Abstract
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.
Highlights
What kind of inherent mathematical structure makes a computational problem tractable, i.e., polynomial time solvable? Finding a general answer to this question is one of the fundamental goals of theoretical computer science
We introduce a new class of problems that can be viewed as algebraic versions of the (Gap) Label Cover problem and show that every promise CSP (PCSP) with a fixed constraint language is equivalent to a problem of this form
We prove below that having polymorphisms satisfying this condition is the only obstacle for a reduction from approximate hypergraph colouring to a given PCSP template
Summary
What kind of inherent mathematical structure makes a computational problem tractable, i.e., polynomial time solvable (assuming P NP)? Finding a general answer to this question is one of the fundamental goals of theoretical computer science. Many computational problems, including various versions of logical satisfiability, graph colouring, and systems of equations can be represented in this form [48, 70] It is well-known [48] that the basic CSP can be cast as a homomorphism problem from one relational structure to another (the latter is often called a template), and we will use this view. Systems of minor identities satisfied by polymorphisms provide a useful measure of how much symmetry a problem has This measure gives a new tool to compare/relate the complexity of PCSPs far beyond what was available before. There is an obvious question whether PCSPs exhibit a dichotomy as CSPs do, but there is not enough evidence yet to conjecture an answer It is not clear whether there is any PCSP whose polymorphisms are not limited enough (in terms of satisfying systems of minor identities) to give NP-hardness, and not strong enough to ensure tractability.
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