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Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials

Authors Bart M. P. Jansen, Astrid Pieterse

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Bart M. P. Jansen
Astrid Pieterse

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Bart M. P. Jansen and Astrid Pieterse. Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 71:1-71:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.MFCS.2016.71

Abstract

This paper analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems, without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP is not contained in coNP/poly, no efficient preprocessing algorithm can reduce n-variable instances of CNF-SAT with d literals per clause, to equivalent instances with O(n^{d-epsilon}) bits for any epsilon > 0. For the Not-All-Equal SAT problem, a compression to size tilde-O(n^{d-1}) exists. We put these results in a common framework by analyzing the compressibility of binary CSPs. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for Exact Satisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n+1, yet no polynomial-time algorithm can reduce to an equivalent instance with O(n^{2-epsilon}) bits for any epsilon > 0, unless NP is contained in coNP/poly.

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Keywords
  • constraint satisfaction problem
  • sparsification
  • satisfiability
  • kernelization

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