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

Authors Marvin Künnemann, Dániel Marx

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Marvin Künnemann
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Dániel Marx
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany

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Marvin Künnemann and Dániel Marx. Finding Small Satisfying Assignments Faster Than Brute Force: A Fine-Grained Perspective into Boolean Constraint Satisfaction. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 27:1-27:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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 ± o(1))k, 2) the best algorithms are as fast as current k-clique algorithms, i.e., g(k) = (ω/3 ± o(1))k, 3) the exponent has sublinear dependence on k with g(k) ∈ [Ω(∛k), O(√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√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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
  • Fine-grained complexity theory
  • algorithmic classification theorem
  • multivariate algorithms and complexity
  • constraint satisfaction problems
  • satisfiability


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