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.
@cite_10 provide a flow-based linear programming (LP) relaxation to approximate the directed spanner problem. Their LP formulation is similar to that in @cite_17 ; however, they provide an approximation algorithm which relaxes their ILP, whereas the previous formulation was used to compute spanners to optimality. Additionally, the LP formulation applies to graphs of unit edge cost; they later take care of it in their rounding algorithm by solving a shortest path arborescence problem. They provide a @math --approximation algorithm for the directed @math --spanner problem for @math , which is the first sublinear approximation algorithm for arbitrary edge lengths. @cite_18 provide a slightly different formulation to approximate @math --spanners as well as other variations of this problem. They provide a polynomial time @math --approximation algorithm for the directed @math --spanner problem. @cite_20 provide an alternative randomized LP rounding schemes that lead to better approximation ratios. They improved the approximation ratio to @math where the approximation ratio of the algorithm provided by @cite_10 was @math . They have also improved the approximation ratio for the important special case of directed 3--spanners with unit edge lengths.
@cite_18 @cite_10 @cite_20 @cite_17
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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 examine directed spanners through flow-based linear programming relaxations. We design an O(n2 3)-approximation algorithm for the directed k-spanner problem that works for all k ≥ 1, which is the first sublinear approximation for arbitrary edge-lengths. Even in the more restricted setting of unit edge-lengths, our algorithm improves over the previous O(n1-1 k) approximation [BGJRW09] when k ≥ 4. For the special case of k=3 we design a different algorithm achieving an O(√n)-approximation, improving the previous O(n2 3) [EP05,BGJRW09] (independently of our work, an O(n1-1 ⌈ k 2⌉) was recently devised [BRR10]). Both of our algorithms easily extend to the fault-tolerant setting, which has recently attracted attention but not from an approximation viewpoint. We also prove a nearly matching integrality gap of Ω(n1 3 - e) for every constant e > 0. A virtue of all our algorithms is that they are relatively simple. Technically, we introduce a new yet natural flow-based relaxation, and show how to approximately solve it even when its size is not polynomial. The main challenge is to design a rounding scheme that "coordinates" the choices of flow-paths between the many demand pairs while using few edges overall. We achieve this, roughly speaking, by randomization at the level of vertices. We present two new approximation algorithms for the problem of finding a k-node connected spanning subgraph (directed or undirected) of minimum cost. The best known approximation guarantees for this problem were @math for both directed and undirected graphs, and @math for undirected graphs with @math , where @math is the number of nodes in the input graph. Our first algorithm has approximation ratio @math , which is @math except for very large values of @math , namely, @math . This algorithm is based on a new result on @math -connected @math -critical graphs, which is of independent interest in the context of graph theory. Our second algorithm uses the primal-dual method and has approximation ratio @math for all values of @math . Combining these two gives an algorithm with approximation ratio @math , which asymptotically improves the best known approximation guarantee for directed graphs for all values of @math , and for undirected graphs for @math . Moreover, this is the first algorithm that has an approximation guarantee better than @math for all values of @math . Our approximation ratio also provides an upper bound on the integrality gap of the standard LP-relaxation. A natural requirement of many distributed structures is fault-tolerance: after some failures, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general graphs that are tolerant to vertex failures, and significantly improve their dependence on the number of faults @math , for all stretch bounds. For stretch @math we design a simple transformation that converts every @math -spanner construction with at most @math edges into an @math -fault-tolerant @math -spanner construction with at most @math edges. Applying this to standard greedy spanner constructions gives @math -fault tolerant @math -spanners with @math edges. The previous construction by Chechik, Langberg, Peleg, and Roddity [STOC 2009] depends similarly on @math but exponentially on @math (approximately like @math ). For the case @math and unit-length edges, an @math -approximation algorithm is known from recent work of Dinitz and Krauthgamer [arXiv 2010], where several spanner results are obtained using a common approach of rounding a natural flow-based linear programming relaxation. Here we use a different (stronger) LP relaxation and improve the approximation ratio to @math , which is, notably, independent of the number of faults @math . We further strengthen this bound in terms of the maximum degree by using the Local Lemma. Finally, we show that most of our constructions are inherently local by designing equivalent distributed algorithms in the LOCAL model of distributed computation.