Better Approximation Algorithms for the Graph Diameter

Chechik, Shiri; Larkin, Daniel H.; Roditty, Liam; Schoenebeck, Grant; Tarjan, Robert E.; Williams, Virginia Vassilevska

doi:10.1137/1.9781611973402.78

Cited by 84 publications

(107 citation statements)

References 33 publications

(50 reference statements)

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“…All the remaining reductions are given in this paper, excluding the one from APSP to Negative Triangle which is taken from [53]. [45] (see also [10] for a refinement of the approximation factor). The authors also show that a 3/2 − ε approximation for Diameter running in time O(m 2−ε ) (for any constant ε > 0) would imply that the Strong Exponential Time Hypothesis (SETH) of [33] fails, thus showing that improving on the 3/2-approximation factor while still using a fast algorithm would be difficult.…”

Section: Approximation Algorithmsmentioning

confidence: 99%

“…Our goal is to show that we can determine whether F is satisfiable in O * (2 (1−δ)n ) time for some constant δ > 0 8 . Using the sparsification lemma of [33] (as, e.g., in [10]), we can assume w.l.o.g. that F contains O(n) clauses.…”

Section: A Monte-carlo Ptas For Large Bmentioning

confidence: 99%

“…Indeed, in terms of approximation algorithms, any known algorithm to approximate the diameter can be converted to also approximate the radius in undirected graphs within the same factor [3,5,10,45]. Furthermore, the exact algorithms for Diameter and Radius in graphs with small integer weights are also extremely similar [13].…”

Section: Introductionmentioning

confidence: 99%

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Subcubic Equivalences Between Graph Centrality Problems, APSP and Diameter

Abboud¹,

Grandoni²,

Williams³

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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315

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Measuring the importance of a node in a network is a major goal in the analysis of social networks, biological systems, transportation networks etc. Different centrality measures have been proposed to capture the notion of node importance. For example, the center of a graph is a node that minimizes the maximum distance to any other node (the latter distance is the radius of the graph). The median of a graph is a node that minimizes the sum of the distances to all other nodes. Informally, the betweenness centrality of a node w measures the fraction of shortest paths that have w as an intermediate node.Finally, the reach centrality of a node w is the smallest distance r such that any s-t shortest path passing through w has either s or t in the ball of radius r around w.The fastest known algorithms to compute the center and the median of a graph, and to compute the betweenness or reach centrality even of a single node take roughly cubic time in the number n of nodes in the input graph. It is open whether these problems admit truly subcubic algorithms, i.e. algorithms with running timeÕ(n 3−δ ) for some constant δ > 0 1 . We relate the complexity of the mentioned centrality problems to two classical problems for which no truly subcubic algorithm is known, namely All Pairs Shortest Paths (APSP) and Diameter. We show that Radius, Median and Betweenness Centrality are equivalent under subcubic reductions to APSP, i.e. that a truly subcubic algorithm for any of these problems implies a truly subcubic algorithm for all of them. We then show that Reach Centrality is equivalent to Diameter under subcubic reductions. The same holds for the problem of approximating Betweenness Centrality within any constant factor. Thus the latter two centrality problems could potentially be solved in truly subcubic time, even if APSP requires essentially cubic time.

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Section: Approximation Algorithmsmentioning

confidence: 99%

Section: A Monte-carlo Ptas For Large Bmentioning

confidence: 99%

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Subcubic Equivalences Between Graph Centrality Problems, APSP and Diameter

Abboud¹,

Grandoni²,

Williams³

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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315

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“…It may also be viewed as the depth of a breadth first search, rooted at v. The graph diameter d is defined as the largest graph eccentricity for all nodes in a graph (i.e., the largest shortest path within a graph) and may be expressed as d = minv∈V (v). All exact algorithms solve the all pairs shortest paths (All-Pairs Shortest Path (APSP)) problem, and it is still an open problem as to whether or not a diameter may be exactly measured without calculating APSP [4]. As a graph metric, the diameter may be combined with other metrics to indicate the overall structure of the graph.…”

Section: Introductionmentioning

confidence: 99%

“…An even simpler estimation is to iteratively find path-lengths between random pairs of vertices and record the longest such path as the approximate diameter. The recent popularity of network science and graph theory as resulted in a slew of new approximation schemes, including an algorithm with time complexity ofÕ(EV 2/3 ) (where V and E are the number of vertices and edges) [4]. Despite the improved performance, the resulting approximations can be wildly inaccurate, although each pseudo-diameter algorithm typically defines a bounded error rate.…”

Section: Introductionmentioning

confidence: 99%

Fast, exact graph diameter computation with vertex programming

Pennycuff¹,

Weninger

2015

1st High Performance Graph Mining Workshop, Sydney, 10 August 2015

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In graph theory the diameter is an important topological metric for understanding size and density of a graph. Unfortunately, the graph diameter is computationally difficult to measure for even moderately-sized graphs, insomuch that approximation algorithms are commonly used instead of exact measurements. In this paper, we present a new algorithm to measure the exact diameter of unweighted graphs using vertex programming, which is easily distributed. We also show the practical performance of the algorithm in comparison to other, widely available algorithms and implementations, as well as the unreliability in accuracy of some pseudo-diameter estimators.

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Fast Approximation of Centrality and Distances in Hyperbolic Graphs

Chepoi¹,

Dragan²,

Habib³

et al. 2018

Combinatorial Optimization and Applications

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We show that the eccentricities (and thus the centrality indices) of all vertices of a δhyperbolic graph G = (V, E) can be computed in linear time with an additive one-sided error of at most cδ, i.e., after a linear time preprocessing, for every vertex v of G one can compute in O(1) time an estimateê(v) of its eccentricity eccG(v) such that eccG(v) ≤ê(v) ≤ eccG(v) + cδ for a small constant c. We prove that every δ-hyperbolic graph G has a shortest path tree, constructible in linear time, such that for every vertexThese results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of G, the smaller its eccentricity is. We also show that the distance matrix of G with an additive one-sided error of at most c ′ δ can be computed in O(|V | 2 log 2 |V |) time, where c ′ < c is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating centrality and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.

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Better Approximation Algorithms for the Graph Diameter

Cited by 84 publications

References 33 publications

Subcubic Equivalences Between Graph Centrality Problems, APSP and Diameter

Subcubic Equivalences Between Graph Centrality Problems, APSP and Diameter

Fast, exact graph diameter computation with vertex programming

Fast Approximation of Centrality and Distances in Hyperbolic Graphs

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