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. [SODA'23] gave an FPT approximation scheme (FPT-AS) with running time $2^{O\left(\left(dk/\epsilon\right)^d\right)} \cdot (n + m)^{O(1)}$ 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 an $(1 - \epsilon)$-approximate kernel with $\left(\frac{d k}{\epsilon}\right)^{O(d)}$ variables. This also implies an improved FPT-AS with running time $(dk/\epsilon)^{O(dk)} \cdot (n + m)^{O(1)}$. Our approximate kernel is based mainly on a couple of greedy strategies together with a sunflower lemma-style reduction rule.

翻译：在最大k-权重SAT问题（即带基数约束的最大SAT问题）中，给定一个包含n个变量和m个子句的CNF公式以及一个正整数k，目标是找到一个最多将k个变量赋值为1的赋值，使得满足尽可能多的约束。近期，Jain等人[SODA'23]针对关联图为$K_{d,d}$-自由的最大k-权重SAT问题，提出了一种运行时间为$2^{O\left(\left(dk/\epsilon\right)^d\right)} \cdot (n + m)^{O(1)}$的FPT逼近方案（FPT-AS）。他们询问是否存在多项式大小的近似核。本文通过给出一个包含$\left(\frac{d k}{\epsilon}\right)^{O(d)}$个变量的$(1 - \epsilon)$-近似核，对该问题给出了肯定回答。这同时蕴含了一个运行时间为$(dk/\epsilon)^{O(dk)} \cdot (n + m)^{O(1)}$的改进型FPT-AS。我们的近似核主要基于若干贪婪策略以及一个向日葵引理风格的归约规则。