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

Max-SAT with cardinality constraint (CC-Max-SAT) is one of the classical NP-complete problems. In this problem, given a CNF-formula Φ on n variables, positive integers k, t, the goal is to find an assignment β with at most k variables set to true (also called a weight k-assignment) such that the number of clauses satisfied by β is at least t. The problem is known to be 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 2O(t)nO(1) algorithm. We present a 2O(t)nO(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 Kd,d-free formulas (that is, the clause-variable incidence bipartite graph of the formula that excludes Kd,d as a subgraph). Recently, in [Jain et al., SODA 2023], an O(dtd+1) kernel has been designed for the Maximum Coverage problem on Kd,d-free incidence graphs. We extend this result to Max-SAT on Kd,d-free formulas and design a O(d4d2td+1) kernel.