Maximum concurrent flows and minimum cuts

In many applications, we need to find a minimum cost partition of a network separating a given pair of nodes. A classical example is the Max-Flow Min-Cut Theorem, where the cost of the partition is defined to be the sum of capacities of arcs connecting the two parts. Other similar concepts such as minimum weighted sparsest cut and flux cut have also been introduced. There is always a cost associated with a cut, and we always seek the min-cost cut separating a given pair of nodes. A natural generalization from the separation of a given pair is to find all minimum cost cuts separating all \(\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\) pairs of nodes, with arbitrary costs associated with all 2^n?1 — 1 cuts. In the present paper, we show thatn — 1 minimum cost cuts are always sufficient to separate all \(\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\) pairs of nodes. A further generalization is to considerk-way partitions rather than two-way partitions. An interesting relationship exists betweenk-way partitions, the multicommodity flow problem, and the minimum weighted sparsest cut. Namely, if the staturated arcs in a multicommodity flow problem form ak-way partition (k ≤ 4), then thek-way partition contains a two-way partition. This two-way partition is the minimum weight sparsest cut.