论文标题
流程授权III：布尔CSP的复杂性二分法参数，由不满意的约束数量
Flow-augmentation III: Complexity dichotomy for Boolean CSPs parameterized by the number of unsatisfied constraints
论文作者
论文摘要
我们研究了在固定的有限布尔约束语言$γ$中，给定公式$ f $的参数化问题，具有或没有权重。更确切地说，对于每种有限的布尔约束语言$γ$，我们考虑以下两个问题。在Min Sat $（γ）$中，输入是$γ$的公式$ f $和一个整数$ k $，其任务是找到一个任务$α\ colon v（f）\ to \ {0,1 \} $，但最满足所有满足$ f $的$ k $约束，或确定没有此类$ f $的$ k $约束。在加权最小的SAT $（γ$）中，输入还包含一个权重函数$ w \ colon f \ to \ mathbb {z} _+$和一个整数$ w $，而任务是找到一项任务$α$，这样（1）$α$最多可满足$ f $ f $ f $ f $和（2）的约束。我们为这些问题的固定参数障碍提供了完整的二分法：我们表明，对于每种布尔约束语言$γ$，加权的min sat $（γ）$都是fpt；或加权的最小SAT $（γ）$是w [1] - hard，但最小sat $（γ）$是fpt;或min sat $（γ）$是w [1] -hard。这概括了Kim等人的最新工作。 （SODA 2021）不考虑加权问题，并且仅考虑了无法表达含义$（u \ to V）$的语言$γ$（如使用模型Digraph削减问题）。我们的结果概括并包含多个先前的结果，包括加权的FPT算法，几乎是2个sat，加权且未加权的$ \ ell $链链，以及耦合的min-cut，以及后者的加权和定向版本。我们算法中使用的主要工具是最近开发的定向流动效能方法（Kim等，STOC 2022）。
We study the parameterized problem of satisfying ``almost all'' constraints of a given formula $F$ over a fixed, finite Boolean constraint language $Γ$, with or without weights. More precisely, for each finite Boolean constraint language $Γ$, we consider the following two problems. In Min SAT$(Γ)$, the input is a formula $F$ over $Γ$ and an integer $k$, and the task is to find an assignment $α\colon V(F) \to \{0,1\}$ that satisfies all but at most $k$ constraints of $F$, or determine that no such assignment exists. In Weighted Min SAT$(Γ$), 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 $α$ such that (1) $α$ 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 $Γ$, either Weighted Min SAT$(Γ)$ is FPT; or Weighted Min SAT$(Γ)$ is W[1]-hard but Min SAT$(Γ)$ is FPT; or Min SAT$(Γ)$ is W[1]-hard. This generalizes recent work of Kim et al. (SODA 2021) which did not consider weighted problems, and only considered languages $Γ$ 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).
