Volume
LIPIcs, Volume 154
37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Theoretical Aspects of Computer Science (STACS)
Event
STACS 2020, March 10-13, 2020, Montpellier, FranceEditors
Publication Details
- published at: 2020-03-04
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-140-5
- DBLP: db/conf/stacs/stacs2020
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Document
Complete Volume
LIPIcs, Vol. 154, STACS 2020, Complete Volume
Abstract
LIPIcs, Vol. 154, STACS 2020, Complete Volume
Cite as
37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 0-1024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@Proceedings{paul_et_al:LIPIcs.STACS.2020,
title = {{LIPIcs, Vol. 154, STACS 2020, Complete Volume}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020},
URN = {urn:nbn:de:0030-drops-118603},
doi = {10.4230/LIPIcs.STACS.2020},
annote = {Keywords: LIPIcs, Vol. 154, STACS 2020, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{paul_et_al:LIPIcs.STACS.2020.0,
author = {Paul, Christophe and Bl\"{a}ser, Markus},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {0:i--0:xii},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.0},
URN = {urn:nbn:de:0030-drops-118614},
doi = {10.4230/LIPIcs.STACS.2020.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
Statistical Physics and Algorithms (Invited Talk)
Abstract
The field of randomized algorithms has benefitted greatly from insights from statistical physics. We give examples in two distinct settings. The first is in the context of Markov chain Monte Carlo algorithms, which have become ubiquitous across science and engineering as a means of exploring large configuration spaces. One of the most striking discoveries was the realization that many natural Markov chains undergo phase transitions, whereby they are efficient for some parameter settings and then suddenly become inefficient as a parameter of the system is slowly modified. The second is in the context of distributed algorithms for programmable matter. Self-organizing particle systems based on statistical models with phase changes have been used to achieve basic tasks involving coordination, movement, and conformation in a fully distributed, local setting. We briefly describe these two settings to demonstrate how computing and statistical physics together provide powerful insights that apply across multiple domains.
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Dana Randall. Statistical Physics and Algorithms (Invited Talk). In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 1:1-1:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{randall:LIPIcs.STACS.2020.1,
author = {Randall, Dana},
title = {{Statistical Physics and Algorithms}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {1:1--1:6},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.1},
URN = {urn:nbn:de:0030-drops-118624},
doi = {10.4230/LIPIcs.STACS.2020.1},
annote = {Keywords: Markov chains, mixing times, phase transitions, programmable matter}
}
Document
Invited Talk
Weisfeiler and Leman’s Unlikely Journey from Graph Isomorphism to Neural Networks (Invited Talk)
Abstract
The Weisfeiler-Leman algorithm is a well-known combinatorial graph isomorphism test going back to work of Weisfeiler and Leman in the late 1960s. The algorithm has a surprising number of seemingly unrelated characterisations in terms of logic, algebra, linear and semi-definite programming, and graph homomorphisms. Due to its simplicity and efficiency, it is an important subroutine of all modern graph isomorphism tools. In recent years, further applications in linear optimisation, probabilistic inference, and machine learning have surfaced. In the first part of my talk, I will give an introduction to the Weisfeiler-Leman algorithm and its various characterisations. In the second part I will speak about its applications, in particular about recent work relating the algorithm to graph neural networks.
Cite as
Martin Grohe. Weisfeiler and Leman’s Unlikely Journey from Graph Isomorphism to Neural Networks (Invited Talk). In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{grohe:LIPIcs.STACS.2020.2,
author = {Grohe, Martin},
title = {{Weisfeiler and Leman’s Unlikely Journey from Graph Isomorphism to Neural Networks}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {2:1--2:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.2},
URN = {urn:nbn:de:0030-drops-118634},
doi = {10.4230/LIPIcs.STACS.2020.2},
annote = {Keywords: Weisfeiler adn Leman algorithm, Graph isomorphism, Neural network, logic, algebra, linear and semi-definite programming}
}
Document
Invited Talk
Computability, Complexity and Programming with Ordinary Differential Equations (Invited Talk)
Abstract
Ordinary Differential Equations (ODEs) appear to be a universally adopted and very natural way for expressing properties for continuous time dynamical systems. They are intensively used, in particular in applied sciences. There exists an abundant literature about the hardness of solving ODEs with numerical methods.
We adopt a dual view: we consider ODEs as a way to program or to describe our mathematical/computer science world. We survey several results considering ODEs under this computational perspective, with a computability and complexity theory point of view. In particular, we provide various reasons why polynomial ODEs should be considered as the continuous time analog of Turing machines for continuous-time computations, or should be used as a way to talk about mathematical logic.
This has already applications in various fields: determining whether analog models of computation can compute faster than classical digital models of computation; solving complexity issues for computations with biochemical reactions in bioinformatics; machine independent characterizations of various computability and complexity classes such as PTIME or NPTIME, or proof of the existence of a universal polynomial ordinary differential equation whose solutions can approximate any continuous function if provided with a suitable well-chosen initial condition.
Cite as
Olivier Bournez. Computability, Complexity and Programming with Ordinary Differential Equations (Invited Talk). In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 3:1-3:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bournez:LIPIcs.STACS.2020.3,
author = {Bournez, Olivier},
title = {{Computability, Complexity and Programming with Ordinary Differential Equations}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {3:1--3:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.3},
URN = {urn:nbn:de:0030-drops-118642},
doi = {10.4230/LIPIcs.STACS.2020.3},
annote = {Keywords: Ordinary differential equations, Models of computation, Analog Computations, discrete ordinary differential equations, Implicit complexity, recursion scheme}
}
Document
Tutorial
Graphical Models: Queries, Complexity, Algorithms (Tutorial)
Abstract
Graphical models (GMs) define a family of mathematical models aimed at the concise description of multivariate functions using decomposability. We restrict ourselves to functions of discrete variables but try to cover a variety of models that are not always considered as "Graphical Models", ranging from functions with Boolean variables and Boolean co-domain (used in automated reasoning) to functions over finite domain variables and integer or real co-domains (usual in machine learning and statistics). We use a simple algebraic semi-ring based framework for generality, define associated queries, relationships between graphical models, complexity results, and families of algorithms, with their associated guarantees.
Cite as
Martin C. Cooper, Simon de Givry, and Thomas Schiex. Graphical Models: Queries, Complexity, Algorithms (Tutorial). In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 4:1-4:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{cooper_et_al:LIPIcs.STACS.2020.4,
author = {Cooper, Martin C. and de Givry, Simon and Schiex, Thomas},
title = {{Graphical Models: Queries, Complexity, Algorithms}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {4:1--4:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.4},
URN = {urn:nbn:de:0030-drops-118654},
doi = {10.4230/LIPIcs.STACS.2020.4},
annote = {Keywords: Computational complexity, tree decomposition, graphical models, submodularity, message passing, local consistency, artificial intelligence, valued constraints, optimization}
}
Document
Inapproximability Results for Scheduling with Interval and Resource Restrictions
Abstract
In the restricted assignment problem, the input consists of a set of machines and a set of jobs each with a processing time and a subset of eligible machines. The goal is to find an assignment of the jobs to the machines minimizing the makespan, that is, the maximum summed up processing time any machine receives. Herein, jobs should only be assigned to those machines on which they are eligible. It is well-known that there is no polynomial time approximation algorithm with an approximation guarantee of less than 1.5 for the restricted assignment problem unless P=NP. In this work, we show hardness results for variants of the restricted assignment problem with particular types of restrictions.
For the case of interval restrictions - where the machines can be totally ordered such that jobs are eligible on consecutive machines - we show that there is no polynomial time approximation scheme (PTAS) unless P=NP. The question of whether a PTAS for this variant exists was stated as an open problem before, and PTAS results for special cases of this variant are known.
Furthermore, we consider a variant with resource restriction where the sets of eligible machines are of the following form: There is a fixed number of (renewable) resources, each machine has a capacity, and each job a demand for each resource. A job is eligible on a machine if its demand is at most as big as the capacity of the machine for each resource. For one resource, this problem has been intensively studied under several different names and is known to admit a PTAS, and for two resources the variant with interval restrictions is contained as a special case. Moreover, the version with multiple resources is closely related to makespan minimization on parallel machines with a low rank processing time matrix. We show that there is no polynomial time approximation algorithm with a rate smaller than 48/47 ≈ 1.02 or 1.5 for scheduling with resource restrictions with 2 or 4 resources, respectively, unless P=NP. All our results can be extended to the so called Santa Claus variants of the problems where the goal is to maximize the minimal processing time any machine receives.
Cite as
Marten Maack and Klaus Jansen. Inapproximability Results for Scheduling with Interval and Resource Restrictions. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{maack_et_al:LIPIcs.STACS.2020.5,
author = {Maack, Marten and Jansen, Klaus},
title = {{Inapproximability Results for Scheduling with Interval and Resource Restrictions}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {5:1--5:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.5},
URN = {urn:nbn:de:0030-drops-118663},
doi = {10.4230/LIPIcs.STACS.2020.5},
annote = {Keywords: Scheduling, Restricted Assignment, Approximation, Inapproximability, PTAS}
}
Document
An Automaton Group with PSPACE-Complete Word Problem
Abstract
We construct an automaton group with a PSPACE-complete word problem, proving a conjecture due to Steinberg. Additionally, the constructed group has a provably more difficult, namely EXPSPACE-complete, compressed word problem. Our construction directly simulates the computation of a Turing machine in an automaton group and, therefore, seems to be quite versatile. It combines two ideas: the first one is a construction used by D'Angeli, Rodaro and the first author to obtain an inverse automaton semigroup with a PSPACE-complete word problem and the second one is to utilize a construction used by Barrington to simulate circuits of bounded degree and logarithmic depth in the group of even permutations over five elements.
Cite as
Jan Philipp Wächter and Armin Weiß. An Automaton Group with PSPACE-Complete Word Problem. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{wachter_et_al:LIPIcs.STACS.2020.6,
author = {W\"{a}chter, Jan Philipp and Wei{\ss}, Armin},
title = {{An Automaton Group with PSPACE-Complete Word Problem}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {6:1--6:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.6},
URN = {urn:nbn:de:0030-drops-118674},
doi = {10.4230/LIPIcs.STACS.2020.6},
annote = {Keywords: automaton group, word problem, PSPACE, compressed word problem}
}
Document
A Trichotomy for Regular Trail Queries
Abstract
Regular path queries (RPQs) are an essential component of graph query languages. Such queries consider a regular expression r and a directed edge-labeled graph G and search for paths in G for which the sequence of labels is in the language of r. In order to avoid having to consider infinitely many paths, some database engines restrict such paths to be trails, that is, they only consider paths without repeated edges. In this paper we consider the evaluation problem for RPQs under trail semantics, in the case where the expression is fixed. We show that, in this setting, there exists a trichotomy. More precisely, the complexity of RPQ evaluation divides the regular languages into the finite languages, the class T_tract (for which the problem is tractable), and the rest. Interestingly, the tractable class in the trichotomy is larger than for the trichotomy for simple paths, discovered by Bagan et al. [Bagan et al., 2013]. In addition to this trichotomy result, we also study characterizations of the tractable class, its expressivity, the recognition problem, closure properties, and show how the decision problem can be extended to the enumeration problem, which is relevant to practice.
Cite as
Wim Martens, Matthias Niewerth, and Tina Trautner. A Trichotomy for Regular Trail Queries. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{martens_et_al:LIPIcs.STACS.2020.7,
author = {Martens, Wim and Niewerth, Matthias and Trautner, Tina},
title = {{A Trichotomy for Regular Trail Queries}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {7:1--7:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.7},
URN = {urn:nbn:de:0030-drops-118681},
doi = {10.4230/LIPIcs.STACS.2020.7},
annote = {Keywords: Regular languages, query languages, path queries, graph databases, databases, complexity, trails, simple paths}
}
Document
Descriptive Complexity on Non-Polish Spaces
Abstract
Represented spaces are the spaces on which computations can be performed. We investigate the descriptive complexity of sets in represented spaces. We prove that the standard representation of a countably-based space preserves the effective descriptive complexity of sets. We prove that some results from descriptive set theory on Polish spaces extend to arbitrary countably-based spaces. We study the larger class of coPolish spaces, showing that their representation does not always preserve the complexity of sets, and we relate this mismatch with the sequential aspects of the space. We study in particular the space of polynomials.
Cite as
Antonin Callard and Mathieu Hoyrup. Descriptive Complexity on Non-Polish Spaces. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{callard_et_al:LIPIcs.STACS.2020.8,
author = {Callard, Antonin and Hoyrup, Mathieu},
title = {{Descriptive Complexity on Non-Polish Spaces}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {8:1--8:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.8},
URN = {urn:nbn:de:0030-drops-118694},
doi = {10.4230/LIPIcs.STACS.2020.8},
annote = {Keywords: Represented space, Computable analysis, Descriptive set theory, CoPolish spaces}
}
Document
NP-Completeness, Proof Systems, and Disjoint NP-Pairs
Abstract
The article investigates the relation between three well-known hypotheses.
