Compact Representations of Cuts

Consider the ${n}\choose{2}$ (or $O(n^{2})$) min-cut problems on a graph with $n$ nodes and nonnegative edge weights. Gomory and Hu [J. Soc. Indust. Appl. Math., 9 (1961), pp. 551--570] showed (essentially) that there are at most n-1 different min-cuts. They also described a compact structure (the flow equivalent tree) of size O(n) with the following property: for any pair of nodes, the value of a min-cut can be obtained from this structure. Furthermore, they showed how this structure can be found by solving only n-1 min-cut problems. This paper contains generalizations of these results. For example, consider a k-terminal cut problem on a graph: for a given set of k nodes, delete a minimum weight set of edges (called a k-cut) so that each of the k nodes is in a different component. There are ${n}\choose{k}$ (or O(nk)) such problems. Hassin [Math. Oper. Res., 13 (1988), pp. 535--542] showed (essentially) that there are at most ${n-1}\choose{k-1}$ (or O(nk-1)) different min k-cuts. We describe a compact structure of size O(nk-1) with the following property: for any k nodes, the value of a min k-cut can be obtained from this structure. We also show how this structure can be found by solving only ${n-1}\choose{k-1}$ k-terminal cut problems. This work builds upon the results of Hassin [Math. Oper. Res., 13 (1988), pp. 535--542], [Oper. Res. Lett., 9 (1990), pp. 315--318], and [Lecture Notes in Comput. Sci. 450, 1990, pp. 228--299].