Approximately Counting H-Colourings is $\#\mathrm {BIS}$-Hard

Abstract

We consider counting H-colourings from an input graph G to a target graph H. We show that for any fixed graph H without trivial components, this is as hard as the well-known problem $\#\mathrm {BIS}$, the problem of (approximately) counting independent sets in a bipartite graph. $\#\mathrm {BIS}$ is a complete problem in an important complexity class for approximate counting, and is believed not to have an FPRAS. If this is so, then our result shows that for every graph H without trivial components, the H-colouring counting problem has no FPRAS. This problem was studied a decade ago by Goldberg, Kelk and Paterson. They were able to show that approximately sampling H-colourings is $\#\mathrm {BIS}$-hard, but it was not known how to get the result for approximate counting. Our solution builds on non-constructive ideas using the work of Lovász. The full version is available at arxiv.org/abs/1502.01335. The theorem numbering here matches the full version.

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) ERC grant agreement no. 334828. The paper reflects only the authors’ views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein.