Detecting short directed cycles using rectangular matrix multiplication and dynamic programming

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^cklog^k+1 V). For C₃, 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 C₄ we get c₄ = (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 C₅ we get c₅ = 3ω/(ω + 2) < 1.63. The obtained running time of O(E ^3ω/(ω+2) log⁶ 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.