🤖 AI Summary
This paper investigates the parameterized complexity of “almost satisfying all constraints” for finite Boolean constraint languages Γ, formalized as Min SAT(Γ) (minimize the number k of unsatisfied constraints) and its weighted variant Weighted Min SAT(Γ) (ensure total violation weight ≤ W). Employing a novel synthesis of directed flow augmentation, algebraic classification of constraint languages, weight-sensitive kernelization, and structural analysis of graph cuts, we establish, for the first time, a unified complexity dichotomy for both unweighted and weighted cases. For every Γ, we completely characterize fixed-parameter tractability: either (i) Weighted Min SAT(Γ) is FPT while Min SAT(Γ) is W[1]-hard, (ii) both are W[1]-hard, or (iii) Weighted Min SAT(Γ) is FPT (implying Min SAT(Γ) is also FPT). Our framework overcomes prior limitations in modeling implication constraints (u → v), and systematically generalizes and unifies landmark results including Almost 2-SAT, ℓ-Chain SAT, and Coupled Min-Cut.
📝 Abstract
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 $alpha colon V(F) o {0,1}$ 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 $w colon F o mathbb{Z}_+$ 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 $(u o v)$ (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).
