## Abstract

We present several new algorithms for detecting short fixed length cycles in digraphs. The new algorithms utilize fast rectangular matrix multiplication algorithms together with a dynamic programming approach similar to the one used in the solution of the classical chain matrix product problem. The new algorithms are instantiations of a generic algorithm that we present for finding a directed C_{k}, i.e., a directed cycle of length k, in a digraph, for any fixed k ≥ 3. This algorithm partitions the prospective C_{k}'s in the input digraph G = (V, E) into O(log^{k} V) classes, according to the degrees of their vertices. For each cycle class we determine, in O(E ^{ck} log V) time, whether G contains a C_{k} from that class, where c_{k}, = c_{k},(ω) is a constant that depends only on ω, the exponent of square matrix multiplication. The search for cycles from a given class is guided by the solution of a small dynamic programming problem. The total running time of the obtained deterministic algorithm is therefore O(E^{ck}log^{k+1} V). For C_{3}, we get c _{3} = 2ω/(ω + 1) < 1.41 where ω < 2.376 is the exponent of square matrix multiplication. This coincides with an existing algorithm of [AYZ97]. For C_{4} we get c_{4} = (4ω - 1)/(2ω + 1) < 1.48. We can dispense, in this case, of the polylogarithmic factor and get an O(E^{(4ω-1)/(2ω,+1)}) = o(E^{1.48}) time algorithm. This improves upon an O(E^{3/2}) time algorithm of [AYZ97]. For C_{5} we get c_{5} = 3ω/(ω + 2) < 1.63. The obtained running time of O(E ^{3ω/(ω+2)} log^{6} V) = o(E^{1.63}) improves upon an O(E^{5/3}) time algorithm of [AYZ97]. Determining c_{k} for k ≥ 6 is a difficult task. We conjecture that c_{k} = (k + 1)ω/(2ω + k - 1), for every odd k. The values of c _{k}, for even k ≥ 6 seem to exhibit a much more complicated dependence on ω.

Original language | English |
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Pages | 247-253 |

Number of pages | 7 |

State | Published - 2004 |

Event | Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms - New Orleans, LA., United States Duration: 11 Jan 2004 → 13 Jan 2004 |

### Conference

Conference | Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms |
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Country/Territory | United States |

City | New Orleans, LA. |

Period | 11/01/04 → 13/01/04 |