Strong Steiner Tree Approximations in Practice

Abstract

In this experimental study we consider Steiner tree approximation algorithms that guarantee a constant approximation ratio smaller than 2. The considered greedy algorithms and approaches based on linear programming involve the incorporation of k-restricted full components for some k ≥ 3. For most of the algorithms, their strongest theoretical approximation bounds are only achieved for k → ∞. However, the running time is also exponentially dependent on k, so only small k are tractable in practice.We investigate different implementation aspects and parameter choices that finally allow us to construct algorithms (somewhat) feasible for practical use. We compare the algorithms against each other, to an exact LP-based algorithm, and to fast and simple 2-approximations. Funded by the German Research Foundation (DFG), project number CH 897/1-1. Preliminary versions of this work appeared as [Chimani and Woste 2011] and [Beyer and Chimani 2014]. 1 arXiv:1409.8318v2 [cs.DS] 9 Dec 2015bound for an approximation ratio is 96/95 ≈ 1.0105 [Chlebík and Chlebíková 2008].The STP has various applications in the fields of VLSI design, routing, network design, computational biology, and computer-aided design. It serves as a basis for generalized problems like prize-collecting and stochastic Steiner trees, Steiner forests, Survivable Network Design problems, discount-augmented problems like Buy-at-Bulk or Rent-or-Buy, and appears as a subproblem in problems like the Steiner packing problem.The versatile applicability of the STP gave rise to a lot of research from virtually all algorithmic points of view: heuristics, metaheuristics, and approximation algorithms [

Approximation algorithms for Steiner forest: An experimental study

“…The closest works to ours consist of experimental studies [7,12,23,[32][33][34]39], software packages implementing various algorithms (GeoSteiner [41], SCIP-Jack [33]), and benchmark suites (SteinLib [26], PACE2018 [31], Vienna [29]) for the Steiner tree problem and its variants. To the best of our knowledge there are no previous experimental studies on approximation algorithms for Steiner forest.…”

“…FIGURE7 Complete graphs with Euclidian weights (selected instances): Running time and cost of solution (The main horizontal axis shows the key parameters of each instance as a pair (n, k), while the secondary horizontal axis shows the corresponding test set. The instances are listed in nonincreasing order of the number of vertices from left to right.…”

Solving Steiner trees: Recent advances, challenges, and perspectives

“…However, Warme [234] showed that if the size of the maximum FSC is bounded by a constant, then the LP‐relaxation of HYP can be obtained in polynomial time. This fact has been extensively exploited in a series of articles that provide LP‐based approximation algorithms for the STP [23, 25, 37, 110, 147] (cf. Section 9).…”

“…Approximation algorithms based on hypergraphic LP‐relaxations have been implemented and empirically tested by Beyer et al [23, 25]; see also Section 9. However, to the best of our knowledge, there are no empirical studies on exact methods for the STP in graphs derived from HYP.…”

“…In a follow‐up article, Beyer and Chimani [24] showed that even after applying reduction tests as a preprocessing step, no implementation of these strong approximation algorithms could compete with state‐of‐the‐art exact and heuristic algorithms. Finally, Beyer and Chimani [25] gave a comprehensive and detailed overview of different implementation aspects and parameter choices for combinatorial and LP‐based strong approximation algorithms for the STP. They concluded that one of the major bottlenecks for the computational efficiency of the method by Goemans et al [110] seems to be the generation and storage of k ‐restricted FSCs and the computation of the LP‐relaxation of HYP.…”