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- W2015382802 abstract "The problem of computing a minimum cost subgraph D′=(V,A′) of a directed graph D=(V,A) so as to contain k edge-disjoint paths from a specified root r0∈V to every other node in V was solved by Edmonds [J. Edmonds, Submodular functions, matroids, and certain polyhedra, in: R. Guy, H. Hanani, N. Sauer, J. Schönheim (Eds.), Combinatorial Structures and their Applications, Gordon and Breach, New York, 1970, pp. 69–87] by an elegant reduction to weighted matroid intersection. A corresponding problem when openly disjoint paths are requested rather than edge-disjoint ones was solved in [A. Frank, É. Tardos, An application of submodular flows, Linear Algebra Appl. 114–115 (1989) 329–348] with the help of submodular flows. Here we show that the use of submodular flows is actually avoidable and even a common generalization of the two rooted k-connection problems reduces to matroid intersection. The approach is based on a new matroid construction extending what Whiteley [W. Whiteley, Some matroids from discrete applied geometry, in: J.E. Bonin, J.G. Oxley, B. Servatius (Eds.), Matroid Theory, in: Contemp. Math., vol. 197, Amer. Math. Soc, Providence, RI, 1996, pp. 171–311] calls count matroids. We also provide a polyhedral description using supermodular functions on bi-sets and this approach enables us to handle more general rooted k-connection problems. For example, with the help of a submodular flow algorithm the following restricted version of the generalized Steiner-network problem is solvable in polynomial time: given a digraph D=(V,A) with a root-node r0, a terminal set T, and a cost function c:A→R+ so that each edge of positive cost has its head in T, find a subgraph D′=(V,A′) of D of minimum cost so that there are k openly disjoint paths in D′ from r0 to every node in T." @default.
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- W2015382802 date "2009-03-01" @default.
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- W2015382802 title "Rooted <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si1.gif display=inline overflow=scroll><mml:mi>k</mml:mi></mml:math>-connections in digraphs" @default.
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- W2015382802 doi "https://doi.org/10.1016/j.dam.2008.03.040" @default.
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