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A sparsification of a given graph G is a sparser graph (typically a subgraph) which aims to approximate or preserve some property of G. Examples of sparsifications include but are not limited to spanning trees, Steiner trees, spanners, emulators, and distance preservers. Each vertex has the same priority in all of these problems. However, real-world graphs typically assign different “priorities” or “levels” to different vertices, in which higher-priority vertices require higher-quality connectivity between them. Multi-priority variants of the Steiner tree problem have been studied in prior literature but this generalization is much less studied for other sparsification problems. In this paper, we define a generalized multi-priority problem and present a rounding-up approach that can be used for a variety of graph sparsifications. Our analysis provides a systematic way to compute approximate solutions to multi-priority variants of a wide range of graph sparsification problems given access to a single-priority subroutine. 

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. 

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 times on these instances. To our knowledge, the composite algorithm achieves the best approximation ratio for up to ℓ=100 levels, which is sufficient for most applications such as network visualization or designing multi-level infrastructure.