Given a weighted graph @math and @math , a subgraph @math is a of @math if the lengths of shortest paths in @math are preserved in @math up to a multiplicative factor of @math . The problem aims to preserve distances in @math 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 (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 @math for the more general minimum , 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.
Sigurd and Zachariasen @cite_17 present an ILP formulation for the minimum-weight pairwise spanner problem (see ), and show that the greedy algorithm @cite_22 performs well on sparse graphs of up to 64 vertices. ' A lvarez-Miranda and Sinnl @cite_1 present a mixed ILP formulation for the tree @math --spanner problem, which asks for a spanning tree of a graph @math with the smallest stretch factor @math .
@cite_1 @cite_22 @cite_17
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Spanners are sparse subgraphs that preserve distances up to a given factor in the underlying graph. Recently spanners have found important practical applications in metric space searching and message distribution in networks. These applications use some variant of the so-called greedy algorithm for constructing the spanner — an algorithm that mimics Kruskal’s minimum spanning tree algorithm. Greedy spanners have nice theoretical properties, but their practical performance with respect to total weight is unknown. In this paper we give an exact algorithm for constructing minimum-weight spanners in arbitrary graphs. By using the solutions (and lower bounds) from this algorithm, we experimentally evaluate the performance of the greedy algorithm for a set of realistic problem instances. The greedy spanner is arguably the simplest and most well-studied spanner construction. Experimental results demonstrate that it is at least as good as any other spanner construction, in terms of both the size and weight parameters. However, a rigorous proof for this statement has remained elusive. In this work we fill in the theoretical gap via a surprisingly simple observation: The greedy spanner is (or existentially near-optimal) for several important graph families. Focusing on the weight parameter, the state-of-the-art spanner constructions for both general graphs (due to Chechik and Wulff-Nilsen [SODA'16]) and doubling metrics (due to Gottlieb [FOCS'15]) are complex. Plugging our observation on these results, we conclude that the greedy spanner achieves near-optimal weight guarantees for both general graphs and doubling metrics, thus resolving two longstanding conjectures in the area. Further, we observe that approximate-greedy algorithms are existentially near-optimal as well. Consequently, we provide an @math -time construction of @math -spanners for doubling metrics with constant lightness and degree. Our construction improves Gottlieb's construction, whose runtime is @math and whose number of edges and degree are unbounded, and remarkably, it matches the state-of-the-art Euclidean result (due to [SICOMP'02]) in all the involved parameters (up to dependencies on @math and the dimension). We present an O(nlogn)-approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G=(V,E) with nonnegative edge lengths d:E->R^>=^0 and a stretchk>=1, a subgraph H=(V,E"H) is a k-spanner of G if for every edge (s,t)@?E, the graph H contains a path from s to t of length at most [email protected]?d(s,t). The previous best approximation ratio was [email protected]?(n^2^ ^3), due to Dinitz and Krauthgamer (STOC @?11). We also improve the approximation ratio for the important special case of directed 3-spanners with unit edge lengths from [email protected]?(n) to O(n^1^ ^3logn). The best previously known algorithms for this problem are due to Berman, Raskhodnikova and Ruan (FSTTCS @?10) and Dinitz and Krauthgamer. The approximation ratio of our algorithm almost matches Dinitz and [email protected]?s lower bound for the integrality gap of a natural linear programming relaxation. Our algorithm directly implies an O(n^1^ ^3logn)-approximation for the 3-spanner problem on undirected graphs with unit lengths. An easy O(n)-approximation algorithm for this problem has been the best known for decades. Finally, we consider the Directed Steiner Forest problem: given a directed graph with edge costs and a collection of ordered vertex pairs, find a minimum-cost subgraph that contains a path between every prescribed pair. We obtain an approximation ratio of O(n^2^ ^3^+^@e) for any constant @e>0, which improves the O(n^@[email protected]?min(n^4^ ^5,m^2^ ^3)) ratio due to Feldman, Kortsarz and Nutov ([email protected]?12). We show that the greedy spanner algorithm constructs a @math -spanner of weight @math for a point set in metrics of doubling dimension @math , resolving an open problem posed by Gottlieb. Our result generalizes the result by Narasimhan and Smid who showed that a point set in @math -dimension Euclidean space has a @math -spanner of weight at most @math . Our proof only uses the packing property of doubling metrics and thus implies a much simpler proof for the same result in Euclidean space.