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A spanner of a graph G is a subgraph H that approximately preserves shortest path distances in G. Spanners are commonly applied to compress computation on metric spaces corresponding to weighted input graphs. Classic spanner constructions can seamlessly handle edge weights, so long as error is measured multiplicatively. In this work, we investigate whether one can similarly extend constructions of spanners with purely additive error to weighted graphs. These extensions are not immediate, due to a key lemma about the size of shortest path neighborhoods that fails for weighted graphs. Despite this, we recover a suitable amortized version, which lets us prove direct extensions of classic +2 and +4 unweighted spanners (both allpairs and pairwise) to +2W and +4W weighted spanners, where W is the maximum edge weight. Specifically, we show that a weighted graph G contains allpairs (pairwise) +2W and +4W weighted spanners of size O(n3/2) and O(n7/5) (O(np1/3) and O(np2/7)) respectively. For a technical reason, the +6 unweighted spanner becomes a +8W weighted spanner; closing this error gap is an interesting remaining open problem. That is, we show that G contains allpairs (pairwise) +8W weighted spanners of size O(n4/3) (O(np1/4)).

We study the multilevel 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 nonproportional 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 multilevel case of the classical (singlelevel) Steiner tree approximation algorithm based on Kruskal’s minimum spanning tree algorithm. Wemore »

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 vertexweighted Steiner tree (VST) problem, each vertex is assigned a nonnegative weight, and the goal is to compute a minimum weight Steiner tree of G. Vertexweighted 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 multilevel graph construction, the vertexweighted gradeofservice Steiner tree problem (VGSST), 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. Multilevel variants such as this one can be useful in network design problems where vertices maymore »

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 minimumcost subsetwise spanner problem to one where vertices appear on multiple levels, which we call the multilevel graph spanner (MLGS) problem, and describe two simple heuristics. Applications of this problem include road/network building and multilevel graph visualization, especially where vertices may require different grades of service. We formulate a 0–1 integer linear program (ILP) of size O(EV 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 polynomialsize 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 nonnegative edge costs and a set of terminals T⊆V, the objective is to find a minimumcost tree E′⊆E that spans the terminals. The problem is APXhard; the best known approximation algorithm has a ratio of ρ=ln(4)+ε<1.39. In this paper, we study a natural generalization, the multilevel 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 Multilevel Network Design, QualityofService Multicast tree, GradeofService Steiner tree, and MultiTier 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 »