Bulatov, AA, Dyer, ME, Goldberg, LA et al. (1 more author) (2012) Log-supermodular functions, functional clones and counting CSPs. In: Dürr, C and Wilke, T, (eds.) 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Symposium on Theoretical Aspects of Computer Science (STACS), 29 Feb - 03 Mar 2012, Paris, France. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik , 302 - 313. ISBN 978-3-939897-35-4
Abstract
Motivated by a desire to understand the computational complexity of counting constraint satisfaction problems (counting CSPs), particularly the complexity of approximation, we study functional clones of functions on the Boolean domain, which are analogous to the familiar relational clones constituting Post's lattice. One of these clones is the collection of log-supermodular (lsm) functions, which turns out to play a significant role in classifying counting CSPs. In our study, we assume that non-negative unary functions (weights) are available. Given this, we prove that there are no functional clones lying strictly between the clone of lsm functions and the total clone (containing all functions). Thus, any counting CSP that contains a single nontrivial non-lsm function is computationally as hard as any problem in #P. Furthermore, any non-trivial functional clone (in a sense that will be made precise below) contains the binary function "implies". As a consequence, all non-trivial counting CSPs (with non-negative unary weights assumed to be available) are computationally at least as difficult as #BIS, the problem of counting independent sets in a bipartite graph. There is empirical evidence that #BIS is hard to solve, even approximately.
Metadata
Item Type: | Proceedings Paper |
---|---|
Authors/Creators: |
|
Editors: |
|
Copyright, Publisher and Additional Information: | © A.A. Bulatov, M. Dyer, L.A. Goldberg and M. Jerrum; This is an open-access article licensed under Creative Commons License NC-ND, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. |
Dates: |
|
Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) > Institute for Computational and Systems Science (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 13 Sep 2013 08:15 |
Last Modified: | 19 Dec 2022 13:25 |
Published Version: | http://dx.doi.org/10.4230/LIPIcs.STACS.2012.302 |
Status: | Published |
Publisher: | Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik |
Identification Number: | 10.4230/LIPIcs.STACS.2012.302 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:76324 |