Hard constraint satisfaction problems have hard gaps at location 1

Abstract

An instance of Max CSP is a finite collection of constraints on a set of variables, and the goal is to assign values to the variables that maximises the number of satisfied constraints. Max CSP captures many well-known problems (such as Max k-SAT and Max Cut) and is consequently NP-hard. Thus, it is natural to study how restrictions on the allowed constraint types (or constraint languages) affect the complexity and approximability of Max CSP. The PCP theorem is equivalent to the existence of a constraint language for which Max CSP has a hard gap at location 1, i.e. it is NP-hard to distinguish between satisfiable instances and instances where at most some constant fraction of the constraints are satisfiable. All constraint languages, for which the CSP problem (i.e., the problem of deciding whether all constraints can be satisfied) is currently known to be NP-hard, have a certain algebraic property. We prove that any constraint language with this algebraic property makes Max CSP have a hard gap at location 1 which, in particular, implies that such problems cannot have a PTAS unless P = NP. We then apply this result to Max CSP restricted to a single constraint type; this class of problems contains, for instance, Max Cut and Max DiCut. Assuming P $\neq$ NP, we show that such problems do not admit PTAS except in some trivial cases. Our results hold even if the number of occurrences of each variable is bounded by a constant. We use these results to partially answer open questions and strengthen results by Engebretsen et al. [Theor. Comput. Sci., 312 (2004), pp. 17--45], Feder et al. [Discrete Math., 307 (2007), pp. 386--392], Krokhin and Larose [Proc. Principles and Practice of Constraint Programming (2005), pp. 388--402], and Jonsson and Krokhin [J. Comput. System Sci., 73 (2007), pp. 691--702]