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 and 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 2^O(t)n^O(1) algorithm. We present a 2^O(t)n^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 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 CC-MAX-SAT on K_d,d-free formulas and design an O(d4^d2t^d+1) kernel.