Improved FPT approximation scheme and approximate kernel for biclique-free max k-weight SAT: Greedy strikes back

Abstract

In the Max k-Weight SAT (aka Max SAT with Cardinality Constraint) problem, we are given a CNF formula with n variables and m clauses together with a positive integer k. The goal is to find an assignment where at most k variables are set to one that satisfies as many constraints as possible. Recently, Jain et al. [20] gave an FPT approximation scheme (FPT-AS) with running time $2^{O\left({(dk/ϵ)}^d\right)}\cdot {(n+m)}^{O\left(1\right)}$ for Max k-Weight SAT when the incidence graph is $K_{d,d}$-free. They asked whether a polynomial-size approximate kernel exists. In this work, we answer this question positively by giving a $(1-ϵ)$-approximate kernel with ${\left(\frac{dk}{ϵ}\right)}^{O\left(d\right)}$ variables. This also implies an improved FPT-AS with running time ${(dk/ϵ)}^{O\left(dk\right)}\cdot {(n+m)}^{O\left(1\right)}$. Our approximate kernel is based mainly on a couple of greedy strategies together with a sunflower lemma-style reduction rule.