TY - GEN
T1 - Max-SAT with Cardinality Constraint Parameterized by the Number of Clauses
AU - Jain, Pallavi
AU - Kanesh, Lawqueen
AU - Panolan, Fahad
AU - Saha, Souvik
AU - Sahu, Abhishek
AU - Saurabh, Saket
AU - Upasana, Anannya
N1 - Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.
PY - 2024/1/1
Y1 - 2024/1/1
N2 - 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.
AB - 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.
KW - FPT
KW - Kernel
KW - Max-SAT
UR - http://www.scopus.com/inward/record.url?scp=85188261870&partnerID=8YFLogxK
U2 - 10.1007/978-3-031-55601-2_15
DO - 10.1007/978-3-031-55601-2_15
M3 - Conference contribution
AN - SCOPUS:85188261870
SN - 9783031556005
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 223
EP - 237
BT - LATIN 2024
A2 - Soto, José A.
A2 - Wiese, Andreas
PB - Springer Science and Business Media Deutschland GmbH
T2 - 16th Latin American Symposium on Theoretical Informatics, LATIN 2042
Y2 - 18 March 2024 through 22 March 2024
ER -