近似计数H-着色是#BIS-困难的
Approximately Counting H-Colourings is #BIS-Hard
摘要：
考虑从一个输入图G到一个目标图H的H-染色的计数问题。我们表明，如果H是任何固定的图没有平凡的组件，那么这个问题是困难的著名的问题#BIS，这是（近似）计数独立集的问题在一个二分图。#BIS是近似计数的重要复杂性类中的完整问题，并且被认为没有FPRAS。如果是这样的话，那么我们的结果表明，对于每个图H没有平凡的组件，H-着色计数问题没有FPRAS。10年前，Goldberg、Kelk和Paterson研究了这个问题。他们能够证明近似采样H着色是#BIS困难的，但不知道如何获得近似计数的结果。我们的解决方案建立在使用Lovasz工作的非建设性想法之上。
英文摘要：
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.
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