NDLI: Solving maximum clique in sparse graphs: an $$O(nm+n2^{d/4})$$ algorithm for $$d$$ -degenerate graphs

Content Provider	Springer Nature Link
Author	Buchanan, Austin Walteros, Jose L. Butenko, Sergiy Pardalos, Pas M.
Copyright Year	2013
Abstract	We describe an algorithm for the maximum clique problem that is parameterized by the graph’s degeneracy $$d$$ . The algorithm runs in $$O\left( nm+n T_d \right) $$ time, where $$T_d$$ is the time to solve the maximum clique problem in an arbitrary graph on $$d$$ vertices. The best bound as of now is $$T_d=O(2^{d/4})$$ by Robson. This shows that the maximum clique problem is solvable in $$O(nm)$$ time in graphs for which $$d \le 4 \log _2 m + O(1)$$ . The analysis of the algorithm’s runtime is simple; the algorithm is easy to implement when given a subroutine for solving maximum clique in small graphs; it is easy to parallelize. In the case of Bianconi-Marsili power-law random graphs, it runs in $$2^{O(\sqrt{n})}$$ time with high probability. We extend the approach for a graph invariant based on common neighbors, generating a second algorithm that has a smaller exponent at the cost of a larger polynomial factor.
Starting Page	1611
Ending Page	1617
Page Count	7
File Format	PDF
ISSN	18624472
Journal	Optimization Letters
Volume Number	8
Issue Number	5
e-ISSN	18624480
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2013-10-18
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Maximum clique Degeneracy Sparse graphs Fixed-parameter tractability $$d$$ -Degenerate graphs Optimization Operations Research/Decision Theory Computational Intelligence Numerical and Computational Physics
Content Type	Text
Resource Type	Article
Subject	Control and Optimization Business, Management and Accounting

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