Counting Independent Sets in Hypergraphs when Strong Spatial Mixing Fails
Ivona Bezáková, Andreas Galanis, Leslie Ann Goldberg, Heng Guo, Daniel Štefankovič · arXiv (Cornell University) · 2015
We study the problem of approximately counting independent sets in hypergraphs with maximum degree $\Delta$ whose hyperedges have arity at least $k\geq 3$. In the graphical case, where the arity of every edge is $2$, it is known that for $\Delta \geq 6$ the problem is hard and for $\Delta \leq 5$ there is an FPTAS. This approximation algorithm uses correlation decay techniques that rely on the fact that strong spatial mixing occurs. Surprisingly, in the hypergraph case we give a correlation decay based algorithm even though strong spatial mixing fails. Our main idea is introducing amortization in the correlation decay proof. In particular, in the correlation decay proof we track not just the decay but also combinatorial properties of the intermediate instances. This enables us to give an FPTAS even when strong spatial mixing fails, for example, when $\Delta=6$ and $k \geq 3$ and also for all $\Delta$ and all sufficiently large $k\geq 1.66\Delta$. We further demonstrate that in the hypergraph independent set model, approximating the partition function is NP-hard even within the uniqueness regime.