Finding a heaviest vertex-weighted triangle is not harder than matrix multiplication
(2009) In SIAM Journal on Computing 39(2). p.431-444- 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... (More)
- 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. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1477508
- author
- Czumaj, Artur and Lingas, Andrzej LU
- organization
- publishing date
- 2009
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- time complexity, graph algorithms, triangle, matrix multiplication, vertex-weighted graph, graph
- in
- SIAM Journal on Computing
- volume
- 39
- issue
- 2
- pages
- 431 - 444
- publisher
- Society for Industrial and Applied Mathematics
- external identifiers
-
- wos:000268859000005
- scopus:67650112691
- ISSN
- 0097-5397
- DOI
- 10.1137/070695149
- project
- VR 2008-4649
- language
- English
- LU publication?
- yes
- id
- c71618dd-556b-4d9b-95a4-b6b5d6ff9b62 (old id 1477508)
- date added to LUP
- 2016-04-01 13:34:50
- date last changed
- 2022-01-27 19:58:27
@article{c71618dd-556b-4d9b-95a4-b6b5d6ff9b62,
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.}},
author = {{Czumaj, Artur and Lingas, Andrzej}},
issn = {{0097-5397}},
keywords = {{time complexity; graph algorithms; triangle; matrix multiplication; vertex-weighted graph; graph}},
language = {{eng}},
number = {{2}},
pages = {{431--444}},
publisher = {{Society for Industrial and Applied Mathematics}},
series = {{SIAM Journal on Computing}},
title = {{Finding a heaviest vertex-weighted triangle is not harder than matrix multiplication}},
url = {{http://dx.doi.org/10.1137/070695149}},
doi = {{10.1137/070695149}},
volume = {{39}},
year = {{2009}},
}