Chicago Journal of Theoretical Computer Science

Volume 2024

Article 1

Published by the Department of Computer Science, The University of Chicago.

Efficient algorithms for the Potts model on small-set expanders

Charlie Carlson
Department of Computer Science
University of California Santa Barbara
Santa Barbara, CA
acr AT cse DOT uconn DOT edu

Evan Davies
Department of Computer Science
Colorado State University
Fort Collins, CO
Ewan DOT Davies AT colostate DOT edu


Alexandra Kolla
Department of Computer Science and Engineering
University of California Santa Cruz
Santa Cruz, CA
akolla AT ucsc DOT edu

March 22, 2024


An emerging trend in approximate counting is to show that certain `low-temperature' problems are easy on typical instances, despite worst-case hardness results. For the class of regular graphs one usually shows that expansion can be exploited algorithmically, and since random regular graphs are good expanders with high probability the problem is typically tractable. Inspired by approaches used in subexponential-time algorithms for Unique Games, we develop an approximation algorithm for the partition function of the ferromagnetic Potts model on graphs with a small-set expansion condition. In such graphs it may not suffice to explore the state space of the model close to ground states, and a novel feature of our method is to efficiently find a larger set of `pseudo-ground states' such that it is enough to explore the model around each pseudo-ground state.

Submitted November 23, 2021, revised March 18, 2024, published March 22, 2024.

DOI: 10.4086/cjtcs.2024.001

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