On the negative cost girth problem in planar networks

In this paper, we discuss an efficient divide-and-conquer algorithm for the negative cost girth (NCG) problem in planar networks. Recall that the girth of an unweighted graph (directed or undirected) is the length of the shortest cycle in the graph. We extend the notion of girth to arbitrarily weighted networks as follows: The negative cost girth of a graph is the number of edges in a shortest negative cost simple cycle, i.e., a negative cost cycle having the fewest number of edges. The NCG problem in general networks has been well-studied, and there exist several algorithms for this problem. Clearly, the extant algorithms for the NCG problem can be used when the input network is restricted to be planar. However, the techniques used in this paper result in an algorithm with a superior running time. Our algorithm is based on the well-known Lipton–Tarjan planar separator theorem. On a directed, weighted, planar network G=〈V,E,c〉G=〈V,E,c〉 with n   vertices, m=O(n)m=O(n) edges, and negative cost girth k  , the algorithm runs in O(n1.5⋅k)O(n1.5⋅k) time. This is a significant improvement over the O(m⋅n⋅k)=O(n2⋅k)O(m⋅n⋅k)=O(n2⋅k) running time that results, when the fastest known topology-oblivious algorithm for the NCG problem is restricted to planar networks. Additionally, our algorithm can be extended to find the NCG in selected generalizations of planar networks.