Fast approximation algorithms for fractional Steiner forest and related problems

We give a fully polynomial time approximation scheme (FPTAS) for the optimum fractional solution to the Steiner forest problem. This can easily be generalized to obtain an FPTAS for a hitting set problem on a collection of clutters. We also identify three other problems on collections of clutters and show how these four problems are related when the clutters have the max-flow min-cut (MFMC) property. Two of these problems which are generalizations of maximum multicommodity flow and maximum concurrent flow have been well studied in the past and this paper is the first attempt at designing efficient algorithms for the other two problems. Our algorithms are very simple to describe and have running times better than those of existing algorithms. For clutters that do not satisfy the MFMC property (e.g., k-spanner, multicommodity flows, T-cuts, T-joins etc.), our algorithms are the only ones known (other than the generic algorithms for linear programming) for solving these hitting set problems.