How well can primal-dual and local-ratio algorithms perform?
ACM Transactions on Algorithms (TALG)
Hypergraphic LP relaxations for steiner trees
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Matroids and integrality gaps for hypergraphic steiner tree relaxations
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Steiner Tree Approximation via Iterative Randomized Rounding
Journal of the ACM (JACM)
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The Steiner tree problem is a classical NP-hard optimization problem with a wide range of practical applications. In an instance of this problem, we are given an undirected graph G = (V, E), a set of terminals $${R\subseteq V}$$, and non-negative costs c e for all edges $${e \in E}$$. Any tree that contains all terminals is called a Steiner tree; the goal is to find a minimum-cost Steiner tree. The vertices $${V \backslash R}$$ are called Steiner vertices. The best approximation algorithm known for the Steiner tree problem is a greedy algorithm due to Robins and Zelikovsky (SIAM J Discrete Math 19(1):122–134, 2005); it achieves a performance guarantee of $${1+\frac{\ln 3}{2}\approx 1.55}$$. The best known linear programming (LP)-based algorithm, on the other hand, is due to Goemans and Bertsimas (Math Program 60:145–166, 1993) and achieves an approximation ratio of 2−2/|R|. In this paper we establish a link between greedy and LP-based approaches by showing that Robins and Zelikovsky’s algorithm can be viewed as an iterated primal-dual algorithm with respect to a novel LP relaxation. The LP used in the first iteration is stronger than the well-known bidirected cut relaxation. An instance is b-quasi-bipartite if each connected component of $${G \backslash R}$$ has at most b vertices. We show that Robins’ and Zelikovsky’s algorithm has an approximation ratio better than $${1+\frac{\ln 3}{2}}$$ for such instances, and we prove that the integrality gap of our LP is between $${\frac{8}{7}}$$ and $${\frac{2b+1}{b+1}}$$.