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We give a randomized algorithm that approximates the number of independentsets in a dense, regular bipartite graph -- in the language of approximatecounting, we give an FPRAS for #BIS on the class of dense, regular bipartitegraphs. Efficient counting algorithms typically apply to ``high-temperature''problems on bounded-degree graphs, and our contribution is a notable exceptionas it applies to dense graphs in a low-temperature setting. Our methods give acounting-focused complement to the long line of work in combinatorialoptimization showing that CSPs such as Max-Cut and Unique Games are easy ondense graphs via spectral arguments. The proof exploits the fact that dense,regular graphs exhibit a kind of small-set expansion (i.e. bounded thresholdrank), which via subspace enumeration lets us enumerate small cuts efficiently.

Approximately counting independent sets in dense bipartite graphs via subspace enumeration