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-ε) for ε > 0 , unless 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-ε) edges, unless 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) 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 ( [ n; d-1 ]) . We show that our kernel is tight under the assumption that 𝖭𝖯⊈𝖼𝗈𝖭𝖯/𝗉𝗈𝗅𝗒 .