Lehman's theorem and the directed Steiner tree problem

Abdi, Ahmad ORCID: 0000-0002-3008-4167, Feldmann, Andreas Emil, Guenin, Bertrand, Könemann, Jochen and Sanita, Laura (2016) Lehman's theorem and the directed Steiner tree problem. SIAM Journal on Discrete Mathematics, 30 (1). pp. 141-153. ISSN 0895-4801

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Abstract

In the directed Steiner tree problem, we are given a digraph, nonnegative arc weights, a subset of vertices called terminals, and a special terminal called the root. The goal is to compute a minimum weight directed tree that connects each terminal to the root. We study the classical directed cut linear programming (LP) formulation which has a variable for every arc, and a constraint for every cut that separates a terminal from the root. For what instances is the directed cut LP integral? In this paper we demonstrate how the celebrated theorem of Lehman [Math. Program., 17 (1979), pp. 403-417] on minimally nonideal clutters provides a framework for deriving answers to this question. Specifically, we show that this framework yields short proofs of the optimum arborescences theorem and the integrality result for series-parallel digraphs. Furthermore, we use this framework to show that the directed cut linear program is integral for digraphs that are acyclic and have at most two nonterminal vertices.

Item Type:	Article
Official URL:	https://epubs.siam.org/loi/sjdmec
Additional Information:	© 2016 Society for Industrial and Applied Mathematics
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	09 Oct 2019 14:15
Last Modified:	11 Sep 2025 10:03
URI:	http://eprints.lse.ac.uk/id/eprint/101880

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