Minimal Forbidden Subgraphs of Unimodular Multicommodity Networks

Abstract

An integral matrix with independent columns is called unimodular if the greatest common divisor of the determinants of its maximum-rank square submatrices is one. An integral matrix A (not necessarily having independent columns) is defined to be unimodular if every basis of A has that property. It is known that this is so if and only if the extreme points of the set of nonnegative vectors x satisfying Ax = b are integral for all integral vectors b. Here the constraint matrix of a k-commodity network flow problem is shown to be unimodular if and only if the associated (undirected) graph does not have a subgraph homeomorphic to the following graphs: (i) the complete graph on four nodes, and if k ≥ 3, (ii) a cycle with three additional edges, each a duplicate of a different edge of the cycle, having no node with degree 2 (these graphs have from three to six nodes). The following algorithm checks whether subgraphs of the above type are present. First find the 2-connected subgraphs that contain at least two edges. Then reduce each such subgraph to a graph with fewer nodes and arcs. The forbidden subgraphs are not present if and only if the algorithm reduces each 2-connected subgraph to a cycle with two edges. When this is not possible, a forbidden subgraph is produced. Suitable implementation of the algorithm results in O(n) complexity, where n is the number of edges of the original graph.