Markov Random Fields, Homomorphism Counting, and Sidorenko’s Conjecture
IEEE Transactions on Information Theory(2022)
摘要
Graph covers and the Bethe free energy (BFE) have been useful theoretical tools for producing lower bounds on a variety of counting problems in graphical models, including the permanent and the ferromagnetic Ising model. Here, we investigate weighted homomorphism counting problems over bipartite graphs that are related to a conjecture of Sidorenko. We show that the BFE does yield a lower bound in a variety of natural settings, and when it does yield a lower bound, it necessarily improves upon the lower bound conjectured by Sidorenko. Conversely, we show that there exist bipartite graphs for which the BFE does not yield a lower bound on the homomorphism number. Finally, we use the characterizations developed as part of this work to provide a simple proof of Sidorenko’s conjecture in a number of special cases.
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关键词
Bethe free energy,graph covers,homomorphism counting,Markov random fields,Sidorenko’s conjecture
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