- H_{union}: the union of disjoint ≤^p_m-complete sets for NP is ≤^p_m-complete
- H_{opps}: there exist optimal propositional proof systems
- H_{cpair}: there exist ≤^{pp}_m-complete disjoint NP-pairs The following results are obtained:
- The hypotheses are pairwise independent under relativizable proofs, except for the known implication H_{opps} ⇒ H_{cpair}.
- An answer to Pudlák’s question for an oracle relative to which ¬H_{cpair}, ¬H_{opps}, and UP has ≤^p_m-complete sets.
- The converse of Köbler, Messner, and Torán’s implication NEE ∩ TALLY ⊆ coNEE ⇒ H_{opps} fails relative to an oracle, where NEE =^{df} NTIME(2^O(2ⁿ)).
- New characterizations of H_{union} and two variants in terms of coNP-completeness and p-producibility of the set of hard formulas of propositional proof systems.
Cite as
Titus Dose and Christian Glaßer. NP-Completeness, Proof Systems, and Disjoint NP-Pairs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{dose_et_al:LIPIcs.STACS.2020.9,
author = {Dose, Titus and Gla{\ss}er, Christian},
title = {{NP-Completeness, Proof Systems, and Disjoint NP-Pairs}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {9:1--9:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.9},
URN = {urn:nbn:de:0030-drops-118707},
doi = {10.4230/LIPIcs.STACS.2020.9},
annote = {Keywords: NP-complete, propositional proof system, disjoint NP-pair, oracle}
}
Document
String Indexing with Compressed Patterns
Abstract
Given a string S of length n, the classic string indexing problem is to preprocess S into a compact data structure that supports efficient subsequent pattern queries. In this paper we consider the basic variant where the pattern is given in compressed form and the goal is to achieve query time that is fast in terms of the compressed size of the pattern. This captures the common client-server scenario, where a client submits a query and communicates it in compressed form to a server. Instead of the server decompressing the query before processing it, we consider how to efficiently process the compressed query directly. Our main result is a novel linear space data structure that achieves near-optimal query time for patterns compressed with the classic Lempel-Ziv 1977 (LZ77) compression scheme. Along the way we develop several data structural techniques of independent interest, including a novel data structure that compactly encodes all LZ77 compressed suffixes of a string in linear space and a general decomposition of tries that reduces the search time from logarithmic in the size of the trie to logarithmic in the length of the pattern.
Cite as
Philip Bille, Inge Li Gørtz, and Teresa Anna Steiner. String Indexing with Compressed Patterns. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 10:1-10:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bille_et_al:LIPIcs.STACS.2020.10,
author = {Bille, Philip and G{\o}rtz, Inge Li and Steiner, Teresa Anna},
title = {{String Indexing with Compressed Patterns}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {10:1--10:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.10},
URN = {urn:nbn:de:0030-drops-118716},
doi = {10.4230/LIPIcs.STACS.2020.10},
annote = {Keywords: string indexing, compression, pattern matching}
}
Document
An FPT Algorithm for Minimum Additive Spanner Problem
Abstract
For a positive integer t and a graph G, an additive t-spanner of G is a spanning subgraph in which the distance between every pair of vertices is at most the original distance plus t. The Minimum Additive t-Spanner Problem is to find an additive t-spanner with the minimum number of edges in a given graph, which is known to be NP-hard. Since we need to care about global properties of graphs when we deal with additive t-spanners, the Minimum Additive t-Spanner Problem is hard to handle and hence only few results are known for it. In this paper, we study the Minimum Additive t-Spanner Problem from the viewpoint of parameterized complexity. We formulate a parameterized version of the problem in which the number of removed edges is regarded as a parameter, and give a fixed-parameter algorithm for it. We also extend our result to the case with both a multiplicative approximation factor α and an additive approximation parameter β, which we call (α, β)-spanners.
Cite as
Yusuke Kobayashi. An FPT Algorithm for Minimum Additive Spanner Problem. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{kobayashi:LIPIcs.STACS.2020.11,
author = {Kobayashi, Yusuke},
title = {{An FPT Algorithm for Minimum Additive Spanner Problem}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {11:1--11:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.11},
URN = {urn:nbn:de:0030-drops-118729},
doi = {10.4230/LIPIcs.STACS.2020.11},
annote = {Keywords: Graph algorithms, Fixed-parameter tractability, Graph spanner}
}
Document
New Bounds for Randomized List Update in the Paid Exchange Model
Abstract
We study the fundamental list update problem in the paid exchange model P^d. This cost model was introduced by Manasse, McGeoch and Sleator [M.S. Manasse et al., 1988] and Reingold, Westbrook and Sleator [N. Reingold et al., 1994]. Here the given list of items may only be rearranged using paid exchanges; each swap of two adjacent items in the list incurs a cost of d. Free exchanges of items are not allowed. The model is motivated by the fact that, when executing search operations on a data structure, key comparisons are less expensive than item swaps.
We develop a new randomized online algorithm that achieves an improved competitive ratio against oblivious adversaries. For large d, the competitiveness tends to 2.2442. Technically, the analysis of the algorithm relies on a new approach of partitioning request sequences and charging expected cost. Furthermore, we devise lower bounds on the competitiveness of randomized algorithms against oblivious adversaries. No such lower bounds were known before. Specifically, we prove that no randomized online algorithm can achieve a competitive ratio smaller than 2 in the partial cost model, where an access to the i-th item in the current list incurs a cost of i-1 rather than i. All algorithms proposed in the literature attain their competitiveness in the partial cost model. Furthermore, we show that no randomized online algorithm can achieve a competitive ratio smaller than 1.8654 in the standard full cost model. Again the lower bounds hold for large d.
Cite as
Susanne Albers and Maximilian Janke. New Bounds for Randomized List Update in the Paid Exchange Model. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{albers_et_al:LIPIcs.STACS.2020.12,
author = {Albers, Susanne and Janke, Maximilian},
title = {{New Bounds for Randomized List Update in the Paid Exchange Model}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {12:1--12:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.12},
URN = {urn:nbn:de:0030-drops-118735},
doi = {10.4230/LIPIcs.STACS.2020.12},
annote = {Keywords: self-organizing lists, online algorithm, competitive analysis, lower bound}
}
Document
On Covering Segments with Unit Intervals
Abstract
We study the problem of covering a set of segments on a line with the minimum number of unit-length intervals, where an interval covers a segment if at least one of the two endpoints of the segment falls in the unit interval. We also study several variants of this problem.
We show that the restrictions of the aforementioned problems to the set of instances in which all the segments have the same length are NP-hard. This result implies several NP-hardness results in the literature for variants and generalizations of the problems under consideration.
We then study the parameterized complexity of the aforementioned problems. We provide tight results for most of them by showing that they are fixed-parameter tractable for the restrictions in which all the segments have the same length, and are W[1]-complete otherwise.
Cite as
Dan Bergren, Eduard Eiben, Robert Ganian, and Iyad Kanj. On Covering Segments with Unit Intervals. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bergren_et_al:LIPIcs.STACS.2020.13,
author = {Bergren, Dan and Eiben, Eduard and Ganian, Robert and Kanj, Iyad},
title = {{On Covering Segments with Unit Intervals}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {13:1--13:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.13},
URN = {urn:nbn:de:0030-drops-118741},
doi = {10.4230/LIPIcs.STACS.2020.13},
annote = {Keywords: Segment covering, unit intervals, NP-completeness, parameterized complexity}
}
Document
Decidability and Periodicity of Low Complexity Tilings
Abstract
In this paper we study low-complexity colorings (or tilings) of the two-dimensional grid ℤ². A coloring is said to be of low complexity with respect to a rectangle if there exists m,n∈ℕ such that there are no more than mn different rectangular m× n patterns in it. Open since it was stated in 1997, Nivat’s conjecture states that such a coloring is necessarily periodic. Suppose we are given at most nm rectangular patterns of size n× m. If Nivat’s conjecture is true, one can only build periodic colorings out of these patterns - meaning that if the m× n rectangular patterns of the coloring are among these mn patterns, it must be periodic. The main contribution of this paper proves that there exists at least one periodic coloring build from these patterns. We use this result to investigate the tiling problem, also known as the domino problem, which is well known to be undecidable in its full generality. However, we show that it is decidable in the low-complexity setting. Finally, we use our result to show that Nivat’s conjecture holds for uniformly recurrent configurations. The results also extend to other convex shapes in place of the rectangle.
Cite as
Jarkko Kari and Etienne Moutot. Decidability and Periodicity of Low Complexity Tilings. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 14:1-14:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{kari_et_al:LIPIcs.STACS.2020.14,
author = {Kari, Jarkko and Moutot, Etienne},
title = {{Decidability and Periodicity of Low Complexity Tilings}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {14:1--14:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.14},
URN = {urn:nbn:de:0030-drops-118752},
doi = {10.4230/LIPIcs.STACS.2020.14},
annote = {Keywords: Nivat’s conjecture, domino problem, decidability, low pattern complexity, 2D subshifts, symbolic dynamics}
}
Document
The Tandem Duplication Distance Is NP-Hard
Abstract
In computational biology, tandem duplication is an important biological phenomenon which can occur either at the genome or at the DNA level. A tandem duplication takes a copy of a genome segment and inserts it right after the segment - this can be represented as the string operation AXB ⇒ AXXB. Tandem exon duplications have been found in many species such as human, fly or worm, and have been largely studied in computational biology.
The Tandem Duplication (TD) distance problem we investigate in this paper is defined as follows: given two strings S and T over the same alphabet, compute the smallest sequence of tandem duplications required to convert S to T. The natural question of whether the TD distance can be computed in polynomial time was posed in 2004 by Leupold et al. and had remained open, despite the fact that tandem duplications have received much attention ever since. In this paper, we prove that this problem is NP-hard, settling the 16-year old open problem. We further show that this hardness holds even if all characters of S are distinct. This is known as the exemplar TD distance, which is of special relevance in bioinformatics. One of the tools we develop for the reduction is a new problem called the Cost-Effective Subgraph, for which we obtain W[1]-hardness results that might be of independent interest. We finally show that computing the exemplar TD distance between S and T is fixed-parameter tractable. Our results open the door to many other questions, and we conclude with several open problems.
Cite as
Manuel Lafond, Binhai Zhu, and Peng Zou. The Tandem Duplication Distance Is NP-Hard. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{lafond_et_al:LIPIcs.STACS.2020.15,
author = {Lafond, Manuel and Zhu, Binhai and Zou, Peng},
title = {{The Tandem Duplication Distance Is NP-Hard}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {15:1--15:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.15},
URN = {urn:nbn:de:0030-drops-118769},
doi = {10.4230/LIPIcs.STACS.2020.15},
annote = {Keywords: Tandem duplication, Text processing, Formal languages, Computational genomics, FPT algorithms}
}
Document
Existential Length Universality
Abstract
We study the following natural variation on the classical universality problem: given a language L(M) represented by M (e.g., a DFA/RE/NFA/PDA), does there exist an integer ? ≥ 0 such that Σ^? ⊆ L(M)? In the case of an NFA, we show that this problem is NEXPTIME-complete, and the smallest such ? can be doubly exponential in the number of states. This particular case was formulated as an open problem in 2009, and our solution uses a novel and involved construction. In the case of a PDA, we show that it is recursively unsolvable, while the smallest such ? is not bounded by any computable function of the number of states. In the case of a DFA, we show that the problem is NP-complete, and e^{√{n log n} (1+o(1))} is an asymptotically tight upper bound for the smallest such ?, where n is the number of states. Finally, we prove that in all these cases, the problem becomes computationally easier when the length ? is also given in binary in the input: it is polynomially solvable for a DFA, PSPACE-complete for an NFA, and co-NEXPTIME-complete for a PDA.
Cite as
Paweł Gawrychowski, Martin Lange, Narad Rampersad, Jeffrey Shallit, and Marek Szykuła. Existential Length Universality. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{gawrychowski_et_al:LIPIcs.STACS.2020.16,
author = {Gawrychowski, Pawe{\l} and Lange, Martin and Rampersad, Narad and Shallit, Jeffrey and Szyku{\l}a, Marek},
title = {{Existential Length Universality}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {16:1--16:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.16},
URN = {urn:nbn:de:0030-drops-118770},
doi = {10.4230/LIPIcs.STACS.2020.16},
annote = {Keywords: decision problem, deterministic automaton, nondeterministic automaton, pushdown automaton, regular expression, regular language, universality}
}
Document
On the Termination of Flooding
Abstract
Flooding is among the simplest and most fundamental of all graph/network algorithms. Consider a (distributed network in the form of a) finite undirected graph G with a distinguished node v that begins flooding by sending copies of the same message to all its neighbours and the neighbours, in the next round, forward the message to all and only the neighbours they did not receive the message from in that round and so on. We assume that nodes do not keep a record of the flooding event, thus, raising the possibility that messages may circulate infinitely even on a finite graph. We call this history-less process amnesiac flooding (to distinguish from a classic distributed implementation of flooding that maintains a history of received messages to ensure a node never sends the same message again). Flooding will terminate when no node in G sends a message in a round, and, thus, subsequent rounds. As far as we know, the question of termination for amnesiac flooding has not been settled - rather, non-termination is implicitly assumed.
