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

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 $$O(n^{2-\varepsilon })$$O(n2-ε) for $$\varepsilon > 0$$ε>0, unless $$\mathsf {NP \subseteq coNP/poly}$$NP⊆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 $$O(k^{2-\varepsilon })$$O(k2-ε) edges, unless $$\mathsf {NP \subseteq coNP/poly}$$NP⊆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 $$O(n^{d-1})$$O(nd-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 $$\left( {\begin{array}{c}n\\ d-1\end{array}}\right) $$nd-1. We show that our kernel is tight under the assumption that $$\mathsf {NP} \nsubseteq \mathsf {coNP}/\mathsf {poly}$$NP⊈coNP/poly.