Dyer, M, Goldberg, LA and Richerby, D (2016) Counting 4×4 matrix partitions of graphs. Discrete Applied Mathematics, 213. pp. 76-92. ISSN 0166-218X
Abstract
Given a symmetric matrix M ∈ {0, 1, ∗}D×D, an M-partition of a graph G is a function from V (G) to D such that no edge of G is mapped to a 0 of M and no non-edge to a 1. We give a computer-assisted proof that, when |D| = 4, the problem of counting the M-partitions of an input graph is either in FP or is #P-complete. Tractability is proved by reduction to the related problem of counting list M-partitions; intractability is shown using a gadget construction and interpolation. We use a computer program to determine which of the two cases holds for all but a small number of matrices, which we resolve manually to establish the dichotomy. We conjecture that the dichotomy also holds for |D| > 4. More specifically, we conjecture that, for any symmetric matrix M ∈ {0, 1, ∗}D×D, the complexity of counting M-partitions is the same as the related problem of counting list M-partitions.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2016 Elsevier B.V. This is an author produced version of a paper published in Discrete Applied Mathematics. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Matrix partitions of graphs; Complexity of counting; Complexity dichotomy; #P-completeness |
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) |
Depositing User: | Symplectic Publications |
Date Deposited: | 13 Jan 2017 10:37 |
Last Modified: | 18 Jul 2017 06:45 |
Published Version: | https://doi.org/10.1016/j.dam.2016.05.001 |
Status: | Published |
Publisher: | Elsevier |
Identification Number: | 10.1016/j.dam.2016.05.001 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:110521 |