Finding Small Satisfying Assignments Faster Than Brute Force: A Fine-grained Perspective into Boolean Constraint Satisfaction

TL;DR

This paper investigates the complexity of finding small satisfying assignments in Boolean constraint satisfaction problems, providing a detailed classification of the optimal algorithms' running times based on the size constraint k.

Contribution

It offers a fine-grained complexity classification for Boolean CSPs with size constraints, extending beyond the traditional FPT/W[1]-hardness dichotomy.

Findings

Characterizes four regimes of algorithmic complexity based on the parameter k.

Provides an almost tight characterization of the exponent g(k) in the running time.

Introduces a novel algorithm for SubsetSum with precedence constraints, generalizing the Frobenius coin problem.

Abstract

To study the question under which circumstances small solutions can be found faster than by exhaustive search (and by how much), we study the fine-grained complexity of Boolean constraint satisfaction with size constraint exactly $k$. More precisely, we aim to determine, for any finite constraint family, the optimal running time $f(k)n^{g(k)}$ required to find satisfying assignments that set precisely $k$ of the $n$ variables to $1$. Under central hardness assumptions on detecting cliques in graphs and 3-uniform hypergraphs, we give an almost tight characterization of $g(k)$ into four regimes: (1) Brute force is essentially best-possible, i.e., $g(k) = (1\pm o(1))k$, (2) the best algorithms are as fast as current $k$-clique algorithms, i.e., $g(k)=(\omega/3\pm o(1))k$, (3) the exponent has sublinear dependence on $k$ with $g(k) \in [\Omega(\sqrt[3]{k}), O(\sqrt{k})]$, or (4) the problem is fixed-parameter tractable, i.e., $g(k) = O(1)$. This yields a more fine-grained perspective than a previous FPT/W[1]-hardness dichotomy (Marx, Computational Complexity 2005). Our most interesting technical contribution is a $f(k)n^{4\sqrt{k}}$-time algorithm for SubsetSum with precedence constraints parameterized by the target $k$ -- particularly the approach, based on generalizing a bound on the Frobenius coin problem to a setting with precedence constraints, might be of independent interest.