of Tree-Depth for Fully Polynomial FPT Algorithms}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {41:1--41:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.41},
URN = {urn:nbn:de:0030-drops-85255},
doi = {10.4230/LIPIcs.STACS.2018.41},
annote = {Keywords: Fully Polynomial FPT Algorithm, Tree-Depth, Divide-and-Conquer}
}
Document
A Unified Polynomial-Time Algorithm for Feedback Vertex Set on Graphs of Bounded Mim-Width
Abstract
We give a first polynomial-time algorithm for (Weighted) Feedback Vertex Set on graphs of bounded maximum induced matching width (mim-width). Explicitly, given a branch decomposition of mim-width w, we give an n^{O(w)}-time algorithm that solves Feedback Vertex Set. This provides a unified algorithm for many well-known classes, such as Interval graphs and Permutation graphs, and furthermore, it gives the first polynomial-time algorithms for other classes of bounded mim-width, such as Circular Permutation and Circular k-Trapezoid graphs for fixed k. In all these classes the decomposition is computable in polynomial time, as shown by Belmonte and Vatshelle [Theor. Comput. Sci. 2013].
We show that powers of graphs of tree-width w-1 or path-width w and powers of graphs of clique-width w have mim-width at most w. These results extensively provide new classes of bounded mim-width. We prove a slight strengthening of the first statement which implies that, surprisingly, Leaf Power graphs which are of importance in the field of phylogenetic studies have mim-width at most 1. Given a tree decomposition of width w-1, a path decomposition of width w, or a clique-width w-expression of a graph G, one can for any value of k find a mim-width decomposition of its k-power in polynomial time, and apply our algorithm to solve Feedback Vertex Set on the k-power in time n^{O(w)}.
In contrast to Feedback Vertex Set, we show that Hamiltonian Cycle is NP-complete even on graphs of linear mim-width 1, which further hints at the expressive power of the mim-width parameter.
Cite as
Lars Jaffke, O-joung Kwon, and Jan Arne Telle. A Unified Polynomial-Time Algorithm for Feedback Vertex Set on Graphs of Bounded Mim-Width. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 42:1-42:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{jaffke_et_al:LIPIcs.STACS.2018.42,
author = {Jaffke, Lars and Kwon, O-joung and Telle, Jan Arne},
title = {{A Unified Polynomial-Time Algorithm for Feedback Vertex Set on Graphs of Bounded Mim-Width}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {42:1--42:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.42},
URN = {urn:nbn:de:0030-drops-85348},
doi = {10.4230/LIPIcs.STACS.2018.42},
annote = {Keywords: graph width parameters, graph classes, feedback vertex set, leaf powers}
}
Document
Generalizing the Kawaguchi-Kyan Bound to Stochastic Parallel Machine Scheduling
Abstract
Minimizing the sum of weighted completion times on m identical
parallel machines is one of the most important and classical
scheduling problems. For the stochastic variant where processing
times of jobs are random variables, Möhring, Schulz, and Uetz (1999) presented the first and still best known approximation result,
achieving, for arbitrarily many machines, performance
ratio 1+1/2(1+Delta), where Delta is an upper bound on the
squared coefficient of variation of the processing times. We prove
performance ratio 1+1/2(sqrt(2)-1)(1+Delta)
for the same
underlying algorithm---the Weighted Shortest Expected Processing
Time (WSEPT) rule. For the special case of deterministic scheduling
(i.e., Delta=0), our bound matches the tight performance
ratio 1/2(1+sqrt(2)) of this algorithm (WSPT rule), derived by
Kawaguchi and Kyan in a 1986 landmark paper. We present several
further improvements for WSEPT's performance ratio, one of them
relying on a carefully refined analysis of WSPT yielding, for every
fixed number of machines m, WSPT's exact performance ratio of
order 1/2(1+sqrt(2))-O(1/m^2).
