In the Steiner forest problem, we are given a set of terminal pairs and need to find the minimum cost subgraph that connects each of the terminal pairs together. Motivated by the recent work on greedy approximation algorithms for the Steiner forest, we provide efficient implementations of existing approximation algorithms and conduct a thorough experimental study to characterize their performance. We consider several approximation algorithms: the influential primal-dual 2-approximation algorithm due to Agrawal, Klein, and Ravi, the greedy algorithm due to Gupta and Kumar, and a randomized algorithm based on probabilistic approximation by tree metrics. We also consider the simplest heuristic greedy algorithm for the problem, which picks the closest unconnected pair of terminals and connects it using the shortest path between the terminals in the current graph. To characterize the performance of the algorithms, we created a new library with more than one thousand Steiner forest problem instances and conducted an extensive experimental analysis on those instances. Our analysis reveals that for the majority of instances the primal-dual algorithm is the fastest among all the algorithms considered here, and obtains solutions that are very close to the optimal solutions obtained by solving the integer program formulation of the problem.

中文翻译：

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Steiner 森林的近似算法：一项实验研究

在施泰纳森林问题中，给定一组终端对，需要找到将每个终端对连接在一起的最小成本子图。受 Steiner 森林贪婪逼近算法最近工作的启发，我们提供了现有逼近算法的有效实现，并进行了彻底的实验研究以表征其性能。我们考虑了几种近似算法：Agrawal、Klein 和 Ravi 的有影响力的原始对偶 2 近似算法，Gupta 和 Kumar 的贪心算法，以及基于树度量的概率近似的随机算法。我们还考虑了该问题的最简单的启发式贪心算法，它选择最近的未连接的一对端子，并使用当前图中端子之间的最短路径将其连接起来。为了表征算法的性能，我们创建了一个包含一千多个 Steiner 森林问题实例的新库，并对这些实例进行了广泛的实验分析。我们的分析表明，对于大多数情况，原始对偶算法是此处考虑的所有算法中最快的，并且获得的解非常接近通过求解问题的整数规划公式获得的最优解。