On the Scope of the Universal-Algebraic Approach to Constraint Satisfaction

Abstract Abductive reasoning is a non-monotonic formalism stemming from the work of Peirce. It describes the process of deriving the most plausible explanations of known facts. Considering the positive version, asking for sets of variables as explanations, we study, besides the problem of wether there exists a set of explanations, two explanation size limited variants of this reasoning problem (less than or equal to, and equal to a given size bound). In this paper, we present a thorough two-dimensional classification of these problems: the first dimension is regarding the parameterized complexity under a wealth of different parameterizations, and the second dimension spans through all possible Boolean fragments of these problems in Schaefer’s constraint satisfaction framework with co-clones (T. J. Schaefer. The complexity of satisfiability problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing, May 1–3, 1978, San Diego, California, USA, R.J. Lipton, W.A. Burkhard, W.J. Savitch, E.P. Friedman, A.V. Aho eds, pp. 216–226. ACM, 1978). Thereby, we almost complete the parameterized complexity classification program initiated by Fellows et al. (The parameterized complexity of abduction. In Proceedings of the Twenty-Sixth AAAI Conference on Articial Intelligence, July 22–26, 2012, Toronto, Ontario, Canada, J. Homann, B. Selman eds. AAAI Press, 2012), partially building on the results by Nordh and Zanuttini (What makes propositional abduction tractable. Artificial Intelligence, 172, 1245–1284, 2008). In this process, we outline a fine-grained analysis of the inherent parameterized intractability of these problems and pinpoint their FPT parts. As the standard algebraic approach is not applicable to our problems, we develop an alternative method that makes the algebraic tools partially available again.

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Algebraic Approach to Promise Constraint Satisfaction

Journal of the ACM ◽

10.1145/3457606 ◽

2021 ◽

Vol 68 (4) ◽

pp. 1-66

Author(s):

Libor Barto ◽

Jakub Bulín ◽

Andrei Krokhin ◽

Jakub Opršal

Keyword(s):

Constraint Satisfaction ◽

Constraint Satisfaction Problem ◽

Algebraic Approach ◽

Graph Colouring ◽

Finite Domain ◽

New Class ◽

The Past ◽

Open Questions ◽

Constraint Language ◽

Special Cases

The complexity and approximability of the constraint satisfaction problem (CSP) has been actively studied over the past 20 years. A new version of the CSP, the promise CSP (PCSP), has recently been proposed, motivated by open questions about the approximability of variants of satisfiability and graph colouring. The PCSP significantly extends the standard decision CSP. The complexity of CSPs with a fixed constraint language on a finite domain has recently been fully classified, greatly guided by the algebraic approach, which uses polymorphisms—high-dimensional symmetries of solution spaces—to analyse the complexity of problems. The corresponding classification for PCSPs is wide open and includes some long-standing open questions, such as the complexity of approximate graph colouring, as special cases. The basic algebraic approach to PCSP was initiated by Brakensiek and Guruswami, and in this article, we significantly extend it and lift it from concrete properties of polymorphisms to their abstract properties. We introduce a new class of problems that can be viewed as algebraic versions of the (Gap) Label Cover problem and show that every PCSP with a fixed constraint language is equivalent to a problem of this form. This allows us to identify a “measure of symmetry” that is well suited for comparing and relating the complexity of different PCSPs via the algebraic approach. We demonstrate how our theory can be applied by giving both general and specific hardness/tractability results. Among other things, we improve the state-of-the-art in approximate graph colouring by showing that, for any k ≥ 3, it is NP-hard to find a (2 k -1)-colouring of a given k -colourable graph.

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Algebraic approach to promise constraint satisfaction

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing - STOC 2019 ◽

10.1145/3313276.3316300 ◽

2019 ◽

Cited By ~ 6

Author(s):

Jakub Bulín ◽

Andrei Krokhin ◽

Jakub Opršal

Keyword(s):

Constraint Satisfaction ◽

Algebraic Approach

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The Complexity of Promise SAT on Non-Boolean Domains

ACM Transactions on Computation Theory ◽

10.1145/3470867 ◽

2021 ◽

Vol 13 (4) ◽

pp. 1-20

Author(s):

Alex Brandts ◽

Marcin Wrochna ◽

Stanislav Živný

Keyword(s):

Polynomial Time ◽

Constraint Satisfaction ◽

Algebraic Approach ◽

Constraint Satisfaction Problems ◽

Hardness Of Approximation ◽

Np Hard ◽

Cover Problem ◽

Finite Domains ◽

Np Hardness ◽

Cnf Formulas

While 3-SAT is NP-hard, 2-SAT is solvable in polynomial time. Austrin et al. [SICOMP’17] proved a result known as “(2+ɛ)-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. The hardness side is proved using the algebraic approach via a new general NP-hardness criterion on polymorphisms, which is based on a gap version of the Layered Label Cover problem. We show that previously used criteria are insufficient—the problem hence gives an interesting benchmark of algebraic techniques for proving hardness of approximation in problems such as PCSPs.

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