33 Search Results for "Zivný, Stanislav"

Volume

Dagstuhl Follow-Ups, Volume 7

The Constraint Satisfaction Problem: Complexity and Approximability

Editors: Andrei Krokhin and Stanislav Zivny

Document
DOI: 10.4230/LIPIcs.ISAAC.2023.57
A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees

Authors: Shuai Shao and Stanislav Živný

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Abstract
General factors are a generalization of matchings. Given a graph G with a set π(v) of feasible degrees, called a degree constraint, for each vertex v of G, the general factor problem is to find a (spanning) subgraph F of G such that deg_F(v) ∈ π(v) for every v of G. When all degree constraints are symmetric Δ-matroids, the problem is solvable in polynomial time. The weighted general factor problem is to find a general factor of the maximum total weight in an edge-weighted graph. Strongly polynomial-time algorithms are only known for weighted general factor problems that are reducible to the weighted matching problem by gadget constructions. In this paper, we present a strongly polynomial-time algorithm for a type of weighted general factor problems with real-valued edge weights that is provably not reducible to the weighted matching problem by gadget constructions. As an application, we obtain a strongly polynomial-time algorithm for the terminal backup problem by reducing it to the weighted general factor problem.
Cite as

Shuai Shao and Stanislav Živný. A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 57:1-57:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{shao_et_al:LIPIcs.ISAAC.2023.57,
  author =	{Shao, Shuai and \v{Z}ivn\'{y}, Stanislav},
  title =	{{A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{57:1--57:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.57},
  URN =		{urn:nbn:de:0030-drops-193597},
  doi =		{10.4230/LIPIcs.ISAAC.2023.57},
  annote =	{Keywords: matchings, factors, edge constraint satisfaction problems, terminal backup problem, delta matroids}
}
Document
DOI: 10.4230/DagRep.12.5.112
The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 22201)

Authors: Martin Grohe, Venkatesan Guruswami, Dániel Marx, and Stanislav Živný

Published in: Dagstuhl Reports, Volume 12, Issue 5 (2022)

Abstract
Constraint satisfaction has always played a central role in computational complexity theory; appropriate versions of CSPs are classical complete problems for most standard complexity classes. CSPs constitute a very rich and yet sufficiently manageable class of problems to give a good perspective on general computational phenomena. For instance, they help to understand which mathematical properties make a computational problem tractable (in a wide sense, e.g., polynomial-time solvable, non-trivially approximable, fixed-parameter tractable, or definable in a weak logic). In the last 15 years, research activity in this area has significantly intensified and hugely impressive progress was made. The Dagstuhl Seminar 22201 "The Constraint Satisfaction Problem: Complexity and Approximability" was aimed at bringing together researchers using all the different techniques in the study of the CSP so that they can share their insights obtained during the past four years. This report documents the material presented during the course of the seminar.
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Martin Grohe, Venkatesan Guruswami, Dániel Marx, and Stanislav Živný. The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 22201). In Dagstuhl Reports, Volume 12, Issue 5, pp. 112-130, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@Article{grohe_et_al:DagRep.12.5.112,
  author =	{Grohe, Martin and Guruswami, Venkatesan and Marx, D\'{a}niel and \v{Z}ivn\'{y}, Stanislav},
  title =	{{The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 22201)}},
  pages =	{112--130},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2022},
  volume =	{12},
  number =	{5},
  editor =	{Grohe, Martin and Guruswami, Venkatesan and Marx, D\'{a}niel and \v{Z}ivn\'{y}, Stanislav},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.12.5.112},
  URN =		{urn:nbn:de:0030-drops-174453},
  doi =		{10.4230/DagRep.12.5.112},
  annote =	{Keywords: Constraint satisfaction problem (CSP); Computational complexity; Hardness of approximation; Universal algebra; Semidefinite programming}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2022.128
Linearly Ordered Colourings of Hypergraphs

