Rooted k-connections in digraphs

Research output: Contribution to journalArticle

25 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)1242-1254
Number of pages13
JournalDiscrete Applied Mathematics
Volume157
Issue number6
DOIs
Publication statusPublished - Mar 28 2009

Fingerprint

Matroid
Digraph
Connection Problem
Matroid Intersection
Disjoint Paths
Subgraph
Costs
Vertex of a graph
Steiner network
Roots
Edge-disjoint Paths
Submodular Function
Linear algebra
Polyhedron
Directed Graph
Cost Function
Polynomial time
Count
Disjoint
Directed graphs

Keywords

  • Arborescences
  • Connectivity of digraphs
  • Matroid intersection

ASJC Scopus subject areas

  • Applied Mathematics
  • Discrete Mathematics and Combinatorics

Cite this

Rooted k-connections in digraphs. / Frank, A.

In: Discrete Applied Mathematics, Vol. 157, No. 6, 28.03.2009, p. 1242-1254.

Research output: Contribution to journalArticle

Frank, A. / Rooted k-connections in digraphs. In: Discrete Applied Mathematics. 2009 ; Vol. 157, No. 6. pp. 1242-1254.
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