Parallel algorithms for finding a suboptimal fundamental-cycle set in a graph

An NP-complete problem of finding a fundamental-cycle set of a graph G with minimum total length is considered. Two parallel algorithms of O(n²/p + n log n log p) and O(m + n²/p + n log(n/p) + n log p) costs to find a suboptimal solution to this problem are presented (p is a number of processors, n is a number of vertices, and m is a number of edges of G). The algorithms partition an edge and vertex set of G among processors, respectively, and use a new heuristic method to solve the problem. A message-based tree-connected MIMD computer is assumed as a model of parallel computations. The algorithms were implemented for a binary tree of 15 transputers, and the experiments were conducted on a wide range of random graphs. The results show that the vertex set partition algorithm with inferior theoretical cost gives better speedups and finds the fundamental-cycle sets of shorter total lenghts as compared to the edge set partition algorithm.