Cite as
Sven Jäger and Martin Skutella. Generalizing the Kawaguchi-Kyan Bound to Stochastic Parallel Machine Scheduling. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 43:1-43:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{jager_et_al:LIPIcs.STACS.2018.43,
author = {J\"{a}ger, Sven and Skutella, Martin},
title = {{Generalizing the Kawaguchi-Kyan Bound to Stochastic Parallel Machine Scheduling}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {43:1--43:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.43},
URN = {urn:nbn:de:0030-drops-85034},
doi = {10.4230/LIPIcs.STACS.2018.43},
annote = {Keywords: Stochastic Scheduling, Parallel Machines, Approximation Algorithm, List Scheduling, Weighted Shortest (Expected) Processing Time Rule}
}
Document
Space-Efficient Algorithms for Longest Increasing Subsequence
Abstract
Given a sequence of integers, we want to find a longest increasing subsequence of the sequence. It is known that this problem can be solved in O(n log n) time and space. Our goal in this paper is to reduce the space consumption while keeping the time complexity small. For sqrt(n) <= s <= n, we present algorithms that use O(s log n) bits and O(1/s n^2 log n) time for computing the length of a longest increasing subsequence, and O(1/s n^2 log^2 n) time for finding an actual subsequence. We also show that the time complexity of our algorithms is optimal up to polylogarithmic factors in the framework of sequential access algorithms with the prescribed amount of space.
Cite as
Masashi Kiyomi, Hirotaka Ono, Yota Otachi, Pascal Schweitzer, and Jun Tarui. Space-Efficient Algorithms for Longest Increasing Subsequence. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{kiyomi_et_al:LIPIcs.STACS.2018.44,
author = {Kiyomi, Masashi and Ono, Hirotaka and Otachi, Yota and Schweitzer, Pascal and Tarui, Jun},
title = {{Space-Efficient Algorithms for Longest Increasing Subsequence}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {44:1--44:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.44},
URN = {urn:nbn:de:0030-drops-84911},
doi = {10.4230/LIPIcs.STACS.2018.44},
annote = {Keywords: longest increasing subsequence, patience sorting, space-efficient algorithm}
}
Document
Rational, Recognizable, and Aperiodic Sets in the Partially Lossy Queue Monoid
Abstract
Partially lossy queue monoids (or plq monoids) model the behavior of queues that can forget arbitrary parts of their content. While many decision problems on recognizable subsets in the plq monoid are decidable, most of them are undecidable if the sets are rational. In particular, in this monoid the classes of rational and recognizable subsets do not coincide. By restricting multiplication and iteration in the construction of rational sets and by allowing complementation we obtain precisely the class of recognizable sets. From these special rational expressions we can obtain an MSO logic describing the recognizable subsets. Moreover, we provide similar results for the class of aperiodic subsets in the plq monoid.
Cite as
Chris Köcher. Rational, Recognizable, and Aperiodic Sets in the Partially Lossy Queue Monoid. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 45:1-45:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{kocher:LIPIcs.STACS.2018.45,
author = {K\"{o}cher, Chris},
title = {{Rational, Recognizable, and Aperiodic Sets in the Partially Lossy Queue Monoid}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {45:1--45:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.45},
URN = {urn:nbn:de:0030-drops-84839},
doi = {10.4230/LIPIcs.STACS.2018.45},
annote = {Keywords: Partially Lossy Queues, Transformation Monoid, Rational Sets, Recognizable Sets, Aperiodic Sets, MSO logic}
}
Document
Relations Between Greedy and Bit-Optimal LZ77 Encodings
Abstract
This paper investigates the size in bits of the LZ77 encoding, which is the most popular and efficient variant of the Lempel-Ziv encodings used in data compression. We prove that, for a wide natural class of variable-length encoders for LZ77 phrases, the size of the greedily constructed LZ77 encoding on constant alphabets is within a factor O(log n / log log log n) of the optimal LZ77 encoding, where n is the length of the processed string. We describe a series of examples showing that, surprisingly, this bound is tight, thus improving both the previously known upper and lower bounds. Further, we obtain a more detailed bound O(min{z, log n / log log z}), which uses the number z of phrases in the greedy LZ77 encoding as a parameter, and construct a series of examples showing that this bound is tight even for binary alphabet. We then investigate the problem on non-constant alphabets: we show that the known O(log n) bound is tight even for alphabets of logarithmic size, and provide tight bounds for some other important cases.