Authors: Tamio-Vesa Nakajima and Stanislav Živný

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
A linearly ordered (LO) k-colouring of an r-uniform hypergraph assigns an integer from {1, …, k} to every vertex so that, in every edge, the (multi)set of colours has a unique maximum. Equivalently, for r = 3, if two vertices in an edge are assigned the same colour, then the third vertex is assigned a larger colour (as opposed to a different colour, as in classic non-monochromatic colouring). Barto, Battistelli, and Berg [STACS'21] studied LO colourings on 3-uniform hypergraphs in the context of promise constraint satisfaction problems (PCSPs). We show two results. First, given a 3-uniform hypergraph that admits an LO 2-colouring, one can find in polynomial time an LO k-colouring with k = O(√{nlog log n}/log n), where n is the number of vertices of the input hypergraph. This is established by building on ideas from algorithms designed for approximate graph colourings. Second, given an r-uniform hypergraph that admits an LO 2-colouring, we establish NP-hardness of finding an LO 3-colouring for every constant uniformity r ≥ 5. In fact, we determine the precise relationship of polymorphism minions for all uniformities r ≥ 3, which reveals a key difference between r = 3,4 and r ≥ 5 and which may be of independent interest. Using the algebraic approach to PCSPs, we actually show a more general result establishing NP-hardness of finding an LO (k+1)-colouring for LO k-colourable r-uniform hypergraphs for k ≥ 2 and r ≥ 5.
Cite as

Tamio-Vesa Nakajima and Stanislav Živný. Linearly Ordered Colourings of Hypergraphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 128:1-128:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{nakajima_et_al:LIPIcs.ICALP.2022.128,
  author =	{Nakajima, Tamio-Vesa and \v{Z}ivn\'{y}, Stanislav},
  title =	{{Linearly Ordered Colourings of Hypergraphs}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{128:1--128:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.128},
  URN =		{urn:nbn:de:0030-drops-164692},
  doi =		{10.4230/LIPIcs.ICALP.2022.128},
  annote =	{Keywords: hypegraph colourings, promise constraint satisfaction, PCSP, polymorphisms, minions, algebraic approach}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.58
QCSP on Reflexive Tournaments

Authors: Benoît Larose, Petar Marković, Barnaby Martin, Daniël Paulusma, Siani Smith, and Stanislav Živný

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract
We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem QCSP(H) when H is a reflexive tournament. It is well-known that reflexive tournaments can be split into a sequence of strongly connected components H₁,…,H_n so that there exists an edge from every vertex of H_i to every vertex of H_j if and only if i < j. We prove that if H has both its initial and final strongly connected component (possibly equal) of size 1, then QCSP(H) is in NL and otherwise QCSP(H) is NP-hard.
Cite as

Benoît Larose, Petar Marković, Barnaby Martin, Daniël Paulusma, Siani Smith, and Stanislav Živný. QCSP on Reflexive Tournaments. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 58:1-58:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{larose_et_al:LIPIcs.ESA.2021.58,
  author =	{Larose, Beno\^{i}t and Markovi\'{c}, Petar and Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani and \v{Z}ivn\'{y}, Stanislav},
  title =	{{QCSP on Reflexive Tournaments}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{58:1--58:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.58},
  URN =		{urn:nbn:de:0030-drops-146392},
  doi =		{10.4230/LIPIcs.ESA.2021.58},
  annote =	{Keywords: computational complexity, algorithmic graph theory, quantified constraints, universal algebra, constraint satisfaction}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.75
Additive Sparsification of CSPs

Authors: Eden Pelleg and Stanislav Živný

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract
Multiplicative cut sparsifiers, introduced by Benczúr and Karger [STOC'96], have proved extremely influential and found various applications. Precise characterisations were established for sparsifiability of graphs with other 2-variable predicates on Boolean domains by Filtser and Krauthgamer [SIDMA'17] and non-Boolean domains by Butti and Živný [SIDMA'20]. Bansal, Svensson and Trevisan [FOCS'19] introduced a weaker notion of sparsification termed "additive sparsification", which does not require weights on the edges of the graph. In particular, Bansal et al. designed algorithms for additive sparsifiers for cuts in graphs and hypergraphs. As our main result, we establish that all Boolean Constraint Satisfaction Problems (CSPs) admit an additive sparsifier; that is, for every Boolean predicate P:{0,1}^k → {0,1} of a fixed arity k, we show that CSP(P)} admits an additive sparsifier. Under our newly introduced notion of all-but-one sparsification for non-Boolean predicates, we show that CSP(P)} admits an additive sparsifier for any predicate P:D^k → {0,1} of a fixed arity k on an arbitrary finite domain D.
Cite as

