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

Abstract

AbstractWe 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.KeywordsBipartite GraphCounting SettingInput GraphComplete Bipartite GraphFull VersionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.