Dyer, M orcid.org/0000-0002-2018-0374, Jerrum, M, Müller, H orcid.org/0000-0002-1123-1774 et al. (1 more author) (2021) Counting Weighted Independent Sets beyond the Permanent. SIAM Journal on Discrete Mathematics, 35 (2). pp. 1503-1524. ISSN 0895-4801
Abstract
Jerrum, Sinclair, and Vigoda [J. ACM, 51 (2004), pp. 671--697] showed that the permanent of any square matrix can be estimated in polynomial time. This computation can be viewed as approximating the partition function of edge-weighted matchings in a bipartite graph. Equivalently, this may be viewed as approximating the partition function of vertex-weighted independent sets in the line graph of a bipartite graph. Line graphs of bipartite graphs are perfect graphs and are known to be precisely the class of (claw, diamond, odd hole)-free graphs. So how far does the result of Jerrum, Sinclair, and Vigoda extend? We first show that it extends to (claw, odd hole)-free graphs, and then show that it extends to the even larger class of (fork, odd hole)-free graphs. Our techniques are based on graph decompositions, which have been the focus of much recent work in structural graph theory, and on structural results of Chvátal and Sbihi [J. Combin. Theory Ser. B, 44 (1988)], Maffray and Reed [J. Combin. Theory Ser. B, 75 (1999)], and Lozin and Milanič [J. Discrete Algorithms, 6 (2008), pp. 595--604].
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2021, Society for Industrial and Applied Mathematics. This is an author produced version of an article published in SIAM Journal on Discrete Mathematics. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | independent set, counting, randomized algorithm, fully polynomial randomized approximation scheme, FPRAS, claw-free graph, fork-free graph, decomposition |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) |
Funding Information: | Funder Grant number EPSRC (Engineering and Physical Sciences Research Council) EP/N019660/1 EPSRC (Engineering and Physical Sciences Research Council) EP/S016562/1 |
Depositing User: | Symplectic Publications |
Date Deposited: | 24 Mar 2021 15:18 |
Last Modified: | 23 Jul 2021 15:00 |
Status: | Published |
Publisher: | Society for Industrial and Applied Mathematics |
Identification Number: | 10.1137/20M1347747 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:172466 |