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The complexity of choosing an H-coloring (nearly) uniformly at random
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Goldberg, Leslie Ann, Kelk, Steven and Paterson, Mike (2004) The complexity of choosing an H-coloring (nearly) uniformly at random. SIAM Journal on Computing, Volume 33 (Number 2). pp. 416-432. doi:10.1137/S0097539702408363 ISSN 0097-5397.
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Official URL: http://dx.doi.org/10.1137/S0097539702408363
Abstract
Cooper, Dyer, and Frieze [J. Algorithms, 39 (2001), pp. 117-134] studied the problem of sampling H-colorings (nearly) uniformly at random. Special cases of this problem include sampling colorings and independent sets and sampling from statistical physics models such as the Widom-Rowlinson model, the Beach model, the Potts model and the hard-core lattice gas model. Cooper et al. considered the family of "cautious" ergodic Markov chains with uniform stationary distribution and showed that, for every fixed connected "nontrivial" graph H, every such chain mixes slowly. In this paper, we give a complexity result for the problem. Namely, we show that for any fixed graph H with no trivial components, there is unlikely to be any polynomial almost uniform sampler (PAUS) for H-colorings. We show that if there were a PAUS for the H-coloring problem, there would also be a PAUS for sampling independent sets in bipartite graphs, and, by the self-reducibility of the latter problem, there would be a fully polynomial randomized approximation scheme (FPRAS) for #BIS-the problem of counting independent sets in bipartite graphs. Dyer, Goldberg, Greenhill, and Jerrum have shown that #BIS is complete in a certain logically defined complexity class. Thus, a PAUS for sampling H-colorings would give an FPRAS for the entire complexity class. In order to achieve our result we introduce the new notion of sampling-preserving reduction which seems to be more useful in certain settings than approximation-preserving reduction.
| Item Type: | Journal Article | ||||
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| Alternative Title: | |||||
| Subjects: | Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software Q Science > QA Mathematics |
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| Divisions: | Faculty of Science, Engineering and Medicine > Science > Computer Science | ||||
| Journal or Publication Title: | SIAM Journal on Computing | ||||
| Publisher: | Society for Industrial and Applied Mathematics | ||||
| ISSN: | 0097-5397 | ||||
| Official Date: | 2004 | ||||
| Dates: |
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| Volume: | Volume 33 | ||||
| Number: | Number 2 | ||||
| Number of Pages: | 17 | ||||
| Page Range: | pp. 416-432 | ||||
| DOI: | 10.1137/S0097539702408363 | ||||
| Status: | Not Peer Reviewed | ||||
| Publication Status: | Published |
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