Minimum mean cycle problem in bidirected and skew-symmetric graphs

The problem of finding, in an edge-weighted bidirected   graph G=(V,E)G=(V,E), a cycle whose mean edge weight is minimum generalizes similar problems for both directed and undirected graphs. (The problem is considered in two variants: for the cycles without repeated edges and for the cycles without repeated nodes.) We develop an algorithm to solve this problem in O(V2min{V2,ElogV})O(V2min{V2,ElogV}) time (to compare: the complexity of an improved version of Barahona’s algorithm for undirected cycles is O(V4)O(V4)). Our algorithm is based on a certain general approach to minimum mean problems and uses, as a subroutine, Gabow’s algorithm for the minimum weight 2-factor problem in a graph. The problem admits a reformulation in terms of regular cycles in a skew-symmetric graph.