In MaxSAT with Cardinality Constraint problem (CC-MaxSAT), we are given a CNF-formula $\Phi$, and $k \ge 0$, and the goal is to find an assignment $\beta$ with at most $k$ variables set to true (also called a weight $k$-assignment) such that the number of clauses satisfied by $\beta$ is maximized. MaxCov can be seen as a special case of CC-MaxSAT, where the formula $\Phi$ is monotone, i.e., does not contain any negative literals. CC-MaxSAT and MaxCov are extremely well-studied problems in the approximation algorithms as well as parameterized complexity literature. Our first contribution is that the two problems are equivalent to each other in the context of FPT-Approximation parameterized by $k$ (approximation is in terms of number of clauses satisfied/elements covered). We give a randomized reduction from CC-MaxSAT to MaxCov in time $O(1/\epsilon)^{k} \cdot (m+n)^{O(1)}$ that preserves the approximation guarantee up to a factor of $1-\epsilon$. Furthermore, this reduction also works in the presence of fairness and matroid constraints. Armed with this reduction, we focus on designing FPT-Approximation schemes (FPT-ASes) for MaxCov and its generalizations. Our algorithms are based on a novel combination of a variety of ideas, including a carefully designed probability distribution that exploits sparse coverage functions. These algorithms substantially generalize the results in Jain et al. [SODA 2023] for CC-MaxSAT and MaxCov for $K_{d,d}$-free set systems (i.e., no $d$ sets share $d$ elements), as well as a recent FPT-AS for Matroid-Constrained MaxCov by Sellier [ESA 2023] for frequency-$d$ set systems.

翻译：在带基数约束的MaxSAT问题（CC-MaxSAT）中，给定一个CNF公式$\Phi$及$k \ge 0$，目标是找到最多将$k$个变量设为真（即权重$k$赋值）的赋值$\beta$，使得$\beta$满足的子句数量最大化。MaxCov可视为CC-MaxSAT的特例，其中公式$\Phi$是单调的，即不含任何负文字。CC-MaxSAT和MaxCov是近似算法与参数化复杂度领域中被广泛研究的问题。我们的首要贡献是：在FPT近似（以$k$为参数，近似比指满足子句数/覆盖元素数）的背景下，这两个问题相互等价。我们给出了一个从CC-MaxSAT到MaxCov的随机化归约，时间复杂度为$O(1/\epsilon)^{k} \cdot (m+n)^{O(1)}$，且近似保证仅损失$1-\epsilon$因子。此外，该归约在存在公平性与拟阵约束时仍然有效。基于此归约，我们致力于为MaxCov及其推广问题设计FPT近似方案（FPT-AS）。我们的算法基于多种创新思想的结合，包括一种精心设计的概率分布以利用稀疏覆盖函数。这些算法显著推广了Jain等人[SODA 2023]针对$K_{d,d}$-自由集系统（即不存在$d$个集合共享$d$个元素）的CC-MaxSAT和MaxCov结果，以及Sellier[ESA 2023]针对频率$d$集系统提出的拟阵约束MaxCov的近期FPT-AS。