Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials

TL;DR

This paper advances graph coloring data reduction by providing optimal kernel sizes for q-Coloring parameterized by vertex cover and twin-cover, and proves non-existence of small kernels for 3-Coloring under certain complexity assumptions.

Contribution

It establishes optimal kernel bounds for q-Coloring with vertex cover and twin-cover parameters, and shows 3-Coloring cannot be sparsified below certain sizes assuming standard complexity hypotheses.

Findings

Kernel of size O(k^{q-1} log k) for q-Coloring parameterized by Vertex Cover.

Kernel of size O(k^{Δ(H)} log k) for H-Coloring parameterized by Twin-Cover.

No polynomial-size kernel for 3-Coloring parameterized by number of vertices, assuming NP not in coNP/poly.

Abstract

The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q-Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural properties, such as the size of a minimum vertex cover. In this paper we settle two open problems about data reduction for q-Coloring. First, we obtain a kernel of bitsize $O(k^{q-1}\log{k})$ for q-Coloring parameterized by Vertex Cover, for any q >= 3. This size bound is optimal up to $k^{o(1)}$ factors assuming NP is not a subset of coNP/poly, and improves on the previous-best kernel of size $O(k^q)$. We generalize this result for deciding q-colorability of a graph G, to deciding the existence of a homomorphism from G to an arbitrary fixed graph H. Furthermore, we can replace the parameter vertex cover by the less restrictive parameter twin-cover. We prove that H-Coloring parameterized by Twin-Cover has a kernel of size $O(k^{\Delta(H)}\log k)$. Our second result shows that 3-Coloring does not admit non-trivial sparsification: assuming NP is not a subset of coNP/poly, the parameterization by the number of vertices n admits no (generalized) kernel of size $O(n^{2-e})$ for any e > 0. Previously, such a lower bound was only known for coloring with q >= 4 colors.