Approximate Integer Decompositions for Undirected Network Design Problems

A well-known theorem of Nash-Williams and Tutte gives a necessary and sufficient condition for the existence of $k$ edge-disjoint spanning trees in an undirected graph. A corollary of this theorem is that every $2k$-edge-connected graph has $k$ edge-disjoint spanning trees. We show that the splitting-off theorem of Mader in undirected graphs implies a generalization of this to finding $k$ edge-disjoint Steiner forests in Eulerian graphs. This leads to new 2-approximation rounding algorithms for certain constrained 0-1 forest problems considered by Goemans and Williamson. These algorithms also produce approximate integer decompositions of fractional solutions. We then discuss open problems and outlets for this approach to the more general class of 0-1 skew supermodular network design problems.

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