For a fixed set $\mathcal{F}$ of Boolean constraint types, a MinCSP$(\mathcal{F})$-instance consists of a formula $F$ that applies $m$ constraints from $\mathcal{F}$ to a set of $n$ Boolean variables. The goal is to remove a minimum subset of constraint applications from $F$ to make the remaining formula satisfiable. Previous work characterized how the choice of $\mathcal{F}$ affects its polynomial-time solvability and approximability. We extend a recently introduced preprocessing framework for graph problems to the problem above. Rephrased in the context of CSPs, this framework defines a constraint application from a given formula $F$ as $c$-essential if it is contained in all $c$-approximate solutions to $F$. Being able to efficiently detect these essential parts of a solution reduces the search space of any follow-up FPT algorithms parameterized by the solution size and yields an immediate asymptotic improvement to the runtime of such algorithms. In this work, we present a dichotomy theorem that distinguishes constraint sets $\mathcal{F}$ for which $c_\mathcal{F}$-essential constraint applications can be detected efficiently for some $c_{\mathcal{F}} \in \mathcal{O}(1)$, from those for which this task is intractable under established complexity-theoretic conjectures. Our results show that for any set $\mathcal{F}$ of bijunctive constraints, there is a polynomial-time algorithm that detects $\mathcal{O}(1)$-essential constraint applications. This contrasts the fact that constant-factor approximating a bijunctive MinCSP$(\mathcal{F})$-problem is intractable under the Unique Games Conjecture.

翻译：对于固定的布尔约束类型集合 $\mathcal{F}$，MinCSP$(\mathcal{F})$ 实例由公式 $F$ 构成，该公式将 $m$ 个来自 $\mathcal{F}$ 的约束应用于 $n$ 个布尔变量。目标是移除 $F$ 中最小子集的约束应用，使得剩余公式可满足。先前工作刻画了 $\mathcal{F}$ 的选择如何影响其多项式时间可解性和可近似性。我们将最近提出的图问题预处理框架扩展到上述问题。在CSP语境下重新表述后，该框架将给定公式 $F$ 中的约束应用定义为 $c$-本质的，当且仅当它包含在 $F$ 的所有 $c$-近似解中。高效检测这些本质解部分能够缩减后续以解大小为参数的FPT算法的搜索空间，并直接导致此类算法运行时间的渐近改进。本文提出一个二分定理，区分那些对于某些 $c_{\mathcal{F}} \in \mathcal{O}(1)$ 可高效检测 $c_\mathcal{F}$-本质约束应用的约束集合 $\mathcal{F}$，与那些在既定复杂性理论猜想下此任务难以处理的集合。结果表明，对于任意二元约束集合 $\mathcal{F}$，存在多项式时间算法可检测 $\mathcal{O}(1)$-本质约束应用。这对比了在唯一游戏猜想下，常数因子近似二元MinCSP$(\mathcal{F})$ 问题不可处理的事实。