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Finding a heaviest triangle is not harder than matrix multiplication

Czumaj, Artur and Lingas, Andrzej LU (2007) Symposium on Discrete Algorithms, 2007 In Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms p.986-994
Abstract
We show that for any ε > 0, a maximum-weight triangle in an undirected graph with n vertices and real weights assigned to vertices can be found in time O(nω + n2+ε), where ω is the exponent of fastest matrix multiplication algorithm. By the currently best bound on ω, the running time of our algorithm is O(n2.376). Our algorithm substantially improves the previous time-bounds for this problem recently established by Vassilevska et al. (STOC 2006, O(n2.688)) and (ICALP 2006, O(n2.575)). Its asymptotic time complexity matches that of the fastest known algorithm for finding a triangle (not necessarily a maximum-weight one) in a graph.



By applying or extending our algorithm, we can also improve the upper bounds on finding... (More)
We show that for any ε > 0, a maximum-weight triangle in an undirected graph with n vertices and real weights assigned to vertices can be found in time O(nω + n2+ε), where ω is the exponent of fastest matrix multiplication algorithm. By the currently best bound on ω, the running time of our algorithm is O(n2.376). Our algorithm substantially improves the previous time-bounds for this problem recently established by Vassilevska et al. (STOC 2006, O(n2.688)) and (ICALP 2006, O(n2.575)). Its asymptotic time complexity matches that of the fastest known algorithm for finding a triangle (not necessarily a maximum-weight one) in a graph.



By applying or extending our algorithm, we can also 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 established in the papers by Vassilevska et al. For example, 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(m1.41). (Less)
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author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
in
Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
pages
986 - 994
publisher
Society for Industrial and Applied Mathematics
conference name
Symposium on Discrete Algorithms, 2007
external identifiers
  • Scopus:84969247244
ISBN
978-0-898716-24-5
project
VR 2005-4085
language
English
LU publication?
yes
id
33b04fb8-7727-4261-bba3-101a82d8404f (old id 629534)
alternative location
http://portal.acm.org/citation.cfm?id=1283489
date added to LUP
2007-11-27 15:06:11
date last changed
2016-10-13 04:43:20
@misc{33b04fb8-7727-4261-bba3-101a82d8404f,
  abstract     = {We show that for any ε &gt; 0, a maximum-weight triangle in an undirected graph with n vertices and real weights assigned to vertices can be found in time O(nω + n2+ε), where ω is the exponent of fastest matrix multiplication algorithm. By the currently best bound on ω, the running time of our algorithm is O(n2.376). Our algorithm substantially improves the previous time-bounds for this problem recently established by Vassilevska et al. (STOC 2006, O(n2.688)) and (ICALP 2006, O(n2.575)). Its asymptotic time complexity matches that of the fastest known algorithm for finding a triangle (not necessarily a maximum-weight one) in a graph.<br/><br>
<br/><br>
By applying or extending our algorithm, we can also 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 established in the papers by Vassilevska et al. For example, 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(m1.41).},
  author       = {Czumaj, Artur and Lingas, Andrzej},
  isbn         = {978-0-898716-24-5},
  language     = {eng},
  pages        = {986--994},
  publisher    = {ARRAY(0xa5e1bd8)},
  series       = {Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms},
  title        = {Finding a heaviest triangle is not harder than matrix multiplication},
  year         = {2007},
}