Robust Weighted Partial Maximum Satisfiability Problem: Challenge to Σ2P -Complete Problem

This paper introduces a new problem called the Robust Maximum Satisfiability problem (R-MaxSAT), as well as its extension called the Robust weighted Partial MaxSAT (R-PMaxSAT). In R-MaxSAT (or R-PMaxSAT), a problem solver called defender hopes to maximize the number of satisfied clauses (or the sum of their weights) as the standard MaxSAT/partial MaxSAT problem, although she must ensure that the obtained solution is robust (In this paper, we use the pronoun “she” for the defender and “he” for the attacker). We assume an adversary called the attacker will flip some variables after the defender selects a solution. R-PMaxSAT can formalize the robust Clique Partitioning Problem (robust CPP), where CPP has many real-life applications. We first demonstrate that the decision version of R-MaxSAT is Σ2P -complete. Then, we develop two algorithms to solve R-PMaxSAT, by utilizing a state-of-the-art SAT solver or a Quantified Boolean Formula (QBF) solver as a subroutine. Our experimental results show that we can obtain optimal solutions within a reasonable amount of time for randomly generated R-MaxSAT instances with 30 variables and 150 clauses (within 40 s) and R-PMaxSAT instances based on CPP benchmark problems with 60 vertices (within 500 s).