In this paper, we show that surprisingly synchronous amnesiac flooding always terminates on any arbitrary finite graph and derive exact termination times which differ sharply in bipartite and non-bipartite graphs. In particular, synchronous flooding terminates in e rounds, where e is the eccentricity of the source node, if and only if G is bipartite, and, otherwise, in j rounds where e < j ≤ e+d+1 and d is the diameter of G. Since e is bounded above by d, this implies termination times of at most d and of at most 2d + 1 for bipartite and non-bipartite graphs respectively. This suggests that if communication/broadcast to all nodes is the motivation, the history-less amnesiac flooding is asymptotically time optimal and obviates the need for construction and maintenance of spanning structures like spanning trees. Moreover, the clear separation in the termination times of bipartite and non-bipartite graphs may suggest possible mechanisms for distributed discovery of the topology/distances in an arbitrary graph.
For comparison, we also show that, for asynchronous networks, however, an adversary can force the process to be non-terminating.
Cite as
Walter Hussak and Amitabh Trehan. On the Termination of Flooding. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 17:1-17:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{hussak_et_al:LIPIcs.STACS.2020.17,
author = {Hussak, Walter and Trehan, Amitabh},
title = {{On the Termination of Flooding}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {17:1--17:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.17},
URN = {urn:nbn:de:0030-drops-118786},
doi = {10.4230/LIPIcs.STACS.2020.17},
annote = {Keywords: Flooding algorithm, Network algorithms, Distributed algorithms, Graph theory, Termination, Bipartiteness, Communication, Broadcast}
}
Document
Generalised Pattern Matching Revisited
Abstract
In the problem of Generalised Pattern Matching (GPM) [STOC'94, Muthukrishnan and Palem], we are given a text T of length n over an alphabet Σ_T, a pattern P of length m over an alphabet Σ_P, and a matching relationship ⊆ Σ_T × Σ_P, and must return all substrings of T that match P (reporting) or the number of mismatches between each substring of T of length m and P (counting). In this work, we improve over all previously known algorithms for this problem:
- For ? being the maximum number of characters that match a fixed character, we show two new Monte Carlo algorithms, a reporting algorithm with time ?(? n log n log m) and a (1-ε)-approximation counting algorithm with time ?(ε^-1 ? n log n log m). We then derive a (1-ε)-approximation deterministic counting algorithm for GPM with ?(ε^-2 ? n log⁶ n) time.
- For ? being the number of pairs of matching characters, we demonstrate Monte Carlo algorithms for reporting and (1-ε)-approximate counting with running time ?(√? n log m √{log n}) and ?(√{ε^-1 ?} n log m √{log n}), respectively, as well as a (1-ε)-approximation deterministic algorithm for the counting variant of GPM with ?(ε^-1 √{?} n log^{7/2} n) time.
- Finally, for ℐ being the total number of disjoint intervals of characters that match the m characters of the pattern P, we show that both the reporting and the counting variants of GPM can be solved exactly and deterministically in ?(n√{ℐ log m} +n log n) time.
At the heart of our new deterministic upper bounds for ? and ? lies a faster construction of superimposed codes, which solves an open problem posed in [FOCS'97, Indyk] and can be of independent interest.
To conclude, we demonstrate first lower bounds for GPM. We start by showing that any deterministic or Monte Carlo algorithm for GPM must use Ω(?) time, and then proceed to show higher lower bounds for combinatorial algorithms. These bounds show that our algorithms are almost optimal, unless a radically new approach is developed.
Cite as
Bartłomiej Dudek, Paweł Gawrychowski, and Tatiana Starikovskaya. Generalised Pattern Matching Revisited. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{dudek_et_al:LIPIcs.STACS.2020.18,
author = {Dudek, Bart{\l}omiej and Gawrychowski, Pawe{\l} and Starikovskaya, Tatiana},
title = {{Generalised Pattern Matching Revisited}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {18:1--18:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.18},
URN = {urn:nbn:de:0030-drops-118798},
doi = {10.4230/LIPIcs.STACS.2020.18},
annote = {Keywords: pattern matching, superimposed codes, conditional lower bounds}
}
Document
Parameterized Pre-Coloring Extension and List Coloring Problems
Abstract
Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by k: (1) Given a graph G, a clique modulator D (a clique modulator is a set of vertices, whose removal results in a clique) of size k for G, and a list L(v) of colors for every v ∈ V(G), decide whether G has a proper list coloring; (2) Given a graph G, a clique modulator D of size k for G, and a pre-coloring λ_P: X → Q for X ⊆ V(G), decide whether λ_P can be extended to a proper coloring of G using only colors from Q. For Problem 1 we design an O*(2^k)-time randomized algorithm and for Problem 2 we obtain a kernel with at most 3k vertices. Banik et al. (IWOCA 2019) proved the following problem is fixed-parameter tractable and asked whether it admits a polynomial kernel: Given a graph G, an integer k, and a list L(v) of exactly n-k colors for every v ∈ V(G), decide whether there is a proper list coloring for G. We obtain a kernel with O(k²) vertices and colors and a compression to a variation of the problem with O(k) vertices and O(k²) colors.
Cite as
Gregory Gutin, Diptapriyo Majumdar, Sebastian Ordyniak, and Magnus Wahlström. Parameterized Pre-Coloring Extension and List Coloring Problems. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{gutin_et_al:LIPIcs.STACS.2020.19,
author = {Gutin, Gregory and Majumdar, Diptapriyo and Ordyniak, Sebastian and Wahlstr\"{o}m, Magnus},
title = {{Parameterized Pre-Coloring Extension and List Coloring Problems}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {19:1--19:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.19},
URN = {urn:nbn:de:0030-drops-118801},
doi = {10.4230/LIPIcs.STACS.2020.19},
annote = {Keywords: Parameterized Algorithms, W-hardness, Kernelization, Graph Coloring, List Coloring}
}
Document
Oracle Complexity Classes and Local Measurements on Physical Hamiltonians
Abstract
The canonical hard problems for NP and its quantum analogue, Quantum Merlin-Arthur (QMA), are MAX-k-SAT and the k-local Hamiltonian problem (k-LH), the quantum generalization of MAX-k-SAT, respectively. In recent years, however, an arguably even more physically motivated problem than k-LH has been formalized - the problem of simulating local measurements on ground states of local Hamiltonians (APX-SIM). Perhaps surprisingly, [Ambainis, CCC 2014] showed that APX-SIM is likely harder than QMA. Indeed, [Ambainis, CCC 2014] showed that APX-SIM is P^{QMA[log]}-complete, for P^{QMA[log]} the class of languages decidable by a P machine making a logarithmic number of adaptive queries to a QMA oracle. In this work, we show that APX-SIM is P^{QMA[log]}-complete even when restricted to physically motivated Hamiltonians, obtaining as intermediate steps a variety of related complexity-theoretic results.
Specifically, we first give a sequence of results which together yield P^{QMA[log]}-hardness for APX-SIM on well-motivated Hamiltonians such as the 2D Heisenberg model:
- We show that for NP, StoqMA, and QMA oracles, a logarithmic number of adaptive queries is equivalent to polynomially many parallel queries. Formally, P^{NP[log]}=P^{||NP}, P^{StoqMA[log]}=P^{||StoqMA}, and P^{QMA[log]}=P^{||QMA}. (The result for NP was previously shown using a different proof technique.) These equalities simplify the proofs of our subsequent results.
- Next, we show that the hardness of APX-SIM is preserved under Hamiltonian simulations (à la [Cubitt, Montanaro, Piddock, 2017]) by studying a seemingly weaker problem, ∀-APX-SIM. As a byproduct, we obtain a full complexity classification of APX-SIM, showing it is complete for P, P^{||NP},P^{||StoqMA}, or P^{||QMA} depending on the Hamiltonians employed.
- Leveraging the above, we show that APX-SIM is P^{QMA[log]}-complete for any family of Hamiltonians which can efficiently simulate spatially sparse Hamiltonians. This implies APX-SIM is P^{QMA[log]}-complete even on physically motivated models such as the 2D Heisenberg model.
Our second focus considers 1D systems: We show that APX-SIM remains P^{QMA[log]}-complete even for local Hamiltonians on a 1D line of 8-dimensional qudits. This uses a number of ideas from above, along with replacing the "query Hamiltonian" of [Ambainis, CCC 2014] with a new "sifter" construction.
Cite as
Sevag Gharibian, Stephen Piddock, and Justin Yirka. Oracle Complexity Classes and Local Measurements on Physical Hamiltonians. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 20:1-20:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{gharibian_et_al:LIPIcs.STACS.2020.20,
author = {Gharibian, Sevag and Piddock, Stephen and Yirka, Justin},
title = {{Oracle Complexity Classes and Local Measurements on Physical Hamiltonians}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {20:1--20:37},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.20},
URN = {urn:nbn:de:0030-drops-118818},
doi = {10.4230/LIPIcs.STACS.2020.20},
annote = {Keywords: Quantum Merlin Arthur (QMA), simulation of local measurement, local Hamiltonian, oracle complexity class, physical Hamiltonians}
}
Document
Secret Key Agreement from Correlated Data, with No Prior Information
Abstract
A fundamental question that has been studied in cryptography and in information theory is whether two parties can communicate confidentially using exclusively an open channel. We consider the model in which the two parties hold inputs that are correlated in a certain sense. This model has been studied extensively in information theory, and communication protocols have been designed which exploit the correlation to extract from the inputs a shared secret key. However, all the existing protocols are not universal in the sense that they require that the two parties also know some attributes of the correlation. In other words, they require that each party knows something about the other party’s input. We present a protocol that does not require any prior additional information. It uses space-bounded Kolmogorov complexity to measure correlation and it allows the two legal parties to obtain a common key that looks random to an eavesdropper that observes the communication and is restricted to use a bounded amount of space for the attack. Thus the protocol achieves complexity-theoretical security, but it does not use any unproven result from computational complexity. On the negative side, the protocol is not efficient in the sense that the computation of the two legal parties uses more space than the space allowed to the adversary.
Cite as
Marius Zimand. Secret Key Agreement from Correlated Data, with No Prior Information. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 21:1-21:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{zimand:LIPIcs.STACS.2020.21,
author = {Zimand, Marius},
title = {{Secret Key Agreement from Correlated Data, with No Prior Information}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {21:1--21:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.21},
URN = {urn:nbn:de:0030-drops-118823},
doi = {10.4230/LIPIcs.STACS.2020.21},
annote = {Keywords: secret key agreement, Kolmogorov complexity, extractors}
}
Document
Using Statistical Encoding to Achieve Tree Succinctness Never Seen Before
Abstract
We propose new entropy measures for trees, the known ones are H_k(?), the k-th order (tree label) entropy (Ferragina at al. 2005), and tree entropy H(?) (Jansson et al. 2006), the former considers only the tree labels and the latter only tree shape. The proposed entropy measures, H_k(?|L) and H_k(L|?), exploit the relation between the labels and the tree shape. We prove that they lower bound label entropy and tree entropy, respectively, i.e. H_k(?|L) ≤ H(?) and H_k(L|?) ≤ H_k(L). Besides being theoretically superior, the new measures are significantly smaller in practice.
We also propose a new succinct representation of labeled trees which represents a tree T using one of the following bounds: |T|(H(?) + H_k(L|?)) or |T|(H_k(?|L) + H_k(L)). The representation is based on a new, simple method of partitioning the tree, which preserves both tree shape and node degrees. The previous state-of-the-art method of compressing the tree achieved |T|(H(?) + H_k(L)) bits, by combining the results of Ferragina at al. 2005 and Jansson et al. 2006; so proposed representation is not worse and often superior. Moreover, our representation supports standard tree navigation in constant time as well as more complex queries. Such a structure achieving this space bounds was not known before: aforementioned solution only worked for compression alone, our structure is the first which achieves H_k(?) for k>0 and supports such queries. Lastly, our data structure is fairly simple, both conceptually and in terms of the implementation, moreover it uses known tools, which is a counter-argument to the claim that methods based on tree-partitioning are impractical.
Cite as
Michał Gańczorz. Using Statistical Encoding to Achieve Tree Succinctness Never Seen Before. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 22:1-22:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{ganczorz:LIPIcs.STACS.2020.22,
author = {Ga\'{n}czorz, Micha{\l}},
title = {{Using Statistical Encoding to Achieve Tree Succinctness Never Seen Before}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {22:1--22:29},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.22},
URN = {urn:nbn:de:0030-drops-118836},
doi = {10.4230/LIPIcs.STACS.2020.22},
annote = {Keywords: succinct data structures, labeled tree, ordered tree, entropy, tree entropy}
}
Document
Quantum Distributed Algorithm for Triangle Finding in the CONGEST Model
Abstract
This paper considers the triangle finding problem in the CONGEST model of distributed computing. Recent works by Izumi and Le Gall (PODC'17), Chang, Pettie and Zhang (SODA'19) and Chang and Saranurak (PODC'19) have successively reduced the classical round complexity of triangle finding (as well as triangle listing) from the trivial upper bound O(n) to Õ(n^{1/3}), where n denotes the number of vertices in the graph. In this paper we present a quantum distributed algorithm that solves the triangle finding problem in Õ(n^{1/4}) rounds in the CONGEST model. This gives another example of quantum algorithm beating the best known classical algorithms in distributed computing. Our result also exhibits an interesting phenomenon: while in the classical setting the best known upper bounds for the triangle finding and listing problems are identical, in the quantum setting the round complexities of these two problems are now Õ(n^{1/4}) and Θ~(n^{1/3}), respectively. Our result thus shows that triangle finding is easier than triangle listing in the quantum CONGEST model.
