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Finding a heaviest vertex-weighted triangle is not harder than matrix multiplication
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Czumaj, Artur and Lingas, Andrzej. (2009) Finding a heaviest vertex-weighted triangle is not harder than matrix multiplication. SIAM Journal on Computing, Vol.39 (No.2). pp. 431-444. ISSN 0097-5397
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Official URL: http://dx.doi.org/10.1137/070695149
Abstract
We show that a maximum-weight triangle in an undirected graph with n vertices and real weights assigned to vertices can be found in time O(n(omega) + n(2+o(1))), where omega is the exponent of the fastest matrix multiplication algorithm. By the currently best bound on omega, the running time of our algorithm is O(n(2.376)). Our algorithm substantially improves the previous time-bounds for this problem, and its asymptotic time complexity matches that of the fastest known algorithm for finding any triangle (not necessarily a maximum-weight one) in a graph. We can extend our algorithm to improve the upper bounds on finding a maximum-weight triangle in a sparse graph and on finding a maximum-weight subgraph isomorphic to a fixed graph. We can find a maximum-weight triangle in a vertex-weighted graph with m edges in asymptotic time required by the fastest algorithm for finding any triangle in a graph with m edges, i.e., in time O(m(1.41)). Our algorithms for a maximum-weight fixed subgraph (in particular any clique of constant size) are asymptotically as fast as the fastest known algorithms for a fixed subgraph.
| Item Type: | Journal Article |
|---|---|
| Subjects: | Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software Q Science > QA Mathematics |
| Divisions: | Faculty of Science > Computer Science |
| Library of Congress Subject Headings (LCSH): | Graph algorithms, Triangle |
| Journal or Publication Title: | SIAM Journal on Computing |
| Publisher: | Society for Industrial and Applied Mathematics |
| ISSN: | 0097-5397 |
| Date: | June 2009 |
| Volume: | Vol.39 |
| Number: | No.2 |
| Number of Pages: | 14 |
| Page Range: | pp. 431-444 |
| Identification Number: | 10.1137/070695149 |
| Status: | Peer Reviewed |
| Publication Status: | Published |
| Access rights to Published version: | Restricted or Subscription Access |
| Funder: | National Science Foundation (U.S.) (NSF), Engineering and Physical Sciences Research Council (EPSRC), University of Warwick |
| Grant number: | CCR-0313219 (NSF), EP/D063191/1 (EPSRC), 621-2005-4085 (VR) |
| URI: | http://wrap.warwick.ac.uk/id/eprint/17465 |
Data sourced from Thomson Reuters' Web of Knowledge
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