## 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(w,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} log n) 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}/log n) time randomized algorithm for finding the smallest cycle of any fixed length.

Original language | English |
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Title of host publication | Automata, Languages and Programming - 33rd International Colloquium, ICALP 2006, Proceedings |

Publisher | Springer Verlag |

Pages | 262-273 |

Number of pages | 12 |

ISBN (Print) | 3540359044, 9783540359043 |

DOIs | |

State | Published - 2006 |

Event | 33rd International Colloquium on Automata, Languages and Programming, ICALP 2006 - Venice, Italy Duration: 10 Jul 2006 → 14 Jul 2006 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 4051 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 33rd International Colloquium on Automata, Languages and Programming, ICALP 2006 |
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Country/Territory | Italy |

City | Venice |

Period | 10/07/06 → 14/07/06 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science