arXiv 2002.064212020-02cs.DS

多层Steiner树问题的Kruskal近似算法

Kruskal-based approximation algorithm for the multi-level Steiner tree problem

Reyan Ahmed, Faryad Darabi Sahneh, Keaton Hamm, Stephen Kobourov, Richard Spence

摘要：

我们研究多层次的斯坦纳树问题：一个泛化的斯坦纳树问题的图形终端$T$需要不同的优先级，水平，或服务质量。在这个问题中，我们试图找到一个最小成本树，包含边的变化率，使任何两个终端$u$，$v$与优先级$P(u)$，$P(v)$连接使用边率$\min\{P(u),P(v)\}$或更好。边缘成本与它们的速率成比例的情况近似于最优解的恒定因子内。对于非比例成本的更一般情况，这个问题很难用比率$c \log \log n$近似，其中$n$是图中的顶点数。Charikar等人提出的一种简单的贪婪算法，然而，在此设置中提供了$\min\{2(\ln |T|+1), \ell ρ\}$近似，其中$ρ$是Steiner树问题的启发式求解器的近似比率，$\ell$是优先级或级别的数量（例如，Byrka等人给出了具有$ρ\approx 1.39$的Steiner树算法）。在本文中，我们描述了一个自然的推广到多层次的情况下的经典（单级）Steiner树近似算法的基础上Kruskal的最小生成树算法。我们证明，该算法实现的近似比至少一样好Charikar等人。并且实验上相对于最优解表现得更好。我们开发了一个整数线性规划公式来计算非比例边成本的多级Steiner树问题的精确解，并使用它来评估我们的算法在来自SteinLib的随机图和多级实例上的性能。

英文摘要：

We study the multi-level Steiner tree problem: a generalization of the Steiner tree problem in graphs where terminals $T$ require varying priority, level, or quality of service. In this problem, we seek to find a minimum cost tree containing edges of varying rates such that any two terminals $u$, $v$ with priorities $P(u)$, $P(v)$ are connected using edges of rate $\min\{P(u),P(v)\}$ or better. The case where edge costs are proportional to their rate is approximable to within a constant factor of the optimal solution. For the more general case of non-proportional costs, this problem is hard to approximate with ratio $c \log \log n$, where $n$ is the number of vertices in the graph. A simple greedy algorithm by Charikar et al., however, provides a $\min\{2(\ln |T|+1), \ell ρ\}$-approximation in this setting, where $ρ$ is an approximation ratio for a heuristic solver for the Steiner tree problem and $\ell$ is the number of priorities or levels (Byrka et al. give a Steiner tree algorithm with $ρ\approx 1.39$, for example). In this paper, we describe a natural generalization to the multi-level case of the classical (single-level) Steiner tree approximation algorithm based on Kruskal's minimum spanning tree algorithm. We prove that this algorithm achieves an approximation ratio at least as good as Charikar et al., and experimentally performs better with respect to the optimum solution. We develop an integer linear programming formulation to compute an exact solution for the multi-level Steiner tree problem with non-proportional edge costs and use it to evaluate the performance of our algorithm on both random graphs and multi-level instances derived from SteinLib.

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