Cite as
Taisuke Izumi, François Le Gall, and Frédéric Magniez. Quantum Distributed Algorithm for Triangle Finding in the CONGEST Model. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 23:1-23:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{izumi_et_al:LIPIcs.STACS.2020.23,
author = {Izumi, Taisuke and Le Gall, Fran\c{c}ois and Magniez, Fr\'{e}d\'{e}ric},
title = {{Quantum Distributed Algorithm for Triangle Finding in the CONGEST Model}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {23:1--23:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.23},
URN = {urn:nbn:de:0030-drops-118840},
doi = {10.4230/LIPIcs.STACS.2020.23},
annote = {Keywords: Quantum computing, distributed computing, CONGEST model}
}
Document
Lower Bounds for Arithmetic Circuits via the Hankel Matrix
Abstract
We study the complexity of representing polynomials by arithmetic circuits in both the commutative and the non-commutative settings. To analyse circuits we count their number of parse trees, which describe the non-associative computations realised by the circuit.
In the non-commutative setting a circuit computing a polynomial of degree d has at most 2^{O(d)} parse trees. Previous superpolynomial lower bounds were known for circuits with up to 2^{d^{1/3-ε}} parse trees, for any ε > 0. Our main result is to reduce the gap by showing a superpolynomial lower bound for circuits with just a small defect in the exponent for the total number of parse trees, that is 2^{d^{1 - ε}}, for any ε > 0.
In the commutative setting a circuit computing a polynomial of degree d has at most 2^{O(d log d)} parse trees. We show a superpolynomial lower bound for circuits with up to 2^{d^{1/3 - ε}} parse trees, for any ε > 0. When d is polylogarithmic in n, we push this further to up to 2^{d^{1 - ε}} parse trees.
While these two main results hold in the associative setting, our approach goes through a precise understanding of the more restricted setting where multiplication is not associative, meaning that we distinguish the polynomials (xy)z and x(yz). Our first and main conceptual result is a characterization result: we show that the size of the smallest circuit computing a given non-associative polynomial is exactly the rank of a matrix constructed from the polynomial and called the Hankel matrix. This result applies to the class of all circuits in both commutative and non-commutative settings, and can be seen as an extension of the seminal result of Nisan giving a similar characterization for non-commutative algebraic branching programs. Our key technical contribution is to provide generic lower bound theorems based on analyzing and decomposing the Hankel matrix, from which we derive the results mentioned above.
The study of the Hankel matrix also provides a unifying approach for proving lower bounds for polynomials in the (classical) associative setting. We demonstrate this by giving alternative proofs of recent lower bounds as corollaries of our generic lower bound results.
Cite as
Nathanaël Fijalkow, Guillaume Lagarde, Pierre Ohlmann, and Olivier Serre. Lower Bounds for Arithmetic Circuits via the Hankel Matrix. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{fijalkow_et_al:LIPIcs.STACS.2020.24,
author = {Fijalkow, Nathana\"{e}l and Lagarde, Guillaume and Ohlmann, Pierre and Serre, Olivier},
title = {{Lower Bounds for Arithmetic Circuits via the Hankel Matrix}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {24:1--24:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.24},
URN = {urn:nbn:de:0030-drops-118859},
doi = {10.4230/LIPIcs.STACS.2020.24},
annote = {Keywords: Arithmetic Circuit Complexity, Lower Bounds, Parse Trees, Hankel Matrix}
}
Document
Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs
Abstract
The VertexCover problem is proven to be computationally hard in different ways: It is NP-complete to find an optimal solution and even NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. Similarly, greedy algorithms deliver very good approximations to the optimal solution in practice.
We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability.
The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice. When utilizing the same structural properties in an adaptive greedy algorithm, further experiments suggest that, on real instances, this leads to better approximations than the standard greedy approach within reasonable time.
Cite as
Thomas Bläsius, Philipp Fischbeck, Tobias Friedrich, and Maximilian Katzmann. Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{blasius_et_al:LIPIcs.STACS.2020.25,
author = {Bl\"{a}sius, Thomas and Fischbeck, Philipp and Friedrich, Tobias and Katzmann, Maximilian},
title = {{Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {25:1--25:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.25},
URN = {urn:nbn:de:0030-drops-118865},
doi = {10.4230/LIPIcs.STACS.2020.25},
annote = {Keywords: vertex cover, random graphs, hyperbolic geometry, efficient algorithm}
}
Document
Domino Problem Under Horizontal Constraints
Abstract
The Domino Problem on ℤ² asks if it is possible to tile the plane with a given set of Wang tiles; it is a classical decision problem which is known to be undecidable. The purpose of this article is to parameterize this problem to explore the frontier between decidability and undecidability. To do so we fix some horizontal constraints H on the tiles and consider a new Domino Problem DP_H: given a vertical constraint, is it possible to tile the plane? We characterize the nearest-neighbor horizontal constraints where DP_H is decidable using graphs combinatorics.
Cite as
Nathalie Aubrun, Julien Esnay, and Mathieu Sablik. Domino Problem Under Horizontal Constraints. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{aubrun_et_al:LIPIcs.STACS.2020.26,
author = {Aubrun, Nathalie and Esnay, Julien and Sablik, Mathieu},
title = {{Domino Problem Under Horizontal Constraints}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {26:1--26:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.26},
URN = {urn:nbn:de:0030-drops-118875},
doi = {10.4230/LIPIcs.STACS.2020.26},
annote = {Keywords: Dynamical Systems, Symbolic Dynamics, Subshifts, Wang tiles, Undecidability, Domino Problem, Combinatorics, Tilings, Subshifts of Finite Type}
}
Document
Computing Maximum Matchings in Temporal Graphs
Abstract
Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph G, a temporal graph is represented by assigning a set of integer time-labels to every edge e of G, indicating the discrete time steps at which e is active. We introduce and study the complexity of a natural temporal extension of the classical graph problem Maximum Matching, taking into account the dynamic nature of temporal graphs. In our problem, Maximum Temporal Matching, we are looking for the largest possible number of time-labeled edges (simply time-edges) (e,t) such that no vertex is matched more than once within any time window of Δ consecutive time slots, where Δ ∈ ℕ is given. The requirement that a vertex cannot be matched twice in any Δ-window models some necessary "recovery" period that needs to pass for an entity (vertex) after being paired up for some activity with another entity. We prove strong computational hardness results for Maximum Temporal Matching, even for elementary cases. To cope with this computational hardness, we mainly focus on fixed-parameter algorithms with respect to natural parameters, as well as on polynomial-time approximation algorithms.
Cite as
George B. Mertzios, Hendrik Molter, Rolf Niedermeier, Viktor Zamaraev, and Philipp Zschoche. Computing Maximum Matchings in Temporal Graphs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{mertzios_et_al:LIPIcs.STACS.2020.27,
author = {Mertzios, George B. and Molter, Hendrik and Niedermeier, Rolf and Zamaraev, Viktor and Zschoche, Philipp},
title = {{Computing Maximum Matchings in Temporal Graphs}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {27:1--27:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.27},
URN = {urn:nbn:de:0030-drops-118881},
doi = {10.4230/LIPIcs.STACS.2020.27},
annote = {Keywords: Temporal Graph, Link Stream, Temporal Line Graph, NP-hardness, APX-hardness, Approximation Algorithm, Fixed-parameter Tractability, Independent Set}
}
Document
Tight Bounds for the Cover Times of Random Walks with Heterogeneous Step Lengths
Abstract
Search patterns of randomly oriented steps of different lengths have been observed on all scales of the biological world, ranging from microscopic to the ecological, including in protein motors, bacteria, T-cells, honeybees, marine predators, and more, see e.g., [Humphries et al., 2010; Jansen et al., 2012; Reynolds et al., 2017; Schuster and Levandowsky, 1996; Humphries et al., 2010; Viswanathan et al., 1996; Viswanathan et al., 1999]. Through different models, it has been demonstrated that adopting a variety in the magnitude of the step lengths can greatly improve the search efficiency. However, the precise connection between the search efficiency and the number of step lengths in the repertoire of the searcher has not been identified.
Motivated by biological examples in one-dimensional terrains, a recent paper studied the best cover time on an n-node cycle that can be achieved by a random walk process that uses k step lengths [Boczkowski et al., 2018]. By tuning the lengths and corresponding probabilities the authors therein showed that the best cover time is roughly n^{1+Θ(1/k)}. While this bound is useful for large values of k, it is hardly informative for small k values, which are of interest in biology [Auger-Méthé et al., 2015; Bénichou et al., 2011; Lomholt et al., 2008; {Reynolds}, 2014]. In this paper, we provide a tight bound for the cover time of such a walk, for every integer k> 1. Specifically, up to lower order polylogarithmic factors, the cover time is n^{1+1/(2k-1)}. For k=2,3, 4 and 5 the bound is thus n^{4/3}, n^{6/5}, n^{8/7}, and n^{10/9}, respectively. Informally, our result implies that, as long as the number of step lengths k is not too large, incorporating an additional step length to the repertoire of the process enables to improve the cover time by a polynomial factor, but the extent of the improvement gradually decreases with k.
Cite as
Brieuc Guinard and Amos Korman. Tight Bounds for the Cover Times of Random Walks with Heterogeneous Step Lengths. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 28:1-28:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{guinard_et_al:LIPIcs.STACS.2020.28,
author = {Guinard, Brieuc and Korman, Amos},
title = {{Tight Bounds for the Cover Times of Random Walks with Heterogeneous Step Lengths}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {28:1--28:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.28},
URN = {urn:nbn:de:0030-drops-118892},
doi = {10.4230/LIPIcs.STACS.2020.28},
annote = {Keywords: Computational Biology, Randomness in Computing, Search Algorithms, Random Walks, L\'{e}vy Flights, Intermittent Search, CCRW}
}
Document
Solving Connectivity Problems Parameterized by Treedepth in Single-Exponential Time and Polynomial Space
Abstract
A breakthrough result of Cygan et al. (FOCS 2011) showed that connectivity problems parameterized by treewidth can be solved much faster than the previously best known time ?^*(2^{?(twlog tw)}). Using their inspired Cut&Count technique, they obtained ?^*(α^tw) time algorithms for many such problems. Moreover, they proved these running times to be optimal assuming the Strong Exponential-Time Hypothesis. Unfortunately, like other dynamic programming algorithms on tree decompositions, these algorithms also require exponential space, and this is widely believed to be unavoidable. In contrast, for the slightly larger parameter called treedepth, there are already several examples of matching the time bounds obtained for treewidth, but using only polynomial space. Nevertheless, this has remained open for connectivity problems.
In the present work, we close this knowledge gap by applying the Cut&Count technique to graphs of small treedepth. While the general idea is unchanged, we have to design novel procedures for counting consistently cut solution candidates using only polynomial space. Concretely, we obtain time ?^*(3^d) and polynomial space for Connected Vertex Cover, Feedback Vertex Set, and Steiner Tree on graphs of treedepth d. Similarly, we obtain time ?^*(4^d) and polynomial space for Connected Dominating Set and Connected Odd Cycle Transversal.
Cite as
Falko Hegerfeld and Stefan Kratsch. Solving Connectivity Problems Parameterized by Treedepth in Single-Exponential Time and Polynomial Space. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{hegerfeld_et_al:LIPIcs.STACS.2020.29,
author = {Hegerfeld, Falko and Kratsch, Stefan},
title = {{Solving Connectivity Problems Parameterized by Treedepth in Single-Exponential Time and Polynomial Space}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {29:1--29:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.29},
URN = {urn:nbn:de:0030-drops-118907},
doi = {10.4230/LIPIcs.STACS.2020.29},
annote = {Keywords: Parameterized Complexity, Connectivity, Treedepth, Cut\&Count, Polynomial Space}
}
Document
Non-Rectangular Convolutions and (Sub-)Cadences with Three Elements
Abstract
The discrete acyclic convolution computes the 2n+1 sums ∑_{i+j=k|(i,j)∈[0,1,2,… ,n]²} a_i b_j in ?(n log n) time. By using suitable offsets and setting some of the variables to zero, this method provides a tool to calculate all non-zero sums ∑_{i+j=k|(i,j)∈ P∩ℤ²} a_i b_j in a rectangle P with perimeter p in ?(p log p) time.
