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Finding the Smallest H-Subgraph in Real Weighted Graphs and Related Problems

  • Virginia Vassilevska
  • Ryan Williams
  • Raphael Yuster
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4051)

Abstract

Let G be a graph with real weights assigned to the vertices (edges). The weight of a subgraph of G is the sum of the weights of its vertices (edges). The MIN H-SUBGRAPH problem is to find a minimum weight subgraph isomorphic to H, if one exists. Our main results are new algorithms for the MIN H-SUBGRAPH problem. The only operations we allow on real numbers are additions and comparisons. Our algorithms are based, in part, on fast matrix multiplication.

For vertex-weighted graphs with n vertices we obtain the following results. We present an O(n t(ω,h)) time algorithm for MIN H-SUBGRAPH in case H is a fixed graph with h vertices and ω< 2.376 is the exponent of matrix multiplication. The value of t(ω,h) is determined by solving a small integer program. In particular, the smallest triangle can be found in O(n 2 + 1/(4 − ω)) ≤o(n 2.616) time, the smallest K 4 in O(n ω + 1) time, the smallest K 7 in O(n 4 + 3/(4 − ω)) time. As h grows, t(ω,h) converges to 3h/(6-ω) < 0.828h. Interestingly, only for h = 4,5,8 the running time of our algorithm essentially matches that of the (unweighted) H-subgraph detection problem. Already for triangles, our results improve upon the main result of [VW06]. Using rectangular matrix multiplication, the value of t(ω,h) can be improved; for example, the runtime for triangles becomes O(n 2.575). We also present an algorithm whose running time is a function of m, the number of edges. In particular, the smallest triangle can be found in O(m (18 − 4ω)/(13 − 3ω)) ≤o(m 1.45) time.

For edge-weighted graphs we present an O(m 2 − 1/k logn) time algorithm that finds the smallest cycle of length 2k or 2k-1. This running time is identical, up to a logarithmic factor, to the running time of the algorithm of Alon et al. for the unweighted case. Using the color coding method and a recent algorithm of Chan for distance products, we obtain an O(n 3/logn) time randomized algorithm for finding the smallest cycle of any fixed length.

Keywords

Minimum Weight Logarithmic Factor Small Triangle Distance Product Fast Matrix Multiplication 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Virginia Vassilevska
    • 1
  • Ryan Williams
    • 1
  • Raphael Yuster
    • 2
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburgh
  2. 2.Department of MathematicsUniversity of HaifaHaifaIsrael

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