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- W4288103432 abstract "We present a randomized algorithm that takes as input an undirected $n$-vertex graph $G$ with maximum degree $Delta$ and an integer $k > 3Delta$, and returns a random proper $k$-coloring of $G$. The distribution of the coloring is emph{perfectly} uniform over the set of all proper $k$-colorings; the expected running time of the algorithm is $mathrm{poly}(k,n)=widetilde{O}(nDelta^2cdot log(k))$. This improves upon a result of Huber~(STOC 1998) who obtained a polynomial time perfect sampling algorithm for $k>Delta^2+2Delta$. Prior to our work, no algorithm with expected running time $mathrm{poly}(k,n)$ was known to guarantee perfectly sampling with sub-quadratic number of colors in general. Our algorithm (like several other perfect sampling algorithms including Huber's) is based on the Coupling from the Past method. Inspired by the emph{bounding chain} approach, pioneered independently by Huber~(STOC 1998) and Haggstrom & Nelander~(Scand.{} J.{} Statist., 1999), we employ a novel bounding chain to derive our result for the graph coloring problem." @default.
- W4288103432 created "2022-07-28" @default.
- W4288103432 creator A5031186823 @default.
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- W4288103432 date "2019-09-23" @default.
- W4288103432 modified "2023-09-24" @default.
- W4288103432 title "Improved Bounds for Perfect Sampling of $k$-Colorings in Graphs" @default.
- W4288103432 doi "https://doi.org/10.48550/arxiv.1909.10323" @default.
- W4288103432 hasPublicationYear "2019" @default.
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