This paper extends this geometric interpretation in order to allow arbitrary convex polygons P with k vertices and perimeter p. Also, this extended algorithm only needs ?(k + p(log p)² log k) time.
Additionally, this paper presents fast algorithms for counting sub-cadences and cadences with 3 elements using this extended method.
Cite as
Mitsuru Funakoshi and Julian Pape-Lange. Non-Rectangular Convolutions and (Sub-)Cadences with Three Elements. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 30:1-30:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{funakoshi_et_al:LIPIcs.STACS.2020.30,
author = {Funakoshi, Mitsuru and Pape-Lange, Julian},
title = {{Non-Rectangular Convolutions and (Sub-)Cadences with Three Elements}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {30:1--30:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.30},
URN = {urn:nbn:de:0030-drops-118911},
doi = {10.4230/LIPIcs.STACS.2020.30},
annote = {Keywords: discrete acyclic convolutions, string-cadences, geometric algorithms, number theoretic transforms}
}
Document
Maximum Matchings in Geometric Intersection Graphs
Abstract
Let G be an intersection graph of n geometric objects in the plane. We show that a maximum matching in G can be found in O(ρ^{3ω/2}n^{ω/2}) time with high probability, where ρ is the density of the geometric objects and ω>2 is a constant such that n × n matrices can be multiplied in O(n^ω) time.
The same result holds for any subgraph of G, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators.
We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in O(n^{ω/2}) time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in [1, Ψ] can be found in O(Ψ⁶log^11 n + Ψ^{12 ω} n^{ω/2}) time with high probability.
Cite as
Édouard Bonnet, Sergio Cabello, and Wolfgang Mulzer. Maximum Matchings in Geometric Intersection Graphs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bonnet_et_al:LIPIcs.STACS.2020.31,
author = {Bonnet, \'{E}douard and Cabello, Sergio and Mulzer, Wolfgang},
title = {{Maximum Matchings in Geometric Intersection Graphs}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {31:1--31:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.31},
URN = {urn:nbn:de:0030-drops-118926},
doi = {10.4230/LIPIcs.STACS.2020.31},
annote = {Keywords: computational geometry, geometric intersection graph, maximum matching, disk graph, unit-disk graph}
}
Document
Unambiguous Separators for Tropical Tree Automata
Abstract
In this paper we show that given a max-plus automaton (over trees, and with real weights) computing a function f and a min-plus automaton (similar) computing a function g such that f ⩽ g, there exists effectively an unambiguous tropical automaton computing h such that f ⩽ h ⩽ g.
This generalizes a result of Lombardy and Mairesse of 2006 stating that series which are both max-plus and min-plus rational are unambiguous. This generalization goes in two directions: trees are considered instead of words, and separation is established instead of characterization (separation implies characterization). The techniques in the two proofs are very different.
Cite as
Thomas Colcombet and Sylvain Lombardy. Unambiguous Separators for Tropical Tree Automata. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{colcombet_et_al:LIPIcs.STACS.2020.32,
author = {Colcombet, Thomas and Lombardy, Sylvain},
title = {{Unambiguous Separators for Tropical Tree Automata}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {32:1--32:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.32},
URN = {urn:nbn:de:0030-drops-118933},
doi = {10.4230/LIPIcs.STACS.2020.32},
annote = {Keywords: Tree automata, Tropical semiring, Separation, Unambiguity}
}
Document
Asymptotic Quasi-Polynomial Time Approximation Scheme for Resource Minimization for Fire Containment
Abstract
Resource Minimization Fire Containment (RMFC) is a natural model for optimal inhibition of harmful spreading phenomena on a graph. In the RMFC problem on trees, we are given an undirected tree G, and a vertex r where the fire starts at, called root. At each time step, the firefighters can protect up to B vertices of the graph while the fire spreads from burning vertices to all their neighbors that have not been protected so far. The task is to find the smallest B that allows for saving all the leaves of the tree. The problem is hard to approximate up to any factor better than 2 even on trees unless P = NP [King and MacGillivray, 2010].
Chalermsook and Chuzhoy [Chalermsook and Chuzhoy, 2010] presented a Linear Programming based O(log^* n) approximation for RMFC on trees that matches the integrality gap of the natural Linear Programming relaxation. This was recently improved by Adjiashvili, Baggio, and Zenklusen [Adjiashvili et al., 2017] to a 12-approximation through a combination of LP rounding along with several new techniques.
In this paper we present an asymptotic QPTAS for RMFC on trees. More specifically, let ε>0, and ℐ be an instance of RMFC where the optimum number of firefighters to save all the leaves is OPT(ℐ). We present an algorithm which uses at most ⌈(1+ε)OPT(ℐ)⌉ many firefighters at each time step and runs in time n^O(log log n/ε). This suggests that the existence of an asymptotic PTAS is plausible especially since the exponent is O(log log n), not O(log n).
Our result combines a more powerful height reduction lemma than the one in [Adjiashvili et al., 2017] with LP rounding and dynamic programming to find the solution. We also apply our height reduction lemma to the algorithm provided in [Adjiashvili et al., 2017] plus a more careful analysis to improve their 12-approximation and provide a polynomial time (5+ε)-approximation.
Cite as
Mirmahdi Rahgoshay and Mohammad R. Salavatipour. Asymptotic Quasi-Polynomial Time Approximation Scheme for Resource Minimization for Fire Containment. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{rahgoshay_et_al:LIPIcs.STACS.2020.33,
author = {Rahgoshay, Mirmahdi and Salavatipour, Mohammad R.},
title = {{Asymptotic Quasi-Polynomial Time Approximation Scheme for Resource Minimization for Fire Containment}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {33:1--33:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.33},
URN = {urn:nbn:de:0030-drops-118946},
doi = {10.4230/LIPIcs.STACS.2020.33},
annote = {Keywords: Firefighter Problem, Resource Management, Fire Containment, Approximation Algorithm, Asymptotic Approximation Scheme}
}
Document
Streaming Complexity of Spanning Tree Computation
Abstract
The semi-streaming model is a variant of the streaming model frequently used for the computation of graph problems. It allows the edges of an n-node input graph to be read sequentially in p passes using Õ(n) space. If the list of edges includes deletions, then the model is called the turnstile model; otherwise it is called the insertion-only model. In both models, some graph problems, such as spanning trees, k-connectivity, densest subgraph, degeneracy, cut-sparsifier, and (Δ+1)-coloring, can be exactly solved or (1+ε)-approximated in a single pass; while other graph problems, such as triangle detection and unweighted all-pairs shortest paths, are known to require Ω̃(n) passes to compute. For many fundamental graph problems, the tractability in these models is open. In this paper, we study the tractability of computing some standard spanning trees, including BFS, DFS, and maximum-leaf spanning trees.
Our results, in both the insertion-only and the turnstile models, are as follows.
- Maximum-Leaf Spanning Trees: This problem is known to be APX-complete with inapproximability constant ρ ∈ [245/244, 2). By constructing an ε-MLST sparsifier, we show that for every constant ε > 0, MLST can be approximated in a single pass to within a factor of 1+ε w.h.p. (albeit in super-polynomial time for ε ≤ ρ-1 assuming P ≠ NP) and can be approximated in polynomial time in a single pass to within a factor of ρ_n+ε w.h.p., where ρ_n is the supremum constant that MLST cannot be approximated to within using polynomial time and Õ(n) space. In the insertion-only model, these algorithms can be deterministic.
- BFS Trees: It is known that BFS trees require ω(1) passes to compute, but the naïve approach needs O(n) passes. We devise a new randomized algorithm that reduces the pass complexity to O(√n), and it offers a smooth tradeoff between pass complexity and space usage. This gives a polynomial separation between single-source and all-pairs shortest paths for unweighted graphs.
- DFS Trees: It is unknown whether DFS trees require more than one pass. The current best algorithm by Khan and Mehta [STACS 2019] takes Õ(h) passes, where h is the height of computed DFS trees. Note that h can be as large as Ω(m/n) for n-node m-edge graphs. Our contribution is twofold. First, we provide a simple alternative proof of this result, via a new connection to sparse certificates for k-node-connectivity. Second, we present a randomized algorithm that reduces the pass complexity to O(√n), and it also offers a smooth tradeoff between pass complexity and space usage.
Cite as
Yi-Jun Chang, Martín Farach-Colton, Tsan-Sheng Hsu, and Meng-Tsung Tsai. Streaming Complexity of Spanning Tree Computation. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 34:1-34:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{chang_et_al:LIPIcs.STACS.2020.34,
author = {Chang, Yi-Jun and Farach-Colton, Mart{\'\i}n and Hsu, Tsan-Sheng and Tsai, Meng-Tsung},
title = {{Streaming Complexity of Spanning Tree Computation}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {34:1--34:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.34},
URN = {urn:nbn:de:0030-drops-118951},
doi = {10.4230/LIPIcs.STACS.2020.34},
annote = {Keywords: Max-Leaf Spanning Trees, BFS Trees, DFS Trees}
}
Document
Shortest Reconfiguration of Colorings Under Kempe Changes
Abstract
A k-coloring of a graph maps each vertex of the graph to a color in {1, 2, …, k}, such that no two adjacent vertices receive the same color. Given a k-coloring of a graph, a Kempe change produces a new k-coloring by swapping the colors in a bicolored connected component. We investigate the complexity of finding the smallest number of Kempe changes needed to transform a given k-coloring into another given k-coloring. We show that this problem admits a polynomial-time dynamic programming algorithm on path graphs, which turns out to be highly non-trivial. Furthermore, the problem is NP-hard even on star graphs and we show that on such graphs it admits a constant-factor approximation algorithm and is fixed-parameter tractable when parameterized by the number k of colors. The hardness result as well as the algorithmic results are based on the notion of a canonical transformation.
Cite as
Marthe Bonamy, Marc Heinrich, Takehiro Ito, Yusuke Kobayashi, Haruka Mizuta, Moritz Mühlenthaler, Akira Suzuki, and Kunihiro Wasa. Shortest Reconfiguration of Colorings Under Kempe Changes. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 35:1-35:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bonamy_et_al:LIPIcs.STACS.2020.35,
author = {Bonamy, Marthe and Heinrich, Marc and Ito, Takehiro and Kobayashi, Yusuke and Mizuta, Haruka and M\"{u}hlenthaler, Moritz and Suzuki, Akira and Wasa, Kunihiro},
title = {{Shortest Reconfiguration of Colorings Under Kempe Changes}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {35:1--35:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.35},
URN = {urn:nbn:de:0030-drops-118961},
doi = {10.4230/LIPIcs.STACS.2020.35},
annote = {Keywords: Combinatorial Reconfiguration, Graph Algorithms, Graph Coloring, Kempe Equivalence}
}
Document
Elimination Distances, Blocking Sets, and Kernels for Vertex Cover
Abstract
The Vertex Cover problem plays an essential role in the study of polynomial kernelization in parameterized complexity, i.e., the study of provable and efficient preprocessing for NP-hard problems. Motivated by the great variety of positive and negative results for kernelization for Vertex Cover subject to different parameters and graph classes, we seek to unify and generalize them using so-called blocking sets. A blocking set is a set of vertices such that no optimal vertex cover contains all vertices in the blocking set, and the study of minimal blocking sets played implicit and explicit roles in many existing results.
We show that in the most-studied setting, parameterized by the size of a deletion set to a specified graph class ?, bounded minimal blocking set size is necessary but not sufficient to get a polynomial kernelization. Under mild technical assumptions, bounded minimal blocking set size is showed to allow an essentially tight efficient reduction in the number of connected components.
We then determine the exact maximum size of minimal blocking sets for graphs of bounded elimination distance to any hereditary class ?, including the case of graphs of bounded treedepth. We get similar but not tight bounds for certain non-hereditary classes ?, including the class ?_{LP} of graphs where integral and fractional vertex cover size coincide. These bounds allow us to derive polynomial kernels for Vertex Cover parameterized by the size of a deletion set to graphs of bounded elimination distance to, e.g., forest, bipartite, or ?_{LP} graphs.
Cite as
Eva-Maria C. Hols, Stefan Kratsch, and Astrid Pieterse. Elimination Distances, Blocking Sets, and Kernels for Vertex Cover. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 36:1-36:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{hols_et_al:LIPIcs.STACS.2020.36,
author = {Hols, Eva-Maria C. and Kratsch, Stefan and Pieterse, Astrid},
title = {{Elimination Distances, Blocking Sets, and Kernels for Vertex Cover}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {36:1--36:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.36},
URN = {urn:nbn:de:0030-drops-118974},
doi = {10.4230/LIPIcs.STACS.2020.36},
annote = {Keywords: Vertex Cover, kernelization, blocking sets, elimination distance, structural parameters}
}
Document
Near-Optimal Complexity Bounds for Fragments of the Skolem Problem
Abstract
Given a linear recurrence sequence (LRS), specified using the initial conditions and the recurrence relation, the Skolem problem asks if zero ever occurs in the infinite sequence generated by the LRS. Despite active research over last few decades, its decidability is known only for a few restricted subclasses, by either restricting the order of the LRS (upto 4) or by restricting the structure of the LRS (e.g., roots of its characteristic polynomial).
