On MAX–SAT with Cardinality Constraint

We consider the weighted MAX–SAT problem with an additional constraint that at most k variables can be set to true. We call this problem k–WMAX–SAT. This problem admits a (1 − 1/e )-factor approximation algorithm in polynomial time [Sviridenko, Algorithmica 2001] and it is proved that there is no (1 − 1/e + ϵ)-factor approximation algorithm in f (k) · no(k) time for Maximum Coverage, the unweighted monotone version of k–WMAX–SAT [Manurangsi, SODA 2020]. Therefore, we study two restricted versions of the problem in the realm of parameterized complexity.

1. When the input is an unweighted 2–CNF formula (the problem is called k–MAX–2SAT), we design an efficient polynomial-size approximate kernelization scheme. That is, we design a polynomial-time algorithm that given a 2–CNF formula ψ and�ϵ > 0, compresses the input instance to a 2–CNF formula ψ such that any c-approximate solution of ψ� can be converted to a c(1 − ϵ)-approximate solution of ψ in polynomial time.

2. When the input is a planar CNF formula, i.e., the variable-clause incident graph is a planar graph, we show the following results:

– There is an FPT algorithm for k–WMAX–SAT on planar CNF formulas that runs in 2O(k) · (C + V ) time.

– There is a polynomial-time approximation scheme for k– WMAX–SAT on planar CNF formulas that runs in time 2O(1/ϵ ) ·k · (C+V ).

The above-mentioned C and V are the number of clauses and variables of the input formula respectively.