We study the parameterized problem of satisfying ``almost all'' constraints of a given formula $F$ over a fixed, finite Boolean constraint language $\Gamma$, with or without weights. More precisely, for each finite Boolean constraint language $\Gamma$, we consider the following two problems. In Min SAT$(\Gamma)$, the input is a formula $F$ over $\Gamma$ and an integer $k$, and the task is to find an assignment $\alpha \colon V(F) \to \{0,1\}$ that satisfies all but at most $k$ constraints of $F$, or determine that no such assignment exists. In Weighted Min SAT$(\Gamma$), the input additionally contains a weight function $w \colon F \to \mathbb{Z}_+$ and an integer $W$, and the task is to find an assignment $\alpha$ such that (1) $\alpha$ satisfies all but at most $k$ constraints of $F$, and (2) the total weight of the violated constraints is at most $W$. We give a complete dichotomy for the fixed-parameter tractability of these problems: We show that for every Boolean constraint language $\Gamma$, either Weighted Min SAT$(\Gamma)$ is FPT; or Weighted Min SAT$(\Gamma)$ is W[1]-hard but Min SAT$(\Gamma)$ is FPT; or Min SAT$(\Gamma)$ is W[1]-hard. This generalizes recent work of Kim et al. (SODA 2021) which did not consider weighted problems, and only considered languages $\Gamma$ that cannot express implications $(u \to v)$ (as is used to, e.g., model digraph cut problems). Our result generalizes and subsumes multiple previous results, including the FPT algorithms for Weighted Almost 2-SAT, weighted and unweighted $\ell$-Chain SAT, and Coupled Min-Cut, as well as weighted and directed versions of the latter. The main tool used in our algorithms is the recently developed method of directed flow-augmentation (Kim et al., STOC 2022).

翻译：我们研究在固定有限布尔约束语言$\Gamma$上，给定公式$F$满足“几乎所有”约束的参数化问题，同时考虑加权与无权重情形。具体而言，对于每个有限布尔约束语言$\Gamma$，我们考虑以下两个问题。在Min SAT$(\Gamma)$中，输入为$\Gamma$上的公式$F$和整数$k$，目标是找到一个赋值$\alpha \colon V(F) \to \{0,1\}$，使得$F$中最多有$k$个约束未被满足，或确定不存在这样的赋值。在加权Min SAT$(\Gamma)$中，输入额外包含权重函数$w \colon F \to \mathbb{Z}_+$和整数$W$，目标是找到赋值$\alpha$，使得(1) $\alpha$满足$F$中除最多$k$个约束外的所有约束，且(2)被违反约束的总权重不超过$W$。我们给出了这些问题的固定参数可处理性的完整二分法：我们证明，对于每个布尔约束语言$\Gamma$，要么加权Min SAT$(\Gamma)$是FPT的；要么加权Min SAT$(\Gamma)$是W[1]-难的但Min SAT$(\Gamma)$是FPT的；要么Min SAT$(\Gamma)$是W[1]-难的。这推广了Kim等人(SODA 2021)的近期工作，该工作未考虑加权问题，且仅考虑不能表达蕴含关系$(u \to v)$的语言$\Gamma$（例如用于建模有向图割问题）。我们的结果推广并包含了多个先前结果，包括加权Almost 2-SAT的FPT算法、加权与无权重$\ell$-Chain SAT算法、Coupled Min-Cut算法，以及后者的加权和有向版本。我们算法中使用的主要工具是最近发展的有向流增强方法(Kim等人，STOC 2022)。