In this paper, we identify a subclass of LRS of arbitrary order for which the Skolem problem is easy, namely LRS all of whose characteristic roots are (possibly complex) roots of real algebraic numbers, i.e., roots satisfying x^d = r for r real algebraic. We show that for this subclass, the Skolem problem can be solved in NP^RP. As a byproduct, we implicitly obtain effective bounds on the zero set of the LRS for this subclass. While prior works in this area often exploit deep results from algebraic and transcendental number theory to get such effective results, our techniques are primarily algorithmic and use linear algebra and Galois theory. We also complement our upper bounds with a NP lower bound for the Skolem problem via a new direct reduction from 3-CNF-SAT, matching the best known lower bounds.
Cite as
S. Akshay, Nikhil Balaji, Aniket Murhekar, Rohith Varma, and Nikhil Vyas. Near-Optimal Complexity Bounds for Fragments of the Skolem Problem. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{akshay_et_al:LIPIcs.STACS.2020.37,
author = {Akshay, S. and Balaji, Nikhil and Murhekar, Aniket and Varma, Rohith and Vyas, Nikhil},
title = {{Near-Optimal Complexity Bounds for Fragments of the Skolem Problem}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {37:1--37:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.37},
URN = {urn:nbn:de:0030-drops-118982},
doi = {10.4230/LIPIcs.STACS.2020.37},
annote = {Keywords: Linear Recurrences, Skolem problem, NP-completeness, Weighted automata}
}
Document
Efficient Parameterized Algorithms for Computing All-Pairs Shortest Paths
Abstract
Computing all-pairs shortest paths is a fundamental and much-studied problem with many applications. Unfortunately, despite intense study, there are still no significantly faster algorithms for it than the ?(n³) time algorithm due to Floyd and Warshall (1962). Somewhat faster algorithms exist for the vertex-weighted version if fast matrix multiplication may be used. Yuster (SODA 2009) gave an algorithm running in time ?(n^2.842), but no combinatorial, truly subcubic algorithm is known.
Motivated by the recent framework of efficient parameterized algorithms (or "FPT in P"), we investigate the influence of the graph parameters clique-width (cw) and modular-width (mw) on the running times of algorithms for solving ALL-PAIRS SHORTEST PATHS. We obtain efficient (and combinatorial) parameterized algorithms on non-negative vertex-weighted graphs of times ?(cw²n²), resp. ?(mw²n + n²). If fast matrix multiplication is allowed then the latter can be improved to ?(mw^{1.842} n + n²) using the algorithm of Yuster as a black box. The algorithm relative to modular-width is adaptive, meaning that the running time matches the best unparameterized algorithm for parameter value mw equal to n, and they outperform them already for mw ∈ ?(n^{1 - ε}) for any ε > 0.
Cite as
Stefan Kratsch and Florian Nelles. Efficient Parameterized Algorithms for Computing All-Pairs Shortest Paths. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{kratsch_et_al:LIPIcs.STACS.2020.38,
author = {Kratsch, Stefan and Nelles, Florian},
title = {{Efficient Parameterized Algorithms for Computing All-Pairs Shortest Paths}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {38:1--38:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.38},
URN = {urn:nbn:de:0030-drops-118992},
doi = {10.4230/LIPIcs.STACS.2020.38},
annote = {Keywords: All-pairs shortest Paths, efficient parameterized Algorithms, parameterized Complexity, Clique-width, Modular-width}
}
Document
Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width
Abstract
Solving the algebraic dichotomy conjecture for constraint satisfaction problems over structures first-order definable in countably infinite finitely bounded homogeneous structures requires understanding the applicability of local-consistency methods in this setting. We study the amount of consistency (measured by relational width) needed to solve CSP(?) for first-order expansions ? of countably infinite homogeneous graphs ℋ := (A; E), which happen all to be finitely bounded. We study our problem for structures ? that additionally have bounded strict width, i.e., for which establishing local consistency of an instance of CSP(?) not only decides if there is a solution but also ensures that every solution may be obtained from a locally consistent instance by greedily assigning values to variables, without backtracking.
Our main result is that the structures ? under consideration have relational width exactly (2, ?_ℋ) where ?_ℋ is the maximal size of a forbidden subgraph of ℋ, but not smaller than 3. It beats the upper bound: (2 m, 3 m) where m = max(arity(?)+1, ?, 3) and arity(?) is the largest arity of a relation in ?, which follows from a sufficient condition implying bounded relational width given in [Manuel Bodirsky and Antoine Mottet, 2018]. Since ?_ℋ may be arbitrarily large, our result contrasts the collapse of the relational bounded width hierarchy for finite structures ?, whose relational width, if finite, is always at most (2,3).
Cite as
Michał Wrona. Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 39:1-39:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{wrona:LIPIcs.STACS.2020.39,
author = {Wrona, Micha{\l}},
title = {{Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {39:1--39:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.39},
URN = {urn:nbn:de:0030-drops-119000},
doi = {10.4230/LIPIcs.STACS.2020.39},
annote = {Keywords: Constraint Satisfaction, Homogeneous Graphs, Bounded Width, Strict Width, Relational Width, Computational Complexity}
}
Document
Succinct Population Protocols for Presburger Arithmetic
Abstract
In [Dana Angluin et al., 2006], Angluin et al. proved that population protocols compute exactly the predicates definable in Presburger arithmetic (PA), the first-order theory of addition. As part of this result, they presented a procedure that translates any formula φ of quantifier-free PA with remainder predicates (which has the same expressive power as full PA) into a population protocol with 2^?(poly(|φ|)) states that computes φ. More precisely, the number of states of the protocol is exponential in both the bit length of the largest coefficient in the formula, and the number of nodes of its syntax tree.
In this paper, we prove that every formula φ of quantifier-free PA with remainder predicates is computable by a leaderless population protocol with ?(poly(|φ|)) states. Our proof is based on several new constructions, which may be of independent interest. Given a formula φ of quantifier-free PA with remainder predicates, a first construction produces a succinct protocol (with ?(|φ|³) leaders) that computes φ; this completes the work initiated in [Michael Blondin et al., 2018], where we constructed such protocols for a fragment of PA. For large enough inputs, we can get rid of these leaders. If the input is not large enough, then it is small, and we design another construction producing a succinct protocol with one leader that computes φ. Our last construction gets rid of this leader for small inputs.
Cite as
Michael Blondin, Javier Esparza, Blaise Genest, Martin Helfrich, and Stefan Jaax. Succinct Population Protocols for Presburger Arithmetic. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{blondin_et_al:LIPIcs.STACS.2020.40,
author = {Blondin, Michael and Esparza, Javier and Genest, Blaise and Helfrich, Martin and Jaax, Stefan},
title = {{Succinct Population Protocols for Presburger Arithmetic}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {40:1--40:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.40},
URN = {urn:nbn:de:0030-drops-119018},
doi = {10.4230/LIPIcs.STACS.2020.40},
annote = {Keywords: Population protocols, Presburger arithmetic, state complexity}
}
Document
A Sub-Quadratic Algorithm for the Longest Common Increasing Subsequence Problem
Abstract
The Longest Common Increasing Subsequence problem (LCIS) is a natural variant of the celebrated Longest Common Subsequence (LCS) problem. For LCIS, as well as for LCS, there is an ?(n²)-time algorithm and a SETH-based conditional lower bound of ?(n^{2-ε}). For LCS, there is also the Masek-Paterson ?(n²/log n)-time algorithm, which does not seem to adapt to LCIS in any obvious way. Hence, a natural question arises: does any (slightly) sub-quadratic algorithm exist for the Longest Common Increasing Subsequence problem? We answer this question positively, presenting a ?(n²/log^a n)-time algorithm for a = 1/6-o(1). The algorithm is not based on memorizing small chunks of data (often used for logarithmic speedups, including the "Four Russians Trick" in LCS), but rather utilizes a new technique, bounding the number of significant symbol matches between the two sequences.
Cite as
Lech Duraj. A Sub-Quadratic Algorithm for the Longest Common Increasing Subsequence Problem. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 41:1-41:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{duraj:LIPIcs.STACS.2020.41,
author = {Duraj, Lech},
title = {{A Sub-Quadratic Algorithm for the Longest Common Increasing Subsequence Problem}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {41:1--41:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.41},
URN = {urn:nbn:de:0030-drops-119020},
doi = {10.4230/LIPIcs.STACS.2020.41},
annote = {Keywords: longest common increasing subsequence, log-shaving, matching pairs}
}
Document
Fixed-Parameter Algorithms for Unsplittable Flow Cover
Abstract
The Unsplittable Flow Cover problem (UFP-cover) models the well-studied general caching problem and various natural resource allocation settings. We are given a path with a demand on each edge and a set of tasks, each task being defined by a subpath and a size. The goal is to select a subset of the tasks of minimum cardinality such that on each edge e the total size of the selected tasks using e is at least the demand of e. There is a polynomial time 4-approximation for the problem [Bar-Noy et al., STOC 2000] and also a QPTAS [Höhn et al., ICALP 2014]. In this paper we study fixed-parameter algorithms for the problem. We show that it is W[1]-hard but it becomes FPT if we can slightly violate the edge demands (resource augmentation) and also if there are at most k different task sizes. Then we present a parameterized approximation scheme (PAS), i.e., an algorithm with a running time of f(k)⋅ n^O_ε(1) that outputs a solution with at most (1+ε)k tasks or assert that there is no solution with at most k tasks. In this algorithm we use a new trick that intuitively allows us to pretend that we can select tasks from OPT multiple times.
Cite as
Andrés Cristi, Mathieu Mari, and Andreas Wiese. Fixed-Parameter Algorithms for Unsplittable Flow Cover. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 42:1-42:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{cristi_et_al:LIPIcs.STACS.2020.42,
author = {Cristi, Andr\'{e}s and Mari, Mathieu and Wiese, Andreas},
title = {{Fixed-Parameter Algorithms for Unsplittable Flow Cover}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {42:1--42:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.42},
URN = {urn:nbn:de:0030-drops-119037},
doi = {10.4230/LIPIcs.STACS.2020.42},
annote = {Keywords: Unsplittable Flow Cover, fixed parameter algorithms, approximation algorithms}
}
Document
Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm
Abstract
It is well known that the isomorphism problem for vertex-colored graphs with color multiplicity at most 3 is solvable by the classical 2-dimensional Weisfeiler-Leman algorithm (2-WL). On the other hand, the prominent Cai-Fürer-Immerman construction shows that even the multidimensional version of the algorithm does not suffice for graphs with color multiplicity 4. We give an efficient decision procedure that, given a graph G of color multiplicity 4, recognizes whether or not G is identifiable by 2-WL, that is, whether or not 2-WL distinguishes G from any non-isomorphic graph. In fact, we solve the more general problem of recognizing whether or not a given coherent configuration of maximum fiber size 4 is separable. This extends our recognition algorithm to directed graphs of color multiplicity 4 with colored edges.
Our decision procedure is based on an explicit description of the class of graphs with color multiplicity 4 that are not identifiable by 2-WL. The Cai-Fürer-Immerman graphs of color multiplicity 4 distinctly appear here as a natural subclass, which demonstrates that the Cai-Fürer-Immerman construction is not ad hoc. Our classification reveals also other types of graphs that are hard for 2-WL. One of them arises from patterns known as (n₃)-configurations in incidence geometry.
Cite as
Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky. Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{fuhlbruck_et_al:LIPIcs.STACS.2020.43,
author = {Fuhlbr\"{u}ck, Frank and K\"{o}bler, Johannes and Verbitsky, Oleg},
title = {{Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {43:1--43:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.43},
URN = {urn:nbn:de:0030-drops-119046},
doi = {10.4230/LIPIcs.STACS.2020.43},
annote = {Keywords: Graph Isomorphism, Weisfeiler-Leman Algorithm, Cai-F\"{u}rer-Immerman Graphs, coherent Configurations}
}
Document
Better Approximations for General Caching and UFP-Cover Under Resource Augmentation
Abstract
In the Unsplittable Flow on a Path Cover (UFP-cover) problem we are given a path with a demand for each edge and a set of tasks where each task is defined by a subpath, a size and a cost. The goal is to select a subset of the tasks of minimum cost that together cover the demand of each edge. This problem models various resource allocation settings and also the general caching problem. The best known polynomial time approximation ratio for it is 4 [Bar-Noy et al., STOC 2000]. In this paper, we study the resource augmentation setting in which we need to cover only a slightly smaller demand on each edge than the compared optimal solution. If the cost of each task equals its size (which represents the natural bit-model in the related general caching problem) we provide a polynomial time algorithm that computes a solution of optimal cost. We extend this result to general caching and to the packing version of Unsplittable Flow on a Path in their respective natural resource augmentation settings. For the case that the cost of each task equals its "area", i.e., the product of its size and its path length, we present a polynomial time (1+ε)-approximation for UFP-cover. If additionally the edge capacities are in a constant range we compute even a solution of optimal cost and also obtain a PTAS without resource augmentation.
