Improved bounds for randomly sampling colorings via linear programming
Name
1810.12980.pdf
Description
Submitted version
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524.43 KB
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27471581de66575ab4a2914cdd83bd50
Author(s)
Moitra, Ankur
Chen, Sitan
Date Issued
2019
Citation
Moitra, Ankur and Chen, Sitan. 2019. "Improved bounds for randomly sampling colorings via linear programming."
Version
Original manuscript
Abstract
Copyright © 2019 by SIAM. A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of k-colorings of a graph G on n vertices with maximum degree ∆ is rapidly mixing for k ≥ ∆ + 2. In FOCS 1999, Vigoda [43] showed that the flip dynamics (and therefore also Glauber dynamics) is rapidly mixing for any k > 116 ∆. It turns out that there is a natural barrier at 116 , below which there is no one-step coupling that is contractive with respect to the Hamming metric, even for the flip dynamics. We use linear programming and duality arguments to fully characterize the obstructions to going beyond 116 . These extremal configurations turn out to be quite brittle, and in this paper we use this to give two proofs that the Glauber dynamics is rapidly mixing for any k ≥ (116 − 0)∆ for some absolute constant 0 > 0. This is the first improvement to Vigoda’s result that holds for general graphs. Our first approach analyzes a variable-length coupling in which these configurations break apart with high probability before the coupling terminates, and our other approach analyzes a one-step path coupling with a new metric that counts the extremal configurations. Additionally, our results extend to list coloring, a widely studied generalization of coloring, where the previously best known results required k > 2∆.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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DOI of Published Version
https://dl.acm.org/citation.cfm?id=3310569
