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

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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Received: June 11, 1998

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Karzanov, A. A Combinatorial Algorithm for the Minimum (2, r)-Metric Problem and Some Generalizations. Combinatorica 18, 549–568 (1998). https://doi.org/10.1007/s004930050040

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  • AMS Subject Classification (1991) Classes:  90C27; 90B10