Algebraic approach to promise constraint satisfaction

Abstract

The complexity and approximability of the constraint satisfaction problem (CSP) has been actively studied over the last 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 paper 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 improving the state-of-the-art in approximate graph colouring: we show that, for any k ≥ 3, it is NP-hard to find a (2k − 1)-colouring of a given k-colourable graph. CCS CONCEPTS• Theory of computation → Problems, reductions and completeness.

Promises Make Finite (Constraint Satisfaction) Problems Infinitary

“…Fortunately, building on the initial insights and results in [1,9,11,10], it was observed in [14] (among many other important results such as the NPhardness of 5-coloring a 3-colorable graph) that the basics of the CSP theory from [4] generalize to PCSPs. In particular, the notions of pp-constructions and polymorphisms have their PCSP counterparts and the connection between relational and algebraic structures works just as well as in the CSP.…”

“…In this section we give formal definitions of the concepts essential for the proof. For an in depth introduction to CSP and PCSP, see [3,14] and references therein.…”

“…The proof is similar to the analogous proof for CSP (see [3]). An interesting alternative way for PCSP was given (but explicitly proved only for finite templates) in [14].…”

“…For example, we want to find a 100-coloring of a 3-colorable graph, or we want to find a valid NAE-3-SAT assignment to a 1-in-3satisfiable instance. The complexity of the former problem is a notorious open question (for a recent development see [9,14], but even 6-coloring a 3-colorable graph is not known to be NP-complete). On the other hand, the latter problem admits an elegant polynomial time algorithm [11,10], which we now describe.…”

Algebraic Theory of Promise Constraint Satisfaction Problems, First Steps

“…It is conjectured that PCSP(K k , K l ) is NP-hard for every k < l, but this conjecture was confirmed only in special cases: for l ≤ 2k − 2 [12] (e.g., 4-coloring a 3-colorable graph) and for a large enough k and l ≤ 2 Ω(k 1/3 ) [23]. The algebraic development discussed in the next subsection helped in improving the former result to l ≤ 2k − 1 [18] (e.g., 5-coloring a 3-colorable graph). Example 3.3.…”