Constraint Satisfaction Problems Solvable by Local Consistency Methods

Abstract

We prove that constraint satisfaction problems without the ability to count are solvable by the local consistency checking algorithm. This settles three (equivalent) conjectures: Feder--Vardi [SICOMP’98], Bulatov [LICS’04] and Larose--Zádori [AU’07].

The collapse of the bounded width hierarchy

“…This theorem was proved in [4] under an additional assumption that P is simple. Such a weaker version would be sufficient for our purposes.…”

“…From today's perspective, the short, slightly imprecise, answer is that "simulates" simply means "positively primitively interprets". 4 Positive primitive interpretability is closely related to central objects of interest in universal algebra -varieties and Mal'tsev conditions. This is the basis for the success of the algebraic approach to the CSP.…”

“…Those 1-minimal instances which satisfy (P2) and (P3) and contain only binary constraints, at most one for each pair of variables, were called weak Prague instances in [4] and an analogue of Theorem 6.4 was proved in that paper. We do not know whether conditions (P2) and (P3) are sufficient in general.…”

“…No language which can, in some sense, simulate q-LIN has thus bounded relational width [18]. In [1,4], it was proved that this is the only obstacle in case of finite languages. The parameters (k, l) depend there on the maximum arity of a relation in Γ.…”

A finer reduction of constraint problems to digraphs

“…Barto, Kozik and Niven [BKN09] extended Hell and Nešetřil's result [HN90] on simple graphs to constraint languages consisting of a finite digraph with no sources and no sinks. Barto and Kozik [BK14] gave a complete algebraic description of the constraint languages over finite domains that are solvable by local consistency methods (these problems are said to be of bounded width) and as a consequence it is decidable to determine whether a constraint language can be solved by such methods.…”