Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials - Jansen, Bart M. P.; Pieterse, Astrid (Algorithmica, 2019)
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 ⊈ 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 Δ ( H ) log k ) . Our second result shows that 3- Coloring does not admit non-trivial sparsification: assuming NP ⊈ coNP / poly , the parameterization by the number of vertices n admits no (generalized) kernel of size O ( n 2 - ε ) for any ε > 0 . Previously, such a lower bound was only known for coloring with q ≥ 4 colors.