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 |
|---|---|
| Period | 4/09/14 → 6/09/14 |
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On the Equivalence of the Bidirected and Hypergraphic Relaxations for Steiner Tree. / Feldmann, A.E.; Könemann, J.; Olver, N.K.; Sanità, L.
Proceedings of APPROX-RANDOM 2014. ed. / K. Jansen; J.D.P. Rolim; N.R. Devanur; C. Moore. Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, 2014. p. 176-191.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 -