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-varepsilon }) for varepsilon > 0, unless mathsf {NP subseteq 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-varepsilon }) edges, unless mathsf {NP subseteq coNP/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 left( {begin{array}{c}n d-1end{array}}right) . We show that our kernel is tight under the assumption that mathsf {NP} nsubseteq mathsf {coNP}/mathsf {poly}.
Highlights
Introduction1.1 BackgroundSparsification refers to the method of reducing an object such as a graph or conjunctive normal form (CNF)-formula to an equivalent object that is less dense, that is, an object in which the ratio of edges to vertices (or clauses to variables) is smaller
1.1 BackgroundSparsification refers to the method of reducing an object such as a graph or conjunctive normal form (CNF)-formula to an equivalent object that is less dense, that is, an object in which the ratio of edges to vertices is smaller
We have added several classic graph problems to a growing list of problems for which non-trivial polynomial-time sparsification is provably impossible under the assumption that NP coNP/poly
Summary
1.1 BackgroundSparsification refers to the method of reducing an object such as a graph or CNF-formula to an equivalent object that is less dense, that is, an object in which the ratio of edges to vertices (or clauses to variables) is smaller. Using tools developed in the last five years, it has become possible to address questions such as: “Is there a polynomial-time algorithm that reduces an n-vertex instance of my favorite graph problem to an equivalent instance with a subquadratic number of edges?”. The impetus for this line of analysis was given by an influential paper by Dell and van Melkebeek [8] (conference version in 2010).
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