Cite as
Dmitry Kosolobov. Relations Between Greedy and Bit-Optimal LZ77 Encodings. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{kosolobov:LIPIcs.STACS.2018.46,
author = {Kosolobov, Dmitry},
title = {{Relations Between Greedy and Bit-Optimal LZ77 Encodings}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {46:1--46:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.46},
URN = {urn:nbn:de:0030-drops-84894},
doi = {10.4230/LIPIcs.STACS.2018.46},
annote = {Keywords: Lempel-Ziv, LZ77 encoding, greedy LZ77, bit optimal LZ77}
}
Document
Width of Non-deterministic Automata
Abstract
We introduce a measure called width, quantifying the amount of nondeterminism in automata. Width generalises the notion of good-for-games (GFG) automata, that correspond to NFAs of width 1, and where an accepting run can be built on-the-fly on any accepted input. We describe an incremental determinisation construction on NFAs, which can be more efficient than the full powerset determinisation, depending on the width of the input NFA. This construction can be generalised to infinite words, and is particularly well-suited to coBüchi automata in this context. For coBüchi automata, this procedure can be used to compute either a deterministic automaton or a GFG one, and it is algorithmically more efficient in this last case. We show this fact by proving that checking whether a coBüchi automaton is determinisable by pruning is NP-complete. On finite or infinite words, we show that computing the width of an automaton is PSPACE-hard.
Cite as
Denis Kuperberg and Anirban Majumdar. Width of Non-deterministic Automata. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{kuperberg_et_al:LIPIcs.STACS.2018.47,
author = {Kuperberg, Denis and Majumdar, Anirban},
title = {{Width of Non-deterministic Automata}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {47:1--47:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.47},
URN = {urn:nbn:de:0030-drops-84963},
doi = {10.4230/LIPIcs.STACS.2018.47},
annote = {Keywords: width, non-deterministic automata, determinisation, good-for-games, complexity}
}
Document
Computing the Longest Common Prefix of a Context-free Language in Polynomial Time
Abstract
We present two structural results concerning the longest common prefixes of non-empty languages.
First, we show that the longest common prefix of the language generated by a context-free grammar of size N
equals the longest common prefix of the same grammar where the heights of the derivation trees are bounded by
4N.
Second, we show that each non-empty language L has a representative subset of at most three elements which behaves
like L w.r.t. the longest common prefix as well as w.r.t. longest common prefixes of L after unions or
concatenations with arbitrary other languages.
From that, we conclude
that the longest common prefix, and thus the longest common suffix, of a context-free language can be computed in polynomial time.
Cite as
Michael Luttenberger, Raphaela Palenta, and Helmut Seidl. Computing the Longest Common Prefix of a Context-free Language in Polynomial Time. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 48:1-48:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{luttenberger_et_al:LIPIcs.STACS.2018.48,
author = {Luttenberger, Michael and Palenta, Raphaela and Seidl, Helmut},
title = {{Computing the Longest Common Prefix of a Context-free Language in Polynomial Time}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {48:1--48:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.48},
URN = {urn:nbn:de:0030-drops-84828},
doi = {10.4230/LIPIcs.STACS.2018.48},
annote = {Keywords: longest common prefix, context-free languages, combinatorics on words}
}
Document
Surjective H-Colouring over Reflexive Digraphs
Abstract
The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs.
Chen [2014] proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL, otherwise it is NP-complete.
By combining this result with some known and new results we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3.
Cite as
Benoit Larose, Barnaby Martin, and Daniel Paulusma. Surjective H-Colouring over Reflexive Digraphs. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 49:1-49:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{larose_et_al:LIPIcs.STACS.2018.49,
author = {Larose, Benoit and Martin, Barnaby and Paulusma, Daniel},
title = {{Surjective H-Colouring over Reflexive Digraphs}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {49:1--49:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.49},
URN = {urn:nbn:de:0030-drops-84882},
doi = {10.4230/LIPIcs.STACS.2018.49},
annote = {Keywords: Surjective H-Coloring, Computational Complexity, Algorithmic Graph Theory, Universal Algebra, Constraint Satisfaction}
}
Document
Pumping Lemmas for Weighted Automata
Abstract
We present three pumping lemmas for three classes of functions definable by fragments of weighted automata over the min-plus semiring and the semiring of natural numbers. As a corollary we show that the hierarchy of functions definable by unambiguous, finitely-ambiguous, polynomially-ambiguous weighted automata, and the full class of weighted automata is strict for the min-plus semiring.