Cite as
Andrés Cristi and Andreas Wiese. Better Approximations for General Caching and UFP-Cover Under Resource Augmentation. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{cristi_et_al:LIPIcs.STACS.2020.44,
author = {Cristi, Andr\'{e}s and Wiese, Andreas},
title = {{Better Approximations for General Caching and UFP-Cover Under Resource Augmentation}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {44:1--44:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.44},
URN = {urn:nbn:de:0030-drops-119053},
doi = {10.4230/LIPIcs.STACS.2020.44},
annote = {Keywords: General caching, unsplittable flow cover, approximation algorithm, resource augmentation}
}
Document
Improved Bounds on Fourier Entropy and Min-Entropy
Abstract
Given a Boolean function f:{-1,1}ⁿ→ {-1,1}, define the Fourier distribution to be the distribution on subsets of [n], where each S ⊆ [n] is sampled with probability f̂(S)². The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [E. Friedgut and G. Kalai, 1996] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C>0 such that ℍ(f̂²)≤ C⋅ Inf(f), where ℍ(f̂²) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f?
In this paper we present three new contributions towards the FEI conjecture:
ii) Our first contribution shows that ℍ(f̂²) ≤ 2⋅ aUC^⊕(f), where aUC^⊕(f) is the average unambiguous parity-certificate complexity of f. This improves upon several bounds shown by Chakraborty et al. [S. Chakraborty et al., 2016]. We further improve this bound for unambiguous DNFs.
iii) We next consider the weaker Fourier Min-entropy-Influence (FMEI) conjecture posed by O'Donnell and others [R. O'Donnell et al., 2011; R. O'Donnell, 2014] which asks if ℍ_{∞}(f̂²) ≤ C⋅ Inf(f), where ℍ_{∞}(f̂²) is the min-entropy of the Fourier distribution. We show ℍ_{∞}(f̂²) ≤ 2⋅?_{min}^⊕(f), where ?_{min}^⊕(f) is the minimum parity certificate complexity of f. We also show that for all ε ≥ 0, we have ℍ_{∞}(f̂²) ≤ 2log (‖f̂‖_{1,ε}/(1-ε)), where ‖f̂‖_{1,ε} is the approximate spectral norm of f. As a corollary, we verify the FMEI conjecture for the class of read-k DNFs (for constant k).
iv) Our third contribution is to better understand implications of the FEI conjecture for the structure of polynomials that 1/3-approximate a Boolean function on the Boolean cube. We pose a conjecture: no flat polynomial (whose non-zero Fourier coefficients have the same magnitude) of degree d and sparsity 2^ω(d) can 1/3-approximate a Boolean function. This conjecture is known to be true assuming FEI and we prove the conjecture unconditionally (i.e., without assuming the FEI conjecture) for a class of polynomials. We discuss an intriguing connection between our conjecture and the constant for the Bohnenblust-Hille inequality, which has been extensively studied in functional analysis.
Cite as
Srinivasan Arunachalam, Sourav Chakraborty, Michal Koucký, Nitin Saurabh, and Ronald de Wolf. Improved Bounds on Fourier Entropy and Min-Entropy. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 45:1-45:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{arunachalam_et_al:LIPIcs.STACS.2020.45,
author = {Arunachalam, Srinivasan and Chakraborty, Sourav and Kouck\'{y}, Michal and Saurabh, Nitin and de Wolf, Ronald},
title = {{Improved Bounds on Fourier Entropy and Min-Entropy}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {45:1--45:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.45},
URN = {urn:nbn:de:0030-drops-119062},
doi = {10.4230/LIPIcs.STACS.2020.45},
annote = {Keywords: Fourier analysis of Boolean functions, FEI conjecture, query complexity, polynomial approximation, approximate degree, certificate complexity}
}
Document
Information Distance Revisited
Abstract
We consider the notion of information distance between two objects x and y introduced by Bennett, Gács, Li, Vitanyi, and Zurek [C. H. Bennett et al., 1998] as the minimal length of a program that computes x from y as well as computing y from x, and study different versions of this notion. In the above paper, it was shown that the prefix version of information distance equals max (K(x|y),K(y|x)) up to additive logarithmic terms. It was claimed by Mahmud [Mahmud, 2009] that this equality holds up to additive O(1)-precision. We show that this claim is false, but does hold if the distance is at least logarithmic. This implies that the original definition provides a metric on strings that are at superlogarithmically separated.
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Bruno Bauwens. Information Distance Revisited. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bauwens:LIPIcs.STACS.2020.46,
author = {Bauwens, Bruno},
title = {{Information Distance Revisited}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {46:1--46:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.46},
URN = {urn:nbn:de:0030-drops-119071},
doi = {10.4230/LIPIcs.STACS.2020.46},
annote = {Keywords: Kolmogorov complexity, algorithmic information distance}
}
Document
On Computing Multilinear Polynomials Using Multi-r-ic Depth Four Circuits
Abstract
In this paper, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which polynomial computed at every node has a bound on the individual degree of r (referred to as multi-r-ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better and better bounds as the value of r increases.
Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing a multilinear polynomial on n^O(1) variables and degree d=o(n), must have size at least (n/r^1.1)^Ω(√{d/r}) when r is o(d) and is strictly less than n^1.1. This bound however deteriorates with increasing r. It is a natural question to ask if we can prove a bound that does not deteriorate with increasing r or a bound that holds for a larger regime of r.
We here prove a lower bound which does not deteriorate with r, however for a specific instance of d = d(n) but for a wider range of r. Formally, we show that there exists an explicit polynomial on n^O(1) variables and degree Θ(log² n) such that any depth four circuit of bounded individual degree r<n^0.2 must have size at least exp(Ω(log² n)). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017).
Cite as
Suryajith Chillara. On Computing Multilinear Polynomials Using Multi-r-ic Depth Four Circuits. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 47:1-47:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{chillara:LIPIcs.STACS.2020.47,
author = {Chillara, Suryajith},
title = {{On Computing Multilinear Polynomials Using Multi-r-ic Depth Four Circuits}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {47:1--47:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.47},
URN = {urn:nbn:de:0030-drops-119084},
doi = {10.4230/LIPIcs.STACS.2020.47},
annote = {Keywords: Lower Bounds, Multilinear, Multi-r-ic}
}
Document
Observation and Distinction. Representing Information in Infinite Games
Abstract
We compare two approaches for modelling imperfect information in infinite games by using finite-state automata. The first, more standard approach views information as the result of an observation process driven by a sequential Mealy machine. In contrast, the second approach features indistinguishability relations described by synchronous two-tape automata.
The indistinguishability-relation model turns out to be strictly more expressive than the one based on observations. We present a characterisation of the indistinguishability relations that admit a representation as a finite-state observation function. We show that the characterisation is decidable, and give a procedure to construct a corresponding Mealy machine whenever one exists.
Cite as
Dietmar Berwanger and Laurent Doyen. Observation and Distinction. Representing Information in Infinite Games. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 48:1-48:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{berwanger_et_al:LIPIcs.STACS.2020.48,
author = {Berwanger, Dietmar and Doyen, Laurent},
title = {{Observation and Distinction. Representing Information in Infinite Games}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {48:1--48:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.48},
URN = {urn:nbn:de:0030-drops-119095},
doi = {10.4230/LIPIcs.STACS.2020.48},
annote = {Keywords: Infinite Games on Finite Graphs, Imperfect Information, Automatic Structures}
}
Document
How Fast Can You Escape a Compact Polytope?
Abstract
The Continuous Polytope Escape Problem (CPEP) asks whether every trajectory of a linear differential equation initialised within a convex polytope eventually escapes the polytope. We provide a polynomial-time algorithm to decide CPEP for compact polytopes. We also establish a quantitative uniform upper bound on the time required for every trajectory to escape the given polytope. In addition, we establish iteration bounds for termination of discrete linear loops via reduction to the continuous case.
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Julian D'Costa, Engel Lefaucheux, Joël Ouaknine, and James Worrell. How Fast Can You Escape a Compact Polytope?. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 49:1-49:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{dcosta_et_al:LIPIcs.STACS.2020.49,
author = {D'Costa, Julian and Lefaucheux, Engel and Ouaknine, Jo\"{e}l and Worrell, James},
title = {{How Fast Can You Escape a Compact Polytope?}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {49:1--49:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.49},
URN = {urn:nbn:de:0030-drops-119105},
doi = {10.4230/LIPIcs.STACS.2020.49},
annote = {Keywords: Continuous linear dynamical systems, termination}
}
Document
The SDP Value for Random Two-Eigenvalue CSPs
Abstract
We precisely determine the SDP value (equivalently, quantum value) of large random instances of certain kinds of constraint satisfaction problems, "two-eigenvalue 2CSPs". We show this SDP value coincides with the spectral relaxation value, possibly indicating a computational threshold. Our analysis extends the previously resolved cases of random regular 2XOR and NAE-3SAT, and includes new cases such as random Sort₄ (equivalently, CHSH) and Forrelation CSPs. Our techniques include new generalizations of the nonbacktracking operator, the Ihara-Bass Formula, and the Friedman/Bordenave proof of Alon’s Conjecture.
Cite as
Sidhanth Mohanty, Ryan O'Donnell, and Pedro Paredes. The SDP Value for Random Two-Eigenvalue CSPs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 50:1-50:45, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{mohanty_et_al:LIPIcs.STACS.2020.50,
author = {Mohanty, Sidhanth and O'Donnell, Ryan and Paredes, Pedro},
title = {{The SDP Value for Random Two-Eigenvalue CSPs}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {50:1--50:45},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.50},
URN = {urn:nbn:de:0030-drops-119110},
doi = {10.4230/LIPIcs.STACS.2020.50},
annote = {Keywords: Semidefinite programming, constraint satisfaction problems}
}
Document
Asymptotic Divergences and Strong Dichotomy
Abstract
The Schnorr-Stimm dichotomy theorem [Schnorr and Stimm, 1972] concerns finite-state gamblers that bet on infinite sequences of symbols taken from a finite alphabet Σ. The theorem asserts that, for any such sequence S, the following two things are true.
(1) If S is not normal in the sense of Borel (meaning that every two strings of equal length appear with equal asymptotic frequency in S), then there is a finite-state gambler that wins money at an infinitely-often exponential rate betting on S.
(2) If S is normal, then any finite-state gambler betting on S loses money at an exponential rate betting on S.
In this paper we use the Kullback-Leibler divergence to formulate the lower asymptotic divergence div(S||α) of a probability measure α on Σ from a sequence S over Σ and the upper asymptotic divergence Div(S||α) of α from S in such a way that a sequence S is α-normal (meaning that every string w has asymptotic frequency α(w) in S) if and only if Div(S||α)=0. We also use the Kullback-Leibler divergence to quantify the total risk Risk_G(w) that a finite-state gambler G takes when betting along a prefix w of S.
Our main theorem is a strong dichotomy theorem that uses the above notions to quantify the exponential rates of winning and losing on the two sides of the Schnorr-Stimm dichotomy theorem (with the latter routinely extended from normality to α-normality). Modulo asymptotic caveats in the paper, our strong dichotomy theorem says that the following two things hold for prefixes w of S.
(1') The infinitely-often exponential rate of winning in 1 is 2^{Div(S||α)|w|}.
(2') The exponential rate of loss in 2 is 2^{-Risk_G(w)}.
We also use (1') to show that 1-Div(S||α)/c, where c= log(1/ min_{a∈Σ} α(a)), is an upper bound on the finite-state α-dimension of S and prove the dual fact that 1-div(S||α)/c is an upper bound on the finite-state strong α-dimension of S.
Cite as
Xiang Huang, Jack H. Lutz, Elvira Mayordomo, and Donald M. Stull. Asymptotic Divergences and Strong Dichotomy. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 51:1-51:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{huang_et_al:LIPIcs.STACS.2020.51,
author = {Huang, Xiang and Lutz, Jack H. and Mayordomo, Elvira and Stull, Donald M.},
title = {{Asymptotic Divergences and Strong Dichotomy}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {51:1--51:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.51},
URN = {urn:nbn:de:0030-drops-119125},
doi = {10.4230/LIPIcs.STACS.2020.51},
annote = {Keywords: finite-state dimension, finite-state gambler, Kullback-Leibler divergence, normal sequences}
}
Document
Perfect Resolution of Conflict-Free Colouring of Interval Hypergraphs
Abstract
Given a hypergraph H, the conflict-free colouring problem is to colour vertices of H using minimum colours so that in every hyperedge e of H, there is a vertex whose colour is different from that of all other vertices in e. Our results are on a variant of the conflict-free colouring problem considered by Cheilaris et al.[Cheilaris et al., 2014], known as the 1-Strong Conflict-Free (1-SCF) colouring problem, for which they presented a polynomial time 2-approximation algorithm for interval hypergraphs. We show that an optimum 1-SCF colouring for interval hypergraphs can be computed in polynomial time. Our results are obtained by considering a different view of conflict-free colouring which we believe could be useful in general. For interval hypergraphs, this different view brings a connection to the theory of perfect graphs which is useful in coming up with an LP formulation to select the vertices that could be coloured to obtain an optimum conflict-free colouring. The perfect graph connection again plays a crucial role in finding a minimum colouring for the vertices selected by the LP formulation.
