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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A Constant Factor Approximation for Navigating Through Connected Obstacles in the Plane
Given two points s and t in the plane and a set of obstacles defined by closed curves, what is the minimum number of obstacles touched by a path connecting s and t? This is a fundamental and well-studied problem arising naturally in computational geometry, graph theory (under the names Min-Color Path and Minimum Label Path), wireless sensor networks (Barrier Resilience) and motion planning (Minimum Constraint Removal). It remains NP-hard even for very simple-shaped obstacles such as unit-length line segments. In this paper we give the first constant factor approximation algorithm for this problem, resolving an open problem of [Chan and Kirkpatrick, TCS, 2014] and [Bandyapadhyay et al., CGTA, 2020]. We also obtain a constant factor approximation for the Minimum Color Prize Collecting Steiner Forest where the goal is to connect multiple request pairs (s1, t1), . . . , (sk, tk) while minimizing the number of obstacles touched by any (si, ti) path plus a fixed cost of wi for each pair (si, ti) left disconnected. This generalizes the classic Steiner Forest and Prize-Collecting Steiner Forest problems on planar graphs, for which intricate PTASes are known. In contrast, no PTAS is possible for Min-Color Path even on planar graphs since the problem is known to be APX- hard [Eiben and Kanj, TALG, 2020]. Additionally, we show that generalizations of the problem to disconnected obstacles in the plane or connected obstacles in higher dimensions are strongly inapproximable assuming some well-known hardness conjectures.
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- Award ID(s):
- 1814172
- NSF-PAR ID:
- 10284625
- Date Published:
- Journal Name:
- Proceedings of the annual ACM-SIAM Symposium on Discrete Algorithms
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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