Extracting maximal information about sets of minimum cuts

Abstract

There are two well-known, elegant, compact, and efficiently computed representations of selected minimum edge cuts in a weighted, undirected graphG=(V, E) withn nodes andm edges: at one extreme, the Gomory-Hu cut tree [12] represents$\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)$ minimum cuts, one for each pair of nodes inG; at the other extreme, the Picard-Queyranne DAG [24] represents all the minimum cuts between a single pair of nodes inG. The GH cut tree is constructed with onlyn−1 max-flow computations, and the PQ DAG is constructed with one max-flow computation, plusO(m) additional time. In this paper we show how to marry these two representations, getting the best features of both. We first show that we can construct all$\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)$ DAGs, one for each fixed pair of nodes, using onlyn−1 max-flow computations as in [12], plusO(m) time per DAG as in [24]. This speeds up the obvious approach by a factor ofn. We then apply this approach to an unweighted graphG, to find all the edge-connectivity cuts inG, i.e., cuts with capacity equal to the connectivity ofG. Matula [22] gave a method to find one connectivity cut inO(nm) time; we show thatO(nm) time suffices to represent all connectivity cuts compactly, and to list all of them explicitly. This improves the previous best time bound ofO(n ² m) [3] for listing the connectivity cuts. The connectivity cuts are central in network reliability calculations. We then show how to find all pairs of nodes that are separated by at least one connectivity cut inO(nm) time.