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

In the Steiner Tree problem we are given an edge weighted undirected graph $G = (V,E)$ and a set of terminals $R \subseteq V$. The task is to find a connected subgraph of $G$ containing $R$ and minimizing the sum of weights of its edges. 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 input graph minimizing the ratio of its weight to the number of contained terminals minus one. It is not hard to see that the well-known MST-approximation seeks the best star to contract among those containing two terminals only. We perform an empirical study of star contractions with the relaxed condition on the number of terminals in each star contraction. Our experiments suggest the following: -- if the algorithm performs star contractions exhaustively, the quality of the solution is usually slightly better than Zelikovsky's 11/6-approximation algorithm, -- on average the quality of the solution returned by the MST-approximation algorithm improves with every star contraction, -- the same holds for iterated MST (MST+), which outperforms MST in every measurement while keeping very fast running time (on average $\sim 3\times$ slower than MST), -- on average the quality of the solution obtained by exhaustively performing star contraction is about 16\% better than the initial MST-approximation, and -- we propose a more precise way to find the so-called improved stars which yield a slightly better solution within a comparable running time (on average $\sim 3\times$ slower). 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.