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Sparsification Upper and Lower Bounds for Graphs Problems and Not-All-Equal SAT

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. Sparsification Upper and Lower Bounds for Graphs Problems and Not-All-Equal SAT. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 163-174, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.IPEC.2015.163

Abstract

We present several sparsification lower and upper bounds for classic problems in graph theory and logic. For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time algorithm that reduces any n-vertex input to an equivalent instance, of an arbitrary problem, with bitsize O(n^{2-epsilon}) for epsilon > 0, unless NP is a subset of coNP/poly and the polynomial-time hierarchy collapses. These results imply that existing linear-vertex kernels for k-Nonblocker and k-Max Leaf Spanning Tree (the parametric duals of (Connected) Dominating Set) cannot be improved to have O(k^{2-epsilon}) edges, unless NP is a subset of NP/poly. We also present a positive result and exhibit a non-trivial sparsification algorithm for d-Not-All-Equal-SAT. We give an algorithm that reduces an n-variable input with clauses of size at most d to an equivalent input with O(n^{d-1}) clauses, for any fixed d. Our algorithm is based on a linear-algebraic proof of Lovász that bounds the number of hyperedges in critically 3-chromatic d-uniform n-vertex hypergraphs by binom{n}{d-1}. We show that our kernel is tight under the assumption that NP is not a subset of NP/poly.

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Keywords
  • sparsification
  • graph coloring
  • Hamiltonian cycle
  • satisfiability

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