Cite as
S. M. Dhannya and N. S. Narayanaswamy. Perfect Resolution of Conflict-Free Colouring of Interval Hypergraphs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 52:1-52:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{dhannya_et_al:LIPIcs.STACS.2020.52,
author = {Dhannya, S. M. and Narayanaswamy, N. S.},
title = {{Perfect Resolution of Conflict-Free Colouring of Interval Hypergraphs}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {52:1--52:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.52},
URN = {urn:nbn:de:0030-drops-119138},
doi = {10.4230/LIPIcs.STACS.2020.52},
annote = {Keywords: Conflict-free Colouring, Interval Hypergraphs}
}
Document
Constant-Time Dynamic (Δ+1)-Coloring
Abstract
We give a fully dynamic (Las-Vegas style) algorithm with constant expected amortized time per update that maintains a proper (Δ+1)-vertex coloring of a graph with maximum degree at most Δ. This improves upon the previous O(log Δ)-time algorithm by Bhattacharya et al. (SODA 2018). Our algorithm uses an approach based on assigning random ranks to vertices and does not need to maintain a hierarchical graph decomposition. We show that our result does not only have optimal running time, but is also optimal in the sense that already deciding whether a Δ-coloring exists in a dynamically changing graph with maximum degree at most Δ takes Ω(log n) time per operation.
Cite as
Monika Henzinger and Pan Peng. Constant-Time Dynamic (Δ+1)-Coloring. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 53:1-53:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{henzinger_et_al:LIPIcs.STACS.2020.53,
author = {Henzinger, Monika and Peng, Pan},
title = {{Constant-Time Dynamic (\Delta+1)-Coloring}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {53:1--53:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.53},
URN = {urn:nbn:de:0030-drops-119145},
doi = {10.4230/LIPIcs.STACS.2020.53},
annote = {Keywords: Dynamic graph algorithms, Graph coloring, Random sampling}
}
Document
Cryptocurrency Mining Games with Economic Discount and Decreasing Rewards
Abstract
In the consensus protocols used in most cryptocurrencies, participants called miners must find valid blocks of transactions and append them to a shared tree-like data structure. Ideally, the rules of the protocol should ensure that miners maximize their gains if they follow a default strategy, which consists on appending blocks only to the longest branch of the tree, called the blockchain. Our goal is to understand under which circumstances are miners encouraged to follow the default strategy. Unfortunately, most of the existing models work with simplified payoff functions, without considering the possibility that rewards decrease over time because of the game rules (like in Bitcoin), nor integrating the fact that a miner naturally prefers to be paid earlier than later (the economic concept of discount). In order to integrate these factors, we consider a more general model where issues such as economic discount and decreasing rewards can be set as parameters of an infinite stochastic game. In this model, we study the limit situation in which a miner does not receive a full reward for a block if it stops being in the blockchain. We show that if rewards are not decreasing, then miners do not have incentives to create new branches, no matter how high their computational power is. On the other hand, when working with decreasing rewards similar to those in Bitcoin, we show that miners have an incentive to create such branches. Nevertheless, this incentive only occurs when a miner controls a proportion of the computational power which is close to half of the computational power of the entire network.
Cite as
Marcelo Arenas, Juan Reutter, Etienne Toussaint, Martín Ugarte, Francisco Vial, and Domagoj Vrgoč. Cryptocurrency Mining Games with Economic Discount and Decreasing Rewards. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{arenas_et_al:LIPIcs.STACS.2020.54,
author = {Arenas, Marcelo and Reutter, Juan and Toussaint, Etienne and Ugarte, Mart{\'\i}n and Vial, Francisco and Vrgo\v{c}, Domagoj},
title = {{Cryptocurrency Mining Games with Economic Discount and Decreasing Rewards}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {54:1--54:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.54},
URN = {urn:nbn:de:0030-drops-119150},
doi = {10.4230/LIPIcs.STACS.2020.54},
annote = {Keywords: cryptocurrency, game theory, cryptomining, economic discount, decreasing rewards}
}
Document
Randomness and Initial Segment Complexity for Probability Measures
Abstract
We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure μ on the space of infinite bit sequences is Martin-Löf absolutely continuous if the non-Martin-Löf random bit sequences form a null set with respect to μ. We think of this as a weak randomness notion for measures. We begin with examples, and a robustness property related to Solovay tests. Our main work connects our property to the growth of the initial segment complexity for measures μ; the latter is defined as a μ-average over the complexity of strings of the same length. We show that a maximal growth implies our weak randomness property, but also that both implications of the Levin-Schnorr theorem fail. We briefly discuss K-triviality for measures, which means that the growth of initial segment complexity is as slow as possible. We show that full Martin-Löf randomness of a measure implies Martin-Löf absolute continuity; the converse fails because only the latter property is compatible with having atoms. In a final section we consider weak randomness relative to a general ergodic computable measure. We seek appropriate effective versions of the Shannon-McMillan-Breiman theorem and the Brudno theorem where the bit sequences are replaced by measures.
Cite as
André Nies and Frank Stephan. Randomness and Initial Segment Complexity for Probability Measures. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 55:1-55:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{nies_et_al:LIPIcs.STACS.2020.55,
author = {Nies, Andr\'{e} and Stephan, Frank},
title = {{Randomness and Initial Segment Complexity for Probability Measures}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {55:1--55:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.55},
URN = {urn:nbn:de:0030-drops-119168},
doi = {10.4230/LIPIcs.STACS.2020.55},
annote = {Keywords: algorithmic randomness, probability measure on Cantor space, Kolmogorov complexity, statistical superposition, quantum states}
}
Document
Computing Shrub-Depth Decompositions
Abstract
Shrub-depth is a width measure of graphs which, roughly speaking, corresponds to the smallest depth of a tree into which a graph can be encoded. It can be thought of as a low-depth variant of clique-width (or rank-width), similarly as treedepth is a low-depth variant of treewidth. We present an fpt algorithm for computing decompositions of graphs of bounded shrub-depth. To the best of our knowledge, this is the first algorithm which computes the decomposition directly, without use of rank-width decompositions and FO or MSO logic.
Cite as
Jakub Gajarský and Stephan Kreutzer. Computing Shrub-Depth Decompositions. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 56:1-56:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{gajarsky_et_al:LIPIcs.STACS.2020.56,
author = {Gajarsk\'{y}, Jakub and Kreutzer, Stephan},
title = {{Computing Shrub-Depth Decompositions}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {56:1--56:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.56},
URN = {urn:nbn:de:0030-drops-119177},
doi = {10.4230/LIPIcs.STACS.2020.56},
annote = {Keywords: shrub-depth, tree-model, decomposition, fixed-parameter tractability}
}
Document
Typical Sequences Revisited - Computing Width Parameters of Graphs
Abstract
In this work, we give a structural lemma on merges of typical sequences, a notion that was introduced in 1991 [Lagergren and Arnborg, Bodlaender and Kloks, both ICALP 1991] to obtain constructive linear time parameterized algorithms for treewidth and pathwidth. The lemma addresses a runtime bottleneck in those algorithms but so far it does not lead to asymptotically faster algorithms. However, we apply the lemma to show that the cutwidth and the modified cutwidth of series parallel digraphs can be computed in ?(n²) time.
Cite as
Hans L. Bodlaender, Lars Jaffke, and Jan Arne Telle. Typical Sequences Revisited - Computing Width Parameters of Graphs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 57:1-57:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bodlaender_et_al:LIPIcs.STACS.2020.57,
author = {Bodlaender, Hans L. and Jaffke, Lars and Telle, Jan Arne},
title = {{Typical Sequences Revisited - Computing Width Parameters of Graphs}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {57:1--57:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.57},
URN = {urn:nbn:de:0030-drops-119189},
doi = {10.4230/LIPIcs.STACS.2020.57},
annote = {Keywords: typical sequences, treewidth, series parallel digraphs, cutwidth, modified cutwidth}
}
Document
Grundy Coloring & Friends, Half-Graphs, Bicliques
Abstract
The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order σ, the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering σ, i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force f(k)n^{2^{k-1}}-time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where k is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on k in the exponent of n can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative.
The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b-Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS '17]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on K_{t,t}-free graphs for b-Chromatic Core and Partial Grundy Coloring, making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest.
Cite as
Pierre Aboulker, Édouard Bonnet, Eun Jung Kim, and Florian Sikora. Grundy Coloring & Friends, Half-Graphs, Bicliques. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 58:1-58:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{aboulker_et_al:LIPIcs.STACS.2020.58,
author = {Aboulker, Pierre and Bonnet, \'{E}douard and Kim, Eun Jung and Sikora, Florian},
title = {{Grundy Coloring \& Friends, Half-Graphs, Bicliques}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {58:1--58:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.58},
URN = {urn:nbn:de:0030-drops-119190},
doi = {10.4230/LIPIcs.STACS.2020.58},
annote = {Keywords: Grundy coloring, parameterized complexity, ETH lower bounds, K\underline\{t,t\}-free graphs, half-graphs}
}
Document
Lower Bounds Against Sparse Symmetric Functions of ACC Circuits: Expanding the Reach of #SAT Algorithms
Abstract
We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in Quasi-NP = NTIME[n^{(log n)^O(1)}] and NEXP do not have small circuits (in the worst case and/or on average) from various circuit classes C, by showing that C admits non-trivial satisfiability and/or #SAT algorithms which beat exhaustive search by a minor amount.
In this paper, we present a new strong lower bound consequence of non-trivial #SAT algorithm for a circuit class {C}. Say a symmetric Boolean function f(x₁,…,x_n) is sparse if it outputs 1 on O(1) values of ∑_i x_i. We show that for every sparse f, and for all "typical" C, faster #SAT algorithms for C circuits actually imply lower bounds against the circuit class f ∘ C, which may be stronger than C itself. In particular:
- #SAT algorithms for n^k-size C-circuits running in 2ⁿ/n^k time (for all k) imply NEXP does not have f ∘ C-circuits of polynomial size.
- #SAT algorithms for 2^{n^ε}-size C-circuits running in 2^{n-n^ε} time (for some ε > 0) imply Quasi-NP does not have f ∘ C-circuits of polynomial size. Applying #SAT algorithms from the literature, one immediate corollary of our results is that Quasi-NP does not have EMAJ ∘ ACC⁰ ∘ THR circuits of polynomial size, where EMAJ is the "exact majority" function, improving previous lower bounds against ACC⁰ [Williams JACM'14] and ACC⁰ ∘ THR [Williams STOC'14], [Murray-Williams STOC'18]. This is the first nontrivial lower bound against such a circuit class.
Cite as
Nikhil Vyas and R. Ryan Williams. Lower Bounds Against Sparse Symmetric Functions of ACC Circuits: Expanding the Reach of #SAT Algorithms. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 59:1-59:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{vyas_et_al:LIPIcs.STACS.2020.59,
author = {Vyas, Nikhil and Williams, R. Ryan},
title = {{Lower Bounds Against Sparse Symmetric Functions of ACC Circuits: Expanding the Reach of #SAT Algorithms}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {59:1--59:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.59},
URN = {urn:nbn:de:0030-drops-119200},
doi = {10.4230/LIPIcs.STACS.2020.59},
annote = {Keywords: #SAT, satisfiability, circuit complexity, exact majority, ACC}
}
Document
Reversible Pebble Games and the Relation Between Tree-Like and General Resolution Space
Abstract
We show a new connection between the space measure in tree-like resolution and the reversible pebble game in graphs. Using this connection, we provide several formula classes for which there is a logarithmic factor separation between the space complexity measure in tree-like and general resolution. We show that these separations are not far from optimal by proving upper bounds for tree-like resolution space in terms of general resolution clause and variable space. In particular we show that for any formula F, its tree-like resolution space is upper bounded by space(π)log(time(π)), where π is any general resolution refutation of F. This holds considering as space(π) the clause space of the refutation as well as considering its variable space. For the concrete case of Tseitin formulas, we are able to improve this bound to the optimal bound space(π)log n, where n is the number of vertices of the corresponding graph.
Cite as
Jacobo Torán and Florian Wörz. Reversible Pebble Games and the Relation Between Tree-Like and General Resolution Space. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 60:1-60:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{toran_et_al:LIPIcs.STACS.2020.60,
author = {Tor\'{a}n, Jacobo and W\"{o}rz, Florian},
title = {{Reversible Pebble Games and the Relation Between Tree-Like and General Resolution Space}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {60:1--60:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-140-5},
ISSN = {1868-8969},
year = {2020},
volume = {154},
editor = {Paul, Christophe and Bl\"{a}ser, Markus},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.60},
URN = {urn:nbn:de:0030-drops-119213},
doi = {10.4230/LIPIcs.STACS.2020.60},
annote = {Keywords: Proof Complexity, Resolution, Tree-like Resolution, Pebble Game, Reversible Pebbling, Prover-Delayer Game, Raz - McKenzie Game, Clause Space, Variable Space}
}