Max-SAT with cardinality constraint parameterized by the number of clauses

• Published in: Theoretical computer science Vol. 1056; p. 115540
• Main Authors: Jain, Pallavi, Kanesh, Lawqueen, Panolan, Fahad, Saha, Souvik, Sahu, Abhishek, Saurabh, Saket, Upasana, Anannya
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 21.11.2025
• Subjects: FPT; Kernel; Max-SAT; Parameterized algorithms; Parameterized algorithms; FPT; Max-SAT; Kernel
• ISSN: 0304-3975
• DOI: 10.1016/j.tcs.2025.115540

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 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 CC-Max-SAT on Kd,d-free formulas and design an O(d4d2td+1) kernel.