Flow-augmentation III: Complexity dichotomy for Boolean CSPs parameterized by the number of unsatisfied constraints
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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 V(F) →{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 F →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 → 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 ℓ-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).
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