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), lρ}-approximation in this setting, where ρ is an approximation ratio for a heuristic solver for the Steiner tree problem and l 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.
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Revisiting Garg's 2-Approximation Algorithm for the k-MST Problem in Graphs
This paper revisits the 2-approximation algorithm for k-MST presented by Garg [9] in light of a recent paper of Paul et al. [14]. In the k-MST problem, the goal is to return a tree spanning k vertices of minimum total edge cost. Paul et al. [14] extend Garg's primal-dual subroutine to improve the approximation ratios for the budgeted prize-collecting traveling salesman and minimum spanning tree problems. We follow their algorithm and analysis to provide a cleaner version of Garg's result. Additionally, we introduce the novel concept of a kernel which allows an easier visualization of the stages of the algorithm and a clearer understanding of the pruning phase. Other notable updates include presenting a linear programming formulation of the k-MST problem, including pseudocode, replacing the coloring scheme used by Garg with the simpler concept of neutral sets, and providing an explicit potential function.
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- Award ID(s):
- 2007009
- PAR ID:
- 10420894
- Editor(s):
- Kavitha, Telikepalli; Mehlhorn, Kurt
- Date Published:
- Journal Name:
- Proceedings of the 2023 Symposium on Simplicity in Algorithms
- Page Range / eLocation ID:
- 56-68
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1137/1.9781611977585.ch6
