Enumerative combinatorics
The Steiner tree polytope and related polyhedra
Mathematical Programming: Series A and B
SIAM Journal on Computing
On the bidirected cut relaxation for the metric Steiner tree problem
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Recent results on approximating the Steiner tree problem and its generalizations
Theoretical Computer Science - Selected papers in honor of Manuel Blum
A comparison of Steiner tree relaxations
Discrete Applied Mathematics - Special issue on the combinatorial optimization symposium
Steiner trees in uniformly quasi-bipartite graphs
Information Processing Letters
Approximation Hardness of the Steiner Tree Problem on Graphs
SWAT '02 Proceedings of the 8th Scandinavian Workshop on Algorithm Theory
A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Spanning trees in hypergraphs with applications to steiner trees
Spanning trees in hypergraphs with applications to steiner trees
Tighter Bounds for Graph Steiner Tree Approximation
SIAM Journal on Discrete Mathematics
Approximating minimum bounded degree spanning trees to within one of optimal
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
New geometry-inspired relaxations and algorithms for the metric steiner tree problem
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
An improved LP-based approximation for steiner tree
Proceedings of the forty-second ACM symposium on Theory of computing
A partition-based relaxation for Steiner trees
Mathematical Programming: Series A and B
On Steiner trees and minimum spanning trees in hypergraphs
Operations Research Letters
How well can primal-dual and local-ratio algorithms perform?
ACM Transactions on Algorithms (TALG)
Matroids and integrality gaps for hypergraphic steiner tree relaxations
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Integrality gap of the hypergraphic relaxation of Steiner trees: A short proof of a 1.55 upper bound
Operations Research Letters
Steiner Tree Approximation via Iterative Randomized Rounding
Journal of the ACM (JACM)
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We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Könemann et al. [Math. Programming, 2009]. Specifically, we are interested in proving upper bounds on the integrality gap of this LP, and studying its relation to other linear relaxations. Our results are the following. Structural results: We extend the technique of uncrossing, usually applied to families of sets, to families of partitions. As a consequence we show that any basic feasible solution to the partition LP formulation has sparse support. Although the number of variables could be exponential, the number of positive variables is at most the number of terminals. Relations with other relaxations: We show the equivalence of the partition LP relaxation with other known hypergraphic relaxations. We also show that these hypergraphic relaxations are equivalent to the well studied bidirected cut relaxation, if the instance is quasibipartite. Integrality gap upper bounds: We show an upper bound of $\sqrt{3} \doteq 1.729$ on the integrality gap of these hypergraph relaxations in general graphs. In the special case of uniformly quasibipartite instances, we show an improved upper bound of 73/60≐1.216. By our equivalence theorem, the latter result implies an improved upper bound for the bidirected cut relaxation as well.