The steiner problem with edge lengths 1 and 2,
Information Processing Letters
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
Applying Parallel Computation Algorithms in the Design of Serial Algorithms
Journal of the ACM (JACM)
A new approximation algorithm for the Steiner tree problem with performance ratio 5/3
Journal of Algorithms
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
Minimum Bounded Degree Spanning Trees
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
The Steiner tree problem on graphs: Inapproximability results
Theoretical Computer Science
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
Hypergraphic LP relaxations for steiner trees
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Integrality gap of the hypergraphic relaxation of Steiner trees: A short proof of a 1.55 upper bound
Operations Research Letters
On Steiner trees and minimum spanning trees in hypergraphs
Operations Research Letters
Steiner Tree Approximation via Iterative Randomized Rounding
Journal of the ACM (JACM)
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Until recently, LP relaxations have only played a very limited role in the design of approximation algorithms for the Steiner tree problem. In particular, no (efficiently solvable) Steiner tree relaxation was known to have an integrality gap bounded away from 2, before Byrka et al. [3] showed an upper bound of ~1.55 of a hypergraphic LP relaxation and presented a ln(4)+ε ~1.39 approximation based on this relaxation. Interestingly, even though their approach is LP based, they do not compare the solution produced against the LP value. We take a fresh look at hypergraphic LP relaxations for the Steiner tree problem---one that heavily exploits methods and results from the theory of matroids and submodular functions---which leads to stronger integrality gaps, faster algorithms, and a variety of structural insights of independent interest. More precisely, along the lines of the algorithm of Byrka et al.[3], we present a deterministic ln(4)+ε approximation that compares against the LP value and therefore proves a matching ln(4) upper bound on the integrality gap of hypergraphic relaxations. Similarly to [3], we iteratively fix one component and update the LP solution. However, whereas in [3] the LP is solved at every iteration after contracting a component, we show how feasibility can be maintained by a greedy procedure on a well-chosen matroid. Apart from avoiding the expensive step of solving a hypergraphic LP at each iteration, our algorithm can be analyzed using a simple potential function. This potential function gives an easy means to determine stronger approximation guarantees and integrality gaps when considering restricted graph topologies. In particular, this readily leads to a 73/60 ~1.217 upper bound on the integrality gap of hypergraphic relaxations for quasi-bipartite graphs. Additionally, for the case of quasi-bipartite graphs, we present a simple algorithm to transform an optimal solution to the bidirected cut relaxation to an optimal solution of the hypergraphic relaxation, leading to a fast 73/60 approximation for quasi-bipartite graphs. Furthermore, we show how the separation problem of the hypergraphic relaxation can be solved by computing maximum flows, which provides a way to obtain a fast independence oracle for the matroids that we use in our approach.