Max-SAT with Cardinality Constraint Parameterized by the Number of Clauses

Abstract

Max-SAT with cardinality constraint (CC-Max-SAT) is one of the classical NP-complete problems. In this problem, given a CNF-formula $\varPhi$ on n variables, positive integers k, t, the goal is to find an assignment $\beta$ with at most k variables set to true (also called a weight k-assignment) such that the number of clauses satisfied by $\beta$ is at least t. The problem is known to be $\textsf{W}[2]$-hard with respect to the parameter k. In this paper, we study the problem with respect to the parameter t. The special case of CC-Max-SAT, when all the clauses contain only positive literals (known as Maximum Coverage), is known to admit a $2^{\mathcal {O}(t)}n^{\mathcal {O}(1)}$ algorithm. We present a $2^{\mathcal {O}(t)}n^{\mathcal {O}(1)}$ algorithm for the general case, CC-Max-SAT. We further study the problem through the lens of kernelization. Since Maximum Coverage does not admit polynomial kernel with respect to the parameter t, we focus our study on $K_{d,d}$-free formulas (that is, the clause-variable incidence bipartite graph of the formula that excludes $K_{d,d}$ as a subgraph). Recently, in [Jain et al., SODA 2023], an $\mathcal {O}(dt^{d+1})$ kernel has been designed for the Maximum Coverage problem on $K_{d,d}$-free incidence graphs. We extend this result to Max-SAT on $K_{d,d}$-free formulas and design a $\mathcal {O}(d4^{d^2}t^{d+1})$ kernel.