Cite as
Filip Mazowiecki and Cristian Riveros. Pumping Lemmas for Weighted Automata. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 50:1-50:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{mazowiecki_et_al:LIPIcs.STACS.2018.50,
author = {Mazowiecki, Filip and Riveros, Cristian},
title = {{Pumping Lemmas for Weighted Automata}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {50:1--50:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.50},
URN = {urn:nbn:de:0030-drops-84984},
doi = {10.4230/LIPIcs.STACS.2018.50},
annote = {Keywords: Weighted automata, regular functions over words, pumping lemmas}
}
Document
Closure of Resource-Bounded Randomness Notions Under Polynomial-Time Permutations
Abstract
An infinite bit sequence is called recursively random if no computable strategy betting along the sequence has unbounded capital. It is well-known that the property of recursive randomness is closed under computable permutations. We investigate analogous statements for randomness notions defined by betting strategies that are computable within resource bounds. Suppose that S is a polynomial time computable permutation of the set of strings over the unary alphabet (identified with the set of natural numbers). If the inverse of S is not polynomially bounded, it is not hard to build a polynomial time random bit sequence Z such that Z o S is not polynomial time random. So one
should only consider permutations S satisfying the extra condition
that the inverse is polynomially bounded. Now the closure depends on additional assumptions in complexity theory.
Our first main result, Theorem 4, shows that if BPP contains a superpolynomial deterministic time class, then polynomial time randomness is not preserved by some permutation S such that in fact both S and its inverse are in P. Our second result, Theorem 11, shows that polynomial space randomness is preserved by polynomial time permutations with polynomially bounded inverse, so if P = PSPACE then polynomial time randomness is preserved.
Cite as
André Nies and Frank Stephan. Closure of Resource-Bounded Randomness Notions Under Polynomial-Time Permutations. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 51:1-51:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{nies_et_al:LIPIcs.STACS.2018.51,
author = {Nies, Andr\'{e} and Stephan, Frank},
title = {{Closure of Resource-Bounded Randomness Notions Under Polynomial-Time Permutations}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {51:1--51:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.51},
URN = {urn:nbn:de:0030-drops-84938},
doi = {10.4230/LIPIcs.STACS.2018.51},
annote = {Keywords: Computational complexity, Randomness via resource-bounded betting strategies, Martingales, Closure under permutations}
}
Document
Succinct Oblivious RAM
Abstract
As online storage services become increasingly common, it is important that users' private information is protected from database access pattern analyses. Oblivious RAM (ORAM) is a cryptographic primitive that enables users to perform arbitrary database accesses without revealing any information about the access pattern to the server. Previous ORAM studies focused mostly on reducing the access overhead. Consequently, the access overhead of the state-of-the-art ORAM constructions are almost at practical levels in certain application scenarios such as secure processors. However, we assume that the server space usage could become a new important issue in the coming big-data era. To enable large-scale computation in security-aware settings, it is necessary to rethink the ORAM server space cost using big-data standards.
In this paper, we introduce "succinctness" as a theoretically tractable and practically relevant criterion of the ORAM server space efficiency in the big-data era. We, then, propose two succinct ORAM constructions that also exhibit state-of-the-art performance in terms of the bandwidth blowup and the user space. We also give non-asymptotic analyses and simulation results which indicate that the proposed ORAM constructions are practically effective.
Cite as
Taku Onodera and Tetsuo Shibuya. Succinct Oblivious RAM. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 52:1-52:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{onodera_et_al:LIPIcs.STACS.2018.52,
author = {Onodera, Taku and Shibuya, Tetsuo},
title = {{Succinct Oblivious RAM}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {52:1--52:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.52},
URN = {urn:nbn:de:0030-drops-85014},
doi = {10.4230/LIPIcs.STACS.2018.52},
annote = {Keywords: Oblivious RAM, Succinct data structure, Balls-into-bins}
}
Document
Recursion Schemes and the WMSO+U Logic
Abstract
We study the weak MSO logic extended by the unbounding quantifier (WMSO+U), expressing the fact that there exist arbitrarily large finite sets satisfying a given property. We prove that it is decidable whether the tree generated by a given higher-order recursion scheme satisfies a given sentence of WMSO+U.
