An asymptotically optimal linear-time algorithm for locally consistent constraint satisfaction problems

Abstract

An instance of a constraint satisfaction problem is l-consistent if any l constraints of it can be simultaneously satisfied. For a set ∏ of constraint types, ρl(∏) denotes the largest ratio of constraints which can be satisfied in any l-consistent instance composed by constraints from the set ∏. We study the asymptotic behavior of ρl (∏) for sets ∏ consisting of Boolean predicates. The value ρ∞(∏) := liml→∞ ρl(∏) is determined for all such sets ∏. Moreover, we design a robust deterministic algorithm (for a fixed set ∏ of predicates) running in time linear in the size of the input and 1/ε which finds either an inconsistent set of constraints (of size bounded by the function of ε) or a truth assignment which satisfies the fraction of at least ρ∞(∏) - ε of the given constraints. Most of our results hold for both the unweighted and weighted versions of the problem. © Springer-Verlag Berlin Heidelberg 2005.