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# Approximation Algorithms for Steiner Tree Based on Star Contractions: A Unified View

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## Acknowledgements

We thank the authors of the Boost library [Jeremy G. Siek et al., 2002], which we use in our code, for their work and effective implementation of many graph algorithms. We thank Tomáš Toufar for consultations in the early stages of preparation of the experiments in the paper as well as for the implementation of a part [Hušek et al., 2018] which was partially used in our code. We thank the PACE challenge which was a good motivation and inspiration for the development of practical algorithms for the Steiner Tree problem. The results of this challenge motivated us to develop the experiments presented in this paper. We also thank anonymous referees for interesting feedback as well as for pointing us to several interesting directions for future improvements and related results. Part of the work was carried out while D. Knop and T. Masařík were at the University of Bergen.

## Cite As

Radek Hušek, Dušan Knop, and Tomáš Masařk. Approximation Algorithms for Steiner Tree Based on Star Contractions: A Unified View. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.IPEC.2020.16

## Abstract

In the Steiner Tree problem, we are given an edge-weighted undirected graph G = (V,E) and a set of terminals R ⊆ V. The task is to find a connected subgraph of G containing R and minimizing the sum of weights of its edges. Steiner Tree is well known to be NP-complete and is undoubtedly one of the most studied problems in (applied) computer science. We observe that many approximation algorithms for Steiner Tree follow a similar scheme (meta-algorithm) and perform (exhaustively) a similar routine which we call star contraction. Here, by a star contraction, we mean finding a star-like subgraph in (the metric closure of) the input graph minimizing the ratio of its weight to the number of contained terminals minus one; and contract. It is not hard to see that the well-known MST-approximation seeks the best star to contract among those containing two terminals only. Zelikovsky’s approximation algorithm follows a similar workflow, finding the best star among those containing three terminals. We perform an empirical study of star contractions with the relaxed condition on the number of terminals in each star contraction motivated by a recent result of Dvořák et al. [Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices, STACS 2018]. Furthermore, we propose two improvements of Zelikovsky’s 11/6-approximation algorithm and we empirically confirm that the quality of the solution returned by any of these is better than the one returned by the former algorithm. However, such an improvement is exchanged for a slower running time (up to a multiplicative factor of the number of terminals).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Graph algorithms analysis
##### Keywords
• Steiner tree
• approximation
• star contractions
• minimum spanning tree

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