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

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}.