Cite as
Pawel Parys. Recursion Schemes and the WMSO+U Logic. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 53:1-53:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{parys:LIPIcs.STACS.2018.53,
author = {Parys, Pawel},
title = {{Recursion Schemes and the WMSO+U Logic}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {53:1--53:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.53},
URN = {urn:nbn:de:0030-drops-85122},
doi = {10.4230/LIPIcs.STACS.2018.53},
annote = {Keywords: higher-order recursion schemes, intersection types, WMSO+U logic, boundedness}
}
Document
Sums of Palindromes: an Approach via Automata
Abstract
Recently, Cilleruelo, Luca, & Baxter proved, for all bases b >= 5, that every natural number is the sum of at most 3 natural numbers whose base-b representation is a palindrome. However, the cases b = 2, 3, 4 were left unresolved. We prove, using a decision procedure based on automata, that every natural number is the sum of at most 4 natural numbers whose base-2 representation is a palindrome. Here the constant 4 is optimal. We obtain similar results for bases 3 and 4, thus completely resolving the problem.
Cite as
Aayush Rajasekaran, Jeffrey Shallit, and Tim Smith. Sums of Palindromes: an Approach via Automata. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 54:1-54:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{rajasekaran_et_al:LIPIcs.STACS.2018.54,
author = {Rajasekaran, Aayush and Shallit, Jeffrey and Smith, Tim},
title = {{Sums of Palindromes: an Approach via Automata}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {54:1--54:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.54},
URN = {urn:nbn:de:0030-drops-84976},
doi = {10.4230/LIPIcs.STACS.2018.54},
annote = {Keywords: finite automaton, nested-word automaton, decision procedure, palindrome, additive number theory}
}
Document
On the Containment Problem for Linear Sets
Abstract
It is well known that the containment problem (as well as the
equivalence problem) for semilinear sets is log-complete at the second level of the polynomial hierarchy (where hardness even holds in dimension 1). It had been shown quite recently that already the containment problem for multi-dimensional linear sets is log-complete at the same level of the hierarchy (where hardness even holds when numbers are encoded in unary). In this paper, we show that already the containment problem for 1-dimensional linear sets (with binary encoding of the numerical input parameters) is log-hard (and therefore also log-complete) at this level. However, combining both restrictions (dimension 1 and unary encoding), the problem becomes solvable in polynomial time.
Cite as
Hans U. Simon. On the Containment Problem for Linear Sets. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 55:1-55:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{simon:LIPIcs.STACS.2018.55,
author = {Simon, Hans U.},
title = {{On the Containment Problem for Linear Sets}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {55:1--55:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.55},
URN = {urn:nbn:de:0030-drops-84842},
doi = {10.4230/LIPIcs.STACS.2018.55},
annote = {Keywords: polynomial hierarchy, completeness, containment problem, linear sets}
}
Document
Improving the Upper Bound on the Length of the Shortest Reset Word
Abstract
We improve the best known upper bound on the length of the shortest reset words of synchronizing automata. The new bound is slightly better than 114 n^3 / 685 + O(n^2). The Cerny conjecture states that (n-1)^2 is an upper bound. So far, the best general upper bound was (n^3-n)/6-1 obtained by J.-E. Pin and P. Frankl in 1982. Despite a number of efforts, it remained unchanged for about 35 years.
To obtain the new upper bound we utilize avoiding words.
A word is avoiding for a state q if after reading the word the automaton cannot be in q. We obtain upper bounds on the length of the shortest avoiding words, and using the approach of Trahtman from 2011 combined with the well-known Frankl theorem from 1982, we improve the general upper bound on the length of the shortest reset words.
For all the bounds, there exist polynomial algorithms finding a word of length not exceeding the bound.
