# Kruskal-based approximation algorithm for the multi-level steiner tree problem

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

Research output: Chapter in Book/Report/Conference proceedingConference contribution

## Abstract

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), `ρ}-approximation in this setting, where ρ is an approximation ratio for a heuristic solver for the Steiner tree problem and ` is the number of priorities or levels (Byrka et al. give a Steiner tree algorithm with ρ ≈ 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.

Original language English (US) 28th Annual European Symposium on Algorithms, ESA 2020 Fabrizio Grandoni, Grzegorz Herman, Peter Sanders Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing 9783959771627 https://doi.org/10.4230/LIPIcs.ESA.2020.4 Published - Aug 1 2020 28th Annual European Symposium on Algorithms, ESA 2020 - Virtual, Pisa, ItalyDuration: Sep 7 2020 → Sep 9 2020

### Publication series

Name Leibniz International Proceedings in Informatics, LIPIcs 173 1868-8969

### Conference

Conference 28th Annual European Symposium on Algorithms, ESA 2020 Italy Virtual, Pisa 9/7/20 → 9/9/20

## Keywords

• Approximation algorithms
• Multi-level
• Steiner tree

• Software