On Sparsification for Computing Treewidth

TL;DR

This paper proves that significantly reducing the size of Treewidth problem instances without changing their answer is unlikely unless major complexity class collapses occur, and it improves kernel size bounds for Treewidth parameterized by vertex cover.

Contribution

It establishes a lower bound on sparsification for Treewidth and presents an improved kernel with O(k^2) vertices using a novel treewidth-invariant set.

Findings

Sparsification of Treewidth instances is unlikely without complexity class consequences.

Kernel size for Treewidth parameterized by vertex cover can be improved to O(k^2) vertices.

Introduces a new method using treewidth-invariant sets and the q-expansion lemma.

Abstract

We investigate whether an n-vertex instance (G,k) of Treewidth, asking whether the graph G has treewidth at most k, can efficiently be made sparse without changing its answer. By giving a special form of OR-cross-composition, we prove that this is unlikely: if there is an e > 0 and a polynomial-time algorithm that reduces n-vertex Treewidth instances to equivalent instances, of an arbitrary problem, with O(n^{2-e}) bits, then NP is in coNP/poly and the polynomial hierarchy collapses to its third level. Our sparsification lower bound has implications for structural parameterizations of Treewidth: parameterizations by measures that do not exceed the vertex count, cannot have kernels with O(k^{2-e}) bits for any e > 0, unless NP is in coNP/poly. Motivated by the question of determining the optimal kernel size for Treewidth parameterized by vertex cover, we improve the O(k^3)-vertex kernel from Bodlaender et al. (STACS 2011) to a kernel with O(k^2) vertices. Our improved kernel is based on a novel form of treewidth-invariant set. We use the q-expansion lemma of Fomin et al. (STACS 2011) to find such sets efficiently in graphs whose vertex count is superquadratic in their vertex cover number.