Efficient algorithms for clique problems

  • Authors:

  • Virginia Vassilevska

  • Affiliations:

  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA

  • Venue:
  • Information Processing Letters
  • Year:
  • 2009

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Abstract

The k-clique problem is a cornerstone of NP-completeness and parametrized complexity. When k is a fixed constant, the asymptotically fastest known algorithm for finding a k-clique in an n-node graph runs in O(n^0^.^7^9^2^k) time (given by Nesetril and Poljak). However, this algorithm is infamously inapplicable, as it relies on Coppersmith and Winograd's fast matrix multiplication. We present good combinatorial algorithms for solving k-clique problems. These algorithms do not require large constants in their runtime, they can be readily implemented in any reasonable random access model, and are very space-efficient compared to their algebraic counterparts. Our results are the following:*We give an algorithm for k-clique that runs in O(n^k/(@elogn)^k^-^1) time and O(n^@e) space, for all @e0, on graphs with n nodes. This is the first algorithm to take o(n^k) time and O(n^c) space for c independent of k. *Let k be even. Define a k-semiclique to be a k-node graph G that can be divided into two disjoint subgraphs U={u"1,...,u"k"/"2} and V={v"1,...,v"k"/"2} such that U and V are cliques, and for all i=