Minimum Cycle and Homology Bases of Surface Embedded Graphs

We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the 1-dimensional (Z2)-homology classes) of an undirected graph embedded on an orientable surface of genus g. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the 1-skeleton of any graph is exactly its minimum cycle basis. For the minimum cycle basis problem, we give a deterministic O(n + 22gn2)-time algorithm. The best known existing algorithms for surface embedded graphs are those for general sparse graphs: an O(n) time Monte Carlo algorithm [2] and a deterministic O(n3) time algorithm [27]. For the minimum homology basis problem, we give an O(g3n logn)-time algorithm, improving on existing algorithms for many values of g and n. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems