Given a graph G = (V, E) and a subset T ⊆ V of terminals, a Steiner tree of G is a tree that spans T. In the vertex-weighted Steiner tree (VST) problem, each vertex is assigned a non-negative weight, and the goal is to compute a minimum weight Steiner tree of G. Vertex-weighted problems have applications in network design and routing, where there are different costs for installing or maintaining facilities at different vertices. We study a natural generalization of the VST problem motivated by multi-level graph construction, the vertex-weighted grade-of-service Steiner tree problem (V-GSST), which can be stated as follows: given a graph G and terminals T, where each terminal v ∈ T requires a facility of a minimum grade of service R(v) ∈ {1, 2, . . . `}, compute a Steiner tree G0 by installing facilities on a subset of vertices, such that any two vertices requiring a certain grade of service are connected by a path in G 0 with the minimum grade of service or better. Facilities of higher grade are more costly than facilities of lower grade. Multi-level variants such as this one can be useful in network design problems where vertices maymore »
Approximation algorithms and an integer program for multi-level graph spanners
Given a weighted graph G(V, E) and t ≥ 1, a subgraph H is a t–spanner of G if the lengths of shortest paths in G are preserved in H up to a multiplicative factor of t. The subsetwise spanner problem aims to preserve distances in G for only a subset of the vertices. We generalize the minimum-cost subsetwise spanner problem to one where vertices appear on multiple levels, which we call the multi-level graph spanner (MLGS) problem, and describe two simple heuristics. Applications of this problem include road/network building and multi-level graph visualization, especially where vertices may require different grades of service.
We formulate a 0–1 integer linear program (ILP) of size O(|E||V |2) for the more general minimum pairwise spanner problem, which resolves an open question by Sigurd and Zachariasen on whether this problem admits a useful polynomial-size ILP. We extend this ILP formulation to the MLGS problem, and evaluate the heuristic and ILP performance on random graphs of up to 100 vertices and 500 edges.
- Publication Date:
- NSF-PAR ID:
- 10109412
- Journal Name:
- Symposium on Experimental Algorithms
- Sponsoring Org:
- National Science Foundation
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In the classical Steiner tree problem, given an undirected, connected graph G=(V,E) with non-negative edge costs and a set of terminals T⊆V, the objective is to find a minimum-cost tree E′⊆E that spans the terminals. The problem is APX-hard; the best known approximation algorithm has a ratio of ρ=ln(4)+ε<1.39. In this paper, we study a natural generalization, the multi-level Steiner tree (MLST) problem: given a nested sequence of terminals Tℓ⊂⋯⊂T1⊆V, compute nested trees Eℓ⊆⋯⊆E1⊆E that span the corresponding terminal sets with minimum total cost. The MLST problem and variants thereof have been studied under various names including Multi-level Network Design, Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, and Multi-Tier tree. Several approximation results are known. We first present two simple O(ℓ)-approximation heuristics. Based on these, we introduce a rudimentary composite algorithm that generalizes the above heuristics, and determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio using at most 2ℓ Steiner tree computations. We compare these heuristics experimentally on various instances of up to 500 vertices using three different network generation models. We also present various integer linear programming (ILP) formulations for the MLST problem, and compare their running timesmore »
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In the classical Steiner tree problem, given an undirected, connected graph G =( V , E ) with non-negative edge costs and a set of terminals T ⊆ V , the objective is to find a minimum-cost tree E &prime ⊆ E that spans the terminals. The problem is APX-hard; the best-known approximation algorithm has a ratio of ρ = ln (4)+ε < 1.39. In this article, we study a natural generalization, the multi-level Steiner tree (MLST) problem: Given a nested sequence of terminals T ℓ ⊂ … ⊂ T 1 ⊆ V , compute nested trees E ℓ ⊆ … ⊆ E 1 ⊆ E that span the corresponding terminal sets with minimum total cost. The MLST problem and variants thereof have been studied under various names, including Multi-level Network Design, Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, and Multi-tier tree. Several approximation results are known. We first present two simple O (ℓ)-approximation heuristics. Based on these, we introduce a rudimentary composite algorithm that generalizes the above heuristics, and determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio using at most 2ℓ Steiner tree computations. We compare these heuristicsmore »
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