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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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Author Details

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

Funding

  • Marx, Dániel: Research of the second author was supported by funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement SYSTEMATICGRAPH (No. 725978).

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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) https://doi.org/10.4230/LIPIcs.CCC.2020.27

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 ± 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
Keywords
  • Fine-grained complexity theory
  • algorithmic classification theorem
  • multivariate algorithms and complexity
  • constraint satisfaction problems
  • satisfiability

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