An upper (lower) bound for Max (Min) CSP

Abstract

The random constraint satisfaction problem (CSP) instances generated by Model RB have been widely used in the field of CSP and have some nice features. In this paper, we consider two optimization versions of CSP, i.e., the maximum constraint satisfaction problem (Max-CSP) and the minimum satisfaction problem (Min-CSP) of Model RB. The problem of the Max-CSP is how to find an assignment to all the variables such that the maximum number of constraints are satisfied and the problem of Min-CSP is how to find an assignment to all the variables such that the minimum number of constraints are satisfied. We use the first moment method to prove that when r > 2α(1/p−1) (or p > 2α/(2α+r)), an upper bound of Max-CSP can be derived. Similarly, we can prove that when r > 2α(1/p−1) (or p > 2α/(2α + r)), a lower bound of Min-CSP can be derived.