Transitive Fork Environments and Minimum Cost Multiflows
نویسندگان
چکیده
The following minimum cost maximum multiiow problem is the focus of the paper: () given an undirected graph G = (V G; EG), a subset T 2 V G (of terminals), and functions c : EG ! ZZ + (of capacities) and a : EG ! ZZ + (of costs), nd a collection f of ows (a multiiow) connecting arbitrary pairs of distinct terminals so that the total ow f (e) through each edge does not exceed its capacity c(e), and: (a) the sum of values of partial ows is maximum; and (b) the total cost P e2EG a(e) f (e) of f is as small as possible, subject to (a). For jTj = 2 this turns into the classical (undirected) min-cost max-ow problem. In Ka1] it was proved that () has a half-integral optimal solution f, and that such an f can be found by a pseudo-polynomial algorithm. In Ka2] a polynomial algorithm to nd a half-integral optimal f was designed; however, it uses a variant of the ellipsoid method to solve the dual linear program. In the present paper we develop two purely combinatorial polynomial-time algorithms for nding a half-integral optimal solution to (). One of them is based on capacity scaling, and the other one is based on cost scaling. To design these algorithms, we introduce certain combinatorial structures, called transitive fork environments, and study general properties of such structures.
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