License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.115
URN: urn:nbn:de:0030-drops-46925
URL: https://drops.dagstuhl.de/opus/volltexte/2014/4692/
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### Improved Approximation Algorithm for Steiner k-Forest with Nearly Uniform Weights

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### Abstract

In the Steiner k-Forest problem we are given an edge weighted graph, a collection D of node pairs, and an integer k \leq |D|. The goal is to find a minimum cost subgraph that connects at least k pairs. The best known ratio for this problem is min{O(sqrt{n}),O(sqrt{k})} [Gupta et al., 2008]. In [Gupta et al., 2008] it is also shown that ratio rho for Steiner k-Forest implies ratio O(rho log^2 n) for the Dial-a-Ride problem: given an edge weighted graph and a set of items with a source and a destination each, find a minimum length tour to move each object from its source to destination, but carrying at most k objects at a time. The only other algorithm known for Dial-a-Ride, besides the one resulting from [Gupta et al., 2008], has ratio O(sqrt{n}) [Charikar and Raghavachari, 1998]. We obtain ratio n^{0.448} for Steiner k-Forest and Dial-a-Ride with unit weights, breaking the O(sqrt{n}) ratio barrier for this natural special case. We also show that if the maximum weight of an edge is O(n^{epsilon}), then one can achieve ratio O(n^{(1+epsilon) 0.448}), which is less than sqrt{n} if epsilon is small enough. To prove our main result we consider the following generalization of the Minimum k-Edge Subgraph (Mk-ES) problem, which we call Min-Cost l-Edge-Profit Subgraph (MCl-EPS): Given a graph G=(V,E) with edge-profits p={p_e: e in E} and node-costs c={c_v: v in V}, and a lower profit bound l, find a minimum node-cost subgraph of G of edge profit at least l. The Mk-ES problem is a special case of MCl-EPS with unit node costs and unit edge profits. The currently best known ratio for Mk-ES is n^{3-2*sqrt{2} + epsilon} (note that 3-2*sqrt{2} < 0.1716). We extend this ratio to MCl-EPS for arbitrary node weights and edge profits that are polynomial in n, which may be of independent interest.

### BibTeX - Entry

@InProceedings{dinitz_et_al:LIPIcs:2014:4692,
author =	{Michael Dinitz and Guy Kortsarz and Zeev Nutov},
title =	{{Improved Approximation Algorithm for Steiner k-Forest with Nearly Uniform Weights}},
booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
pages =	{115--127},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-74-3},
ISSN =	{1868-8969},
year =	{2014},
volume =	{28},
editor =	{Klaus Jansen and Jos{\'e} D. P. Rolim and Nikhil R. Devanur and Cristopher Moore},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2014/4692},
URN =		{urn:nbn:de:0030-drops-46925},
doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.115},
annote =	{Keywords: k-Steiner Forest; Uniform weights; Densest k-Subgraph; Approximation algorithms}
}


 Keywords: k-Steiner Forest; Uniform weights; Densest k-Subgraph; Approximation algorithms Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014) Issue Date: 2014 Date of publication: 04.09.2014

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