Beating a random assignment

• Authors: Gustav Hast
• Venue: APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
• Year: 2005

Max CSP(P) is the problem of maximizing the weight of satisfied constraints, where each constraint acts over a k-tuple of literals and is evaluated using the predicate P. The approximation ratio of a random assignment is equal to the fraction of satisfying inputs to P. If it is NP-hard to achieve a better approximation ratio for Max CSP(P), then we say that P is approximation resistant. Our goal is to characterize which predicates that have this property. A general approximation algorithm for Max CSP(P) is introduced. For a multitude of different P, it is shown that the algorithm beats the random assignment algorithm, thus implying that P is not approximation resistant. In particular, over 2/3 of the predicates on four binary inputs are proved not to be approximation resistant, as well as all predicates on 2s binary inputs, that have at most 2s+1 accepting inputs. We also prove a large number of predicates to be approximation resistant. In particular, all predicates of arity 2s+s2 with less than $2^{s^2}$ non-accepting inputs are proved to be approximation resistant, as well as almost 1/5 of the predicates on four binary inputs.