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On the equivalence of the bidirected and hypergraphic relaxations for Steiner tree

Feldmann, Andreas Emil, Könemann, Jochen, Olver, Neil and Sanitàa, Laura (2016) On the equivalence of the bidirected and hypergraphic relaxations for Steiner tree. Mathematical Programming, 160 (1-2). 379 - 406. ISSN 0025-5610

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Identification Number: 10.1007/s10107-016-0987-5

Abstract

The bottleneck of the currently best (ln(4)+ε) -approximation algorithm for the NP-hard Steiner tree problem is the solution of its large, so called hypergraphic, linear programming relaxation (HYP). Hypergraphic LPs are strongly NP-hard to solve exactly, and it is a formidable computational task to even approximate them sufficiently well. We focus on another well-studied but poorly understood LP relaxation of the problem: the bidirected cut relaxation (BCR). This LP is compact, and can therefore be solved efficiently. Its integrality gap is known to be greater than 1.16, and while this is widely conjectured to be close to the real answer, only a (trivial) upper bound of 2 is known. In this article, we give an efficient constructive proof that BCR and HYP are polyhedrally equivalent in instances that do not have an (edge-induced) claw on Steiner vertices, i.e., they do not contain a Steiner vertex with three Steiner neighbours. This implies faster ln(4) -approximations for these graphs, and is a significant step forward from the previously known equivalence for (so called quasi-bipartite) instances in which Steiner vertices form an independent set. We complement our results by showing that even restricting to instances where Steiner vertices induce one single star, determining whether the two relaxations are equivalent is NP-hard.

Item Type: Article
Official URL: https://link.springer.com/journal/10107
Additional Information: © 2016 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 20 Jan 2020 10:03
Last Modified: 20 May 2020 04:11
URI: http://eprints.lse.ac.uk/id/eprint/103084

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