Eden Pelleg and Stanislav Živný. Additive Sparsification of CSPs. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 75:1-75:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{pelleg_et_al:LIPIcs.ESA.2021.75,
  author =	{Pelleg, Eden and \v{Z}ivn\'{y}, Stanislav},
  title =	{{Additive Sparsification of CSPs}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{75:1--75:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.75},
  URN =		{urn:nbn:de:0030-drops-146562},
  doi =		{10.4230/LIPIcs.ESA.2021.75},
  annote =	{Keywords: additive sparsification, graphs, hypergraphs, minimum cuts}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.37
Conditional Dichotomy of Boolean Ordered Promise CSPs

Authors: Joshua Brakensiek, Venkatesan Guruswami, and Sai Sandeep

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
Promise Constraint Satisfaction Problems (PCSPs) are a generalization of Constraint Satisfaction Problems (CSPs) where each predicate has a strong and a weak form and given a CSP instance, the objective is to distinguish if the strong form can be satisfied vs. even the weak form cannot be satisfied. Since their formal introduction by Austrin, Guruswami, and Håstad [Per Austrin et al., 2017], there has been a flurry of works on PCSPs, including recent breakthroughs in approximate graph coloring [Barto et al., 2018; Andrei A. Krokhin and Jakub Opršal, 2019; Marcin Wrochna and Stanislav Zivný, 2020]. The key tool in studying PCSPs is the algebraic framework developed in the context of CSPs where the closure properties of the satisfying solutions known as polymorphisms are analyzed. The polymorphisms of PCSPs are significantly richer than CSPs - even in the Boolean case, we still do not know if there exists a dichotomy result for PCSPs analogous to Schaefer’s dichotomy result [Thomas J. Schaefer, 1978] for CSPs. In this paper, we study a special case of Boolean PCSPs, namely Boolean Ordered PCSPs where the Boolean PCSPs have the predicate x ≤ y. In the algebraic framework, this is the special case of Boolean PCSPs when the polymorphisms are monotone functions. We prove that Boolean Ordered PCSPs exhibit a computational dichotomy assuming the Rich 2-to-1 Conjecture [Mark Braverman et al., 2021] which is a perfect completeness surrogate of the Unique Games Conjecture. In particular, assuming the Rich 2-to-1 Conjecture, we prove that a Boolean Ordered PCSP can be solved in polynomial time if for every ε > 0, it has polymorphisms where each coordinate has Shapley value at most ε, else it is NP-hard. The algorithmic part of our dichotomy result is based on a structural lemma showing that Boolean monotone functions with each coordinate having low Shapley value have arbitrarily large threshold functions as minors. The hardness part proceeds by showing that the Shapley value is consistent under a uniformly random 2-to-1 minor. As a structural result of independent interest, we construct an example to show that the Shapley value can be inconsistent under an adversarial 2-to-1 minor.
Cite as

Joshua Brakensiek, Venkatesan Guruswami, and Sai Sandeep. Conditional Dichotomy of Boolean Ordered Promise CSPs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 37:1-37:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{brakensiek_et_al:LIPIcs.ICALP.2021.37,
  author =	{Brakensiek, Joshua and Guruswami, Venkatesan and Sandeep, Sai},
  title =	{{Conditional Dichotomy of Boolean Ordered Promise CSPs}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{37:1--37:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.37},
  URN =		{urn:nbn:de:0030-drops-141060},
  doi =		{10.4230/LIPIcs.ICALP.2021.37},
  annote =	{Keywords: promise constraint satisfaction, Boolean ordered PCSP, Shapley value, rich 2-to-1 conjecture, random minor}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2021.121
Beyond PCSP(1-in-3, NAE)

