# A Combinatorial Algorithm for the Minimum (2, *r*)-Metric Problem and Some Generalizations

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be a graph with nonnegative integer capacities *c*(*e*) of the edges \(\), and let μ be a metric that establishes distances on the pairs of elements of a subset \(\). In the *minimum 0-extension problem* (*), one is asked for finding a (semi)metric *m* on *V* such that *m* coincides with μ within *T*, each \(\) is at zero distance from some \(\), and the value \(\) is as small as possible. This is the classical minimum (undirected) cut problem when \(\) and \(\), and the *minimum *(2, *r*)*-metric problem* when μ is the path metric of the complete bipartite graph \(\). It is known that the latter problem can be solved in strongly polynomial time by use of the ellipsoid method.

We develop a polynomial time algorithm for the minimum (2, *r*)-metric problem, using only ``purely combinatorial'' means. The algorithm simultaneously solves a certain associated integer multiflow problem. We then apply this algorithm to solve (*) for a wider class of metrics μ, present other results and raise open questions.

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