Flow-augmentation III: Complexity dichotomy for Boolean CSPs parameterized by the number of unsatisfied constraints

TL;DR

This work studies the parameterized problem of satisfying almost all constraints in Boolean CSPs over a fixed constraint language Γ, with and without weights. It introduces a unified flow-augmentation framework and reduces the optimization problems to structured graph-cut instances (Generalized Bundled Cut, Generalized Digraph Pair Cut, and Clause Cut) that are tractable under 2K2-freeness assumptions. The authors prove a complete dichotomy: Weighted Min SAT(Γ) is FPT, or Min SAT(Γ) is FPT while Weighted Min SAT(Γ) is W[1]-hard, or Min SAT(Γ) is W[1]-hard, depending on Γ’s position in Post’s lattice (bijunctive vs IHS-B and the 2K2-freeness of Gaifman/arrow graphs). The results generalize known FPT algorithms (e.g., Almost 2-SAT, ℓ-Chain SAT) and integrate weighted and unweighted settings, thereby advancing the understanding of tractability in parameterized CSPs. The techniques provide a robust toolkit for future CSP dichotomies and related graph-separation problems, with potential impact on algorithm design for schedule/planar CSP variants and other structured constraint languages.

Abstract

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).

Figures (4)

• Figure 1: An example of the reversing process of the ultimately bipartite instance, where we reverse the arcs in $V_1$ (bottom half of the graph). Clauses going across become arcs.
• Figure 2: An exemplary situation as in Case 2 of Claim \ref{['claim:detectactive']} and its later usage. The dashed lines depict (crisp) clauses, whose projections on $P_i$ and $P_j$ form an antichain. Red crosses depict a solution $Z$. The clause $\{u_3,v_2\}$ is an active clause, with $u_3$ being an active vertex. Adding the blue arcs to the graph increases the size of a solution by one (the arc $(u_2,u_3)$ needs to be cut as well), but moves the arc $(\pi_i(u_3),u_3)$ from $Z \setminus Z_{s,t}$ to $Z_{s,t}$, decreasing $\lvert Z \setminus Z_{s,t}\rvert$.
• Figure 3: Reversing a path $P_j$ and its influence on a pair $\{u_1,v_1\} \in \mathcal{C}'_{i,j}$. To maintain the role of $\{u_1,v_1\}$ as a constraint meaning "cut $P_i$ before $u_1$ or cut $P_j$ before $v_1$", after reversing $P_j$ the pair $\{u_1,v_1\}$ should become an arc $(u_1,v_1)$ in $E'_{i,j}$. The four columns above correspond to four cases where the paths $P_i$ and $P_j$ are cut in the sought solution; in the last column the pair $\{u_1,v_1\}$ and the corresponding arc $(u_1,v_1)$ are violated.
• Figure 4: An example of the reversing process, where we reverse the paths in $I_1$ (bottom half of the graph). Clauses going across become arcs and arcs in the bottom half reverse.

Theorems & Definitions (109)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 1.4
• Lemma 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8: directed flow-augmentation, Theorem 3.1 of dfl-arxiv
• Lemma 1.8
• Theorem 1.9