Authors: Alex Brandts and Stanislav Živný

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
The promise constraint satisfaction problem (PCSP) is a recently introduced vast generalisation of the constraint satisfaction problem (CSP) that captures approximability of satisfiable instances. A PCSP instance comes with two forms of each constraint: a strict one and a weak one. Given the promise that a solution exists using the strict constraints, the task is to find a solution using the weak constraints. While there are by now several dichotomy results for fragments of PCSPs, they all consider (in some way) symmetric PCSPs. 1-in-3-SAT and Not-All-Equal-3-SAT are classic examples of Boolean symmetric (non-promise) CSPs. While both problems are NP-hard, Brakensiek and Guruswami showed [SODA'18] that given a satisfiable instance of 1-in-3-SAT one can find a solution to the corresponding instance of (weaker) Not-All-Equal-3-SAT. In other words, the PCSP template (𝟏-in-𝟑,NAE) is tractable. We focus on non-symmetric PCSPs. In particular, we study PCSP templates obtained from the Boolean template (𝐭-in-𝐤, NAE) by either adding tuples to 𝐭-in-𝐤 or removing tuples from NAE. For the former, we classify all templates as either tractable or not solvable by the currently strongest known algorithm for PCSPs, the combined basic LP and affine IP relaxation of Brakensiek and Guruswami [SODA'20]. For the latter, we classify all templates as either tractable or NP-hard.
Cite as

Alex Brandts and Stanislav Živný. Beyond PCSP(1-in-3, NAE). In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 121:1-121:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{brandts_et_al:LIPIcs.ICALP.2021.121,
  author =	{Brandts, Alex and \v{Z}ivn\'{y}, Stanislav},
  title =	{{Beyond PCSP(1-in-3, NAE)}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{121:1--121:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.121},
  URN =		{urn:nbn:de:0030-drops-141902},
  doi =		{10.4230/LIPIcs.ICALP.2021.121},
  annote =	{Keywords: promise constraint satisfaction, PCSP, polymorphisms, algebraic approach}
}
Document
PACE Solver Description
DOI: 10.4230/LIPIcs.IPEC.2020.36
PACE Solver Description: Sallow: A Heuristic Algorithm for Treedepth Decompositions

Authors: Marcin Wrochna

Published in: LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

Abstract
We describe a heuristic algorithm for computing treedepth decompositions, submitted for the https://pacechallenge.org/2020 challenge. It relies on a variety of greedy algorithms computing elimination orderings, as well as a Divide & Conquer approach on balanced cuts obtained using a from-scratch reimplementation of the 2016 FlowCutter algorithm by Hamann & Strasser [Michael Hamann and Ben Strasser, 2018].
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Marcin Wrochna. PACE Solver Description: Sallow: A Heuristic Algorithm for Treedepth Decompositions. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 36:1-36:4, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{wrochna:LIPIcs.IPEC.2020.36,
  author =	{Wrochna, Marcin},
  title =	{{PACE Solver Description: Sallow: A Heuristic Algorithm for Treedepth Decompositions}},
  booktitle =	{15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
  pages =	{36:1--36:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-172-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{180},
  editor =	{Cao, Yixin and Pilipczuk, Marcin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.36},
  URN =		{urn:nbn:de:0030-drops-133391},
  doi =		{10.4230/LIPIcs.IPEC.2020.36},
  annote =	{Keywords: treedepth, decomposition, heuristic, weak colouring numbers}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.85
The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains

Authors: Caterina Viola and Stanislav Živný

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract
Convex relaxations have been instrumental in solvability of constraint satisfaction problems (CSPs), as well as in the three different generalisations of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In this work, we extend an existing tractability result to the three generalisations of CSPs combined: We give a sufficient condition for the combined basic linear programming and affine integer programming relaxation for exact solvability of promise valued CSPs over infinite-domains. This extends a result of Brakensiek and Guruswami [SODA'20] for promise (non-valued) CSPs (on finite domains).
Cite as

Caterina Viola and Stanislav Živný. The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 85:1-85:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{viola_et_al:LIPIcs.MFCS.2020.85,
  author =	{Viola, Caterina and \v{Z}ivn\'{y}, Stanislav},
  title =	{{The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{85:1--85:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.85},
  URN =		{urn:nbn:de:0030-drops-127566},
  doi =		{10.4230/LIPIcs.MFCS.2020.85},
  annote =	{Keywords: promise constraint satisfaction, valued constraint satisfaction, convex relaxations, polymorphisms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.17
The Complexity of Promise SAT on Non-Boolean Domains

