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Approximate k-Steiner Forests via the Lagrangian Relaxation Technique with Internal Preprocessing

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Abstract

An instance of the k -Steiner forest problem consists of an undirected graph G=(V,E), the edges of which are associated with non-negative costs, and a collection \(\mathcal{D}=\{(s_{1},t_{1}),\ldots,(s_{d},t_{d})\}\) of distinct pairs of vertices, interchangeably referred to as demands. We say that a forest ℱ⊆G connects a demand (s i ,t i ) when it contains an s i -t i path. Given a profit k i for each demand (s i ,t i ) and a requirement parameter k, the goal is to find a minimum cost forest that connects a subset of demands whose combined profit is at least k. This problem has recently been studied by Hajiaghayi and Jain (SODA ’06), whose main contribution in this context was to relate the inapproximability of k-Steiner forest to that of the dense k -subgraph problem. However, Hajiaghayi and Jain did not provide any algorithmic result for the respective settings, and posed this objective as an important direction for future research.

In this paper, we present the first non-trivial approximation algorithm for the k-Steiner forest problem, which is based on a novel extension of the Lagrangian relaxation technique. Specifically, our algorithm constructs a feasible forest whose cost is within a factor of \(O(\min \{n^{2/3},\sqrt{d}\}\cdot \log d)\) of optimal, where n is the number of vertices in the input graph and d is the number of demands. We believe that the approach illustrated in the current writing is of independent interest, and may be applicable in other settings as well.

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Authors and Affiliations

  1. Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Danny Segev

  2. Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel

    Gil Segev

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  1. Danny Segev
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  2. Gil Segev
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Correspondence to Danny Segev.

Additional information

An extended abstract of this paper appeared in Proceedings of the 14th Annual European Symposium on Algorithms, 2006.

This work is part of D. Segev’s Ph.D. thesis prepared at Tel-Aviv University under the supervision of Prof. Refael Hassin.

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Segev, D., Segev, G. Approximate k-Steiner Forests via the Lagrangian Relaxation Technique with Internal Preprocessing. Algorithmica 56, 529–549 (2010). https://doi.org/10.1007/s00453-008-9189-8

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