### Abstract

Original language | English |
---|---|

Title of host publication | Proceedings of APPROX-RANDOM 2014 |

Editors | K. Jansen, J.D.P. Rolim, N.R. Devanur, C. Moore |

Publisher | Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik |

Pages | 176-191 |

ISBN (Print) | 9783939897743 |

DOIs | |

Publication status | Published - 2014 |

Event | APPROX - Duration: 4 Sep 2014 → 6 Sep 2014 |

### Conference

Conference | APPROX |
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Period | 4/09/14 → 6/09/14 |

### Fingerprint

### Cite this

*Proceedings of APPROX-RANDOM 2014*(pp. 176-191). Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.176

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*Proceedings of APPROX-RANDOM 2014.*Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, pp. 176-191, APPROX, 4/09/14. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.176

**On the Equivalence of the Bidirected and Hypergraphic Relaxations for Steiner Tree.** / Feldmann, A.E.; Könemann, J.; Olver, N.K.; Sanità, L.

Research output: Chapter in Book / Report / Conference proceeding › Conference contribution › Academic › peer-review

TY - GEN

T1 - On the Equivalence of the Bidirected and Hypergraphic Relaxations for Steiner Tree

AU - Feldmann, A.E.

AU - Könemann, J.

AU - Olver, N.K.

AU - Sanità, L.

PY - 2014

Y1 - 2014

N2 - 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 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 paper, 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 3 Steiner neighbors. 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.

AB - 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 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 paper, 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 3 Steiner neighbors. 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.

U2 - 10.4230/LIPIcs.APPROX-RANDOM.2014.176

DO - 10.4230/LIPIcs.APPROX-RANDOM.2014.176

M3 - Conference contribution

SN - 9783939897743

SP - 176

EP - 191

BT - Proceedings of APPROX-RANDOM 2014

A2 - Jansen, K.

A2 - Rolim, J.D.P.

A2 - Devanur, N.R.

A2 - Moore, C.

PB - Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik

ER -