Approximately Counting H-Colourings is #BIS-Hard

TL;DR

This paper proves that counting H-colourings for any fixed graph H without trivial components is as hard as #BIS, indicating no efficient approximation scheme exists unless a major complexity assumption fails.

Contribution

It establishes the #BIS-hardness of approximately counting H-colourings for all non-trivial fixed graphs H, extending previous sampling hardness results to counting.

Findings

Counting H-colourings is #BIS-hard for all non-trivial H.

No known FPRAS exists for this counting problem under standard complexity assumptions.

The proof uses non-constructive methods based on Lovasz's work.

Abstract

We consider the problem of counting H-colourings from an input graph G to a target graph H. We show that if H is any fixed graph without trivial components, then the problem is as hard as the well-known problem #BIS, which is the problem of (approximately) counting independent sets in a bipartite graph. #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 #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 Lovasz.