Cite as
Marek Szykula. Improving the Upper Bound on the Length of the Shortest Reset Word. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 56:1-56:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{szykula:LIPIcs.STACS.2018.56,
author = {Szykula, Marek},
title = {{Improving the Upper Bound on the Length of the Shortest Reset Word}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {56:1--56:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.56},
URN = {urn:nbn:de:0030-drops-85160},
doi = {10.4230/LIPIcs.STACS.2018.56},
annote = {Keywords: avoiding word, Cerny conjecture, reset length, reset threshold, reset word, synchronizing automaton, synchronizing word}
}
Document
Power of Uninitialized Qubits in Shallow Quantum Circuits
Abstract
We study the computational power of shallow quantum circuits
with O(log n) initialized and n^{O(1)} uninitialized ancillary
qubits, where n is the input length and the initial state of
the uninitialized ancillary qubits is arbitrary. First, we show
that such a circuit can compute any symmetric function on n bits
that is classically computable in polynomial time. Then, we
regard such a circuit as an oracle and show that a
polynomial-time classical algorithm with the oracle can estimate
the elements of any unitary matrix corresponding to a
constant-depth quantum circuit on n qubits. Since it seems unlikely
that these tasks can be done with only O(log n) initialized
ancillary qubits, our results give evidences that adding
uninitialized ancillary qubits increases the computational power
of shallow quantum circuits with only O(log n) initialized
ancillary qubits. Lastly, to understand the limitations of
uninitialized ancillary qubits, we focus on
near-logarithmic-depth quantum circuits with them and show
the impossibility of computing the parity function on n bits.
Cite as
Yasuhiro Takahashi and Seiichiro Tani. Power of Uninitialized Qubits in Shallow Quantum Circuits. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 57:1-57:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{takahashi_et_al:LIPIcs.STACS.2018.57,
author = {Takahashi, Yasuhiro and Tani, Seiichiro},
title = {{Power of Uninitialized Qubits in Shallow Quantum Circuits}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {57:1--57:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.57},
URN = {urn:nbn:de:0030-drops-84907},
doi = {10.4230/LIPIcs.STACS.2018.57},
annote = {Keywords: quantum circuit complexity, shallow quantum circuit, uninitialized qubit}
}
Document
Lower Bounds on Black-Box Reductions of Hitting to Density Estimation
Abstract
Consider a deterministic algorithm that tries to find a string in an unknown set S\subseteq{0,1}^n, under the promise that S has large density. The only information that the algorithm can obtain about S is estimates of the density of S in adaptively chosen subsets of {0,1}^n, up to an additive error of mu>0. This problem is appealing as a derandomization problem, when S is the set of satisfying inputs for a circuit C:{0,1}^n->{0,1} that accepts many inputs: In this context, an algorithm as above constitutes a deterministic black-box reduction of the problem of hitting C (i.e., finding a satisfying input for C) to the problem of approximately counting the number of satisfying inputs for C on subsets of {0,1}^n.
We prove tight lower bounds for this problem, demonstrating that naive approaches to solve the problem cannot be improved upon, in general. First, we show a tight trade-off between the estimation error mu and the required number of queries to solve the problem: When mu=O(log(n)/n) a polynomial number of queries suffices, and when mu>=(4log(n)/n) the required number of queries is 2^{Theta(mu \cdot n)}. Secondly, we show that the problem "resists" parallelization: Any algorithm that works in iterations, and can obtain p=p(n) density estimates "in parallel" in each iteration, still requires Omega( frac{n}{log(p)+log(1/mu)} ) iterations to solve the problem.
This work extends the well-known work of Karp, Upfal, and Wigderson (1988), who studied the setting in which S is only guaranteed to be non-empty (rather than dense), and the algorithm can only probe subsets for the existence of a solution in them. In addition, our lower bound on parallel algorithms affirms a weak version of a conjecture of Motwani, Naor, and Naor (1994); we also make progress on a stronger version of their conjecture.
Cite as
Roei Tell. Lower Bounds on Black-Box Reductions of Hitting to Density Estimation. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 58:1-58:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
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@InProceedings{tell:LIPIcs.STACS.2018.58,
author = {Tell, Roei},
title = {{Lower Bounds on Black-Box Reductions of Hitting to Density Estimation}},
booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
pages = {58:1--58:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-062-0},
ISSN = {1868-8969},
year = {2018},
volume = {96},
editor = {Niedermeier, Rolf 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.2018.58},
URN = {urn:nbn:de:0030-drops-85005},
doi = {10.4230/LIPIcs.STACS.2018.58},
annote = {Keywords: Approximate Counting, Lower Bounds, Derandomization, Parallel Algorithms, Query Complexity}
}