Authors: Alex Brandts, Marcin Wrochna, and Stanislav Živný

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
While 3-SAT is NP-hard, 2-SAT is solvable in polynomial time. Austrin, Guruswami, and Håstad [FOCS'14/SICOMP'17] proved a result known as "(2+ε)-SAT is NP-hard". They showed that the problem of distinguishing k-CNF formulas that are g-satisfiable (i.e. some assignment satisfies at least g literals in every clause) from those that are not even 1-satisfiable is NP-hard if g/k < 1/2 and is in P otherwise. We study a generalisation of SAT on arbitrary finite domains, with clauses that are disjunctions of unary constraints, and establish analogous behaviour. Thus we give a dichotomy for a natural fragment of promise constraint satisfaction problems (PCSPs) on arbitrary finite domains.
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Alex Brandts, Marcin Wrochna, and Stanislav Živný. The Complexity of Promise SAT on Non-Boolean Domains. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 17:1-17:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{brandts_et_al:LIPIcs.ICALP.2020.17,
  author =	{Brandts, Alex and Wrochna, Marcin and \v{Z}ivn\'{y}, Stanislav},
  title =	{{The Complexity of Promise SAT on Non-Boolean Domains}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{17:1--17:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.17},
  URN =		{urn:nbn:de:0030-drops-124241},
  doi =		{10.4230/LIPIcs.ICALP.2020.17},
  annote =	{Keywords: promise constraint satisfaction, PCSP, polymorphisms, algebraic approach, label cover}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2019.60
Approximate Counting CSP Seen from the Other Side

Authors: Andrei A. Bulatov and Stanislav Živný

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Abstract
In this paper we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP(C,-), in which the goal is, given a relational structure A from a class C of structures and an arbitrary structure B, to find the number of homomorphisms from A to B. Flum and Grohe showed that #CSP(C,-) is solvable in polynomial time if C has bounded treewidth [FOCS'02]. Building on the work of Grohe [JACM'07] on decision CSPs, Dalmau and Jonsson then showed that, if C is a recursively enumerable class of relational structures of bounded arity, then assuming FPT != #W[1], there are no other cases of #CSP(C,-) solvable exactly in polynomial time (or even fixed-parameter time) [TCS'04]. We show that, assuming FPT != W[1] (under randomised parametrised reductions) and for C satisfying certain general conditions, #CSP(C,-) is not solvable even approximately for C of unbounded treewidth; that is, there is no fixed parameter tractable (and thus also not fully polynomial) randomised approximation scheme for #CSP(C,-). In particular, our condition generalises the case when C is closed under taking minors.
Cite as

Andrei A. Bulatov and Stanislav Živný. Approximate Counting CSP Seen from the Other Side. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 60:1-60:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bulatov_et_al:LIPIcs.MFCS.2019.60,
  author =	{Bulatov, Andrei A. and \v{Z}ivn\'{y}, Stanislav},
  title =	{{Approximate Counting CSP Seen from the Other Side}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{60:1--60:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.60},
  URN =		{urn:nbn:de:0030-drops-110041},
  doi =		{10.4230/LIPIcs.MFCS.2019.60},
  annote =	{Keywords: constraint satisfaction, approximate counting, homomorphisms}
}
Document
DOI: 10.4230/LIPIcs.STACS.2019.17
Sparsification of Binary CSPs

Authors: Silvia Butti and Stanislav Živný

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract
A cut epsilon-sparsifier of a weighted graph G is a re-weighted subgraph of G of (quasi)linear size that preserves the size of all cuts up to a multiplicative factor of epsilon. Since their introduction by Benczúr and Karger [STOC'96], cut sparsifiers have proved extremely influential and found various applications. Going beyond cut sparsifiers, Filtser and Krauthgamer [SIDMA'17] gave a precise classification of which binary Boolean CSPs are sparsifiable. In this paper, we extend their result to binary CSPs on arbitrary finite domains.
Cite as

