Robustly Solvable Constraint Satisfaction Problems

An algorithm for a constraint satisfaction problem is called robust if it outputs an assignment satisfying at least (1 − g(ε))-fraction of the constraints given a (1 − ε)-satisfiable instance, where g(ε) → 0 as ε → 0. Guruswami and Zhou conjectured a characterization of constraint languages for which the corresponding constraint satisfaction problem admits an efficient robust algorithm. This paper confirms their conjecture. 1. Introduction. The constraint satisfaction problem (CSP) provides a common framework for many theoretical problems in computer science as well as for many applications. An instance of the CSP consists of variables and constraints imposed on them and the goal is to find (or decide whether it exists) an assignment of variables which is " best " for given constraints. In the decision problem for CSP we want to decide if there is an assignment satisfying all the constraints. In Max-CSP we wish to find an assignment satisfying maximal number of constraints. In the approximation version of Max-CSP we seek an assignment which is, in some sense, close to the optimal one. This paper deals with a special case of approximation: robust solv-ability of the CSP. Given an instance which is almost satisfiable (say (1 − ε)-fraction of the constraint can be satisfied), the goal is to efficiently find an almost satisfying assignment (which satisfies at least (1 − g(ε))-fraction of the constraints, where the error function g satisfies lim ε→0 g(ε) = 0). Most of the computational problems connected to CSP are hard in general. Therefore , when developing algorithms, one usually restricts the set of allowed instances. Most often the instances are restricted in two ways: one restricts the way in which the variables are constrained (e.g. the shape of the hypergraph of constrained variables), or restricts the allowed constraint relations (defining constraint language). In this paper we use the second approach, i.e. all constraint relations must come from a fixed, finite set of relations on a domain. Robust solvability for a fixed constraint language was first studied in a paper by Zwick [30]. The motivation behind this approach was that, in certain practical situations, instances might be close to satisfiable – for example, a small fraction of constraints might have been corrupted by noise. An algorithm that is able to satisfy, in such a case, most of the constraints could be useful. Zwick [30] concentrated on Boolean CSPs. He designed a semidefinite programming …