Adversarial satisfiability problem

Abstract

We study the adversarial satisfiability problem, where the adversary can choose whetherthe variables are negated in clauses or not, in order to make the resulting formulaunsatisfiable. This problem belongs to a general class of adversarial optimizationproblems that often arise in practice and are algorithmically much harder than thestandard optimization problems. We use the cavity method to compute largedeviations of the entropy in the random satisfiability problem with respect to theconfigurations of negations. We conclude that in the thermodynamic limit thebest strategy the adversary can adopt is to simply balance the number of timesevery variable is negated and the number of times it is not negated. We alsoconduct a numerical study of the problem, and find that there are very strongpre-asymptotic effects that may be due to the fact that for small sizes exponentialand factorial growth is hardly distinguishable. As a side result we compute thesatisfiability threshold for balanced configurations of negations, and also the randomregular satisfiability, i.e. when all variables belong to the same number of clauses.