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

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. Steiner Tree has been analyzed within many theoretical frameworks as well as from the practical perspective.We observe that many approximation algorithms for Steiner Tree 1. follow a similar scheme (meta-algorithm) and 2. 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. 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. 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 ∼ 3× 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 ∼ 3× 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. However, such an improvement is exchanged for a slower running time (up to a multiplicative factor of the number of terminals).