TY - GEN

T1 - Kruskal-based approximation algorithm for the multi-level steiner tree problem

AU - Ahmed, Reyan

AU - Sahneh, Faryad Darabi

AU - Hamm, Keaton

AU - Kobourov, Stephen

AU - Spence, Richard

N1 - Funding Information:
Funding The research for this paper was partially supported by NSF grants CCF-1740858, CCF-1712119, and DMS-1839274.

PY - 2020/8/1

Y1 - 2020/8/1

N2 - 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.

AB - 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.

KW - Approximation algorithms

KW - Multi-level

KW - Steiner tree

UR - http://www.scopus.com/inward/record.url?scp=85092457155&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85092457155&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2020.4

DO - 10.4230/LIPIcs.ESA.2020.4

M3 - Conference contribution

AN - SCOPUS:85092457155

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 28th Annual European Symposium on Algorithms, ESA 2020

A2 - Grandoni, Fabrizio

A2 - Herman, Grzegorz

A2 - Sanders, Peter

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 28th Annual European Symposium on Algorithms, ESA 2020

Y2 - 7 September 2020 through 9 September 2020

ER -