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 $\Phi$ on $n$ variables, positive integers $k$ and $t$, 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 $W\left[2\right]$-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 $2^{O\left(t\right)}{n}^{O\left(1\right)}$ algorithm. We present a $2^{O\left(t\right)}{n}^{O\left(1\right)}$ 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 $O\left(d{t}^{d+1}\right)$ kernel has been designed for the Maximum Coverage problem on $K_{d,d}$-free incidence graphs. We extend this result to CC-Max-SAT on $K_{d,d}$-free formulas and design an $O\left(d{4}^{d^2}{t}^{d+1}\right)$ kernel.