Silvia Butti and Stanislav Živný. Sparsification of Binary CSPs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 17:1-17:8, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{butti_et_al:LIPIcs.STACS.2019.17,
  author =	{Butti, Silvia and \v{Z}ivn\'{y}, Stanislav},
  title =	{{Sparsification of Binary CSPs}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{17:1--17:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.17},
  URN =		{urn:nbn:de:0030-drops-102564},
  doi =		{10.4230/LIPIcs.STACS.2019.17},
  annote =	{Keywords: constraint satisfaction problems, minimum cuts, sparsification}
}
Document
DOI: 10.4230/LIPIcs.STACS.2019.52
Beyond Boolean Surjective VCSPs

Authors: Gregor Matl and Stanislav Živný

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract
Fulla, Uppman, and Živný [ACM ToCT'18] established a dichotomy theorem for Boolean surjective general-valued constraint satisfaction problems (VCSPs), i.e., VCSPs on two-element domains in which both labels have to be used in a solution. This result, in addition to identifying the complexity frontier, features the discovery of a new non-trivial tractable case (called EDS) that does not appear in the non-surjective setting. In this work, we go beyond Boolean domains. As our main result, we introduce a generalisation of EDS to arbitrary finite domains called SEDS (similar to EDS) and establish a conditional complexity classification of SEDS VCSPs based on a reduction to smaller domains. This gives a complete classification of SEDS VCSPs on three-element domains. The basis of our tractability result is a natural generalisation of the Min-Cut problem, in which only solutions of certain size (given by a lower and upper bound) are permitted. We show that all near-optimal solutions to this problem can be enumerated in polynomial time, which might be of independent interest.
Cite as

Gregor Matl and Stanislav Živný. Beyond Boolean Surjective VCSPs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 52:1-52:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{matl_et_al:LIPIcs.STACS.2019.52,
  author =	{Matl, Gregor and \v{Z}ivn\'{y}, Stanislav},
  title =	{{Beyond Boolean Surjective VCSPs}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{52:1--52:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.52},
  URN =		{urn:nbn:de:0030-drops-102911},
  doi =		{10.4230/LIPIcs.STACS.2019.52},
  annote =	{Keywords: constraint satisfaction problems, valued constraint satisfaction, surjective constraint satisfaction, graph cuts}
}
Document
DOI: 10.4230/DagRep.8.6.1
The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 18231)

Authors: Martin Grohe, Venkatesan Guruswami, and Stanislav Zivny

Published in: Dagstuhl Reports, Volume 8, Issue 6 (2019)

Abstract
Constraint satisfaction has always played a central role in computational complexity theory; appropriate versions of CSPs are classical complete problems for most standard complexity classes. CSPs constitute a very rich and yet sufficiently manageable class of problems to give a good perspective on general computational phenomena. For instance, they help to understand which mathematical properties make a computational problem tractable (in a wide sense, e.g., polynomial-time solvable, non-trivially approximable, fixed-parameter tractable, or definable in a weak logic). In the last decade, research activity in this area has significantly intensified and hugely impressive progress was made. The Dagstuhl Seminar 18231 "The Constraint Satisfaction Problem: Complexity and Approximability" was aimed at bringing together researchers using all the different techniques in the study of the CSP so that they can share their insights obtained during the past three years. This report documents the material presented during the course of the seminar.
Cite as

Martin Grohe, Venkatesan Guruswami, and Stanislav Zivny. The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 18231). In Dagstuhl Reports, Volume 8, Issue 6, pp. 1-18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@Article{grohe_et_al:DagRep.8.6.1,
  author =	{Grohe, Martin and Guruswami, Venkatesan and Zivny, Stanislav},
  title =	{{The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 18231)}},
  pages =	{1--18},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2018},
  volume =	{8},
  number =	{6},
  editor =	{Grohe, Martin and Guruswami, Venkatesan and Zivny, Stanislav},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.8.6.1},
  URN =		{urn:nbn:de:0030-drops-100251},
  doi =		{10.4230/DagRep.8.6.1},
  annote =	{Keywords: Constraint satisfaction problem (CSP); Computational complexity; CSP dichotomy conjecture; Hardness of approximation; Unique games conjecture; Parameterised complexity; Descriptive complexity; Universal algebra; Logic; Semidefinite programming}
}
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