Minimum mean cycle problem in bidirected and skew-symmetric graphs

Authors:
Maxim A. Babenko;Alexander V. Karzanov
Affiliations:
Department of Mechanics and Mathematics, Moscow State University, Vorob'yovy Gory, 119899 Moscow, Russia;Institute for System Analysis, 9, Prospect 60 Let Oktyabrya, 117312 Moscow, Russia
Venue:
Discrete Optimization
Year:
2009

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Abstract

The problem of finding, in an edge-weighted bidirected graph 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(V^2min{V^2,ElogV}) time (to compare: the complexity of an improved version of Barahona's algorithm for undirected cycles is O(V^4)). 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.