Finding, minimizing, and counting weighted subgraphs

  • Authors:

  • Virginia Vassilevska;Ryan Williams

  • Affiliations:

  • Institute for Advanced Study, Princeton, NJ, USA;Institute for Advanced Study, Princeton, NJ, USA

  • Venue:
  • Proceedings of the forty-first annual ACM symposium on Theory of computing
  • Year:
  • 2009

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Abstract

For a pattern graph H on k nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of H in a given large graph G on n nodes, as well as finding minimum weight copies in both node-weighted and edge-weighted graphs. Our results include: The number of copies of an H with an independent set of size s can be computed exactly in O*(2s nk-s+3) time. A minimum weight copy of such an H (with arbitrary real weights on nodes and edges) can be found in O(4s+o(s) nk-s+3) time. (The O* notation omits (k) factors.) These algorithms rely on fast algorithms for computing the permanent of a k x n matrix, over rings and semirings. The number of copies of any H having minimum (or maximum) node-weight (with arbitrary real weights on nodes) can be found in O(nω k/3 + n2k/3+o(1)) time, where ω edge-weighted triangle of weight exactly 0 in general graphs requires Ω(n2.5-ε) time for all ε 0, unless the 3SUM problem on N numbers can be solved in O(N2 - ε) time. This suggests that the edge-weighted problem is much harder than its node-weighted version.