Augmenting Graphs to Minimize the Diameter

Frati, Fabrizio; Gaspers, Serge; Gudmundsson, Joachim; Mathieson, Luke

doi:10.1007/978-3-642-45030-3_36

Cited by 25 publications

(32 citation statements)

References 19 publications

(25 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There are many further questions here, notably about the minimum diameters achievable by adding some specified number of edges (cf. [18,22,32]). The 'δ-perspective' seems of considerable interest for further study.…”

Section: Discussionmentioning

confidence: 99%

“…This problem is computationally hard, even as an approximation problem [18,22,32]. In Section 8 we discuss this problem in detail, including the case in which the degree constraint is taken into account.…”

Section: Optimalitymentioning

confidence: 99%

“…[10,16,22,24,38]). For shortcutting without the degree constraint, the problem in which one is just asked to reduce the diameter by at least 1 by adding at most k edges, is known to be NP-complete and W[2]-hard [22,24]. We aim to show that these facts hold even with the degree constraint imposed, i.e.…”

Section: Complexity Of Shortcuttingmentioning

confidence: 99%

See 2 more Smart Citations

Shortcutting directed and undirected networks with a degree constraint

Tan

Leeuwen

2017

Discrete Applied Mathematics

View full text Add to dashboard Cite

a b s t r a c tShortcutting is the operation of adding edges to a network with the intent to decrease its diameter. We are interested in shortcutting networks while keeping degree increases in the network bounded, a problem first posed by Chung and Garey. Improving on a result of Bokhari and Raza we show that, for any δ ≥ 1, every undirected graph G can be shortcut in linear time to a diameter of at most O(log 1+δ n) by adding no more than O(n/ log 1+δ n) edges such that degree increases remain bounded by δ. The result extends an estimate due to Alon et al. for the unconstrained case. Degree increases can be limited to 1 at only a small extra cost. For strongly connected, bounded-degree directed graphs Flaxman and Frieze proved that, if ϵn random arcs are added, then the resulting graph has diameter O(ln n) with high probability. We prove that O(n/ ln n) edges suffice to shortcut any strongly connected directed graph to a graph with diameter less than O(ln n) while keeping the degree increases bounded by O(1) per node. The result is proved in a stronger, parameterized form. For general directed graphs with stability number α, we show that all distances can be shortcut to O(α⌈ln n α ⌉) by adding only 4n ln n/α + αφ edges while keeping degree increases bounded by at most O(1) per node, where φ is equal to the so-called feedback-dimension of the graph. Finally, we prove bounds for various special classes of graphs, including graphs with Hamiltonian cycles or paths. Shortcutting with a degree constraint is proved to be strongly NP-complete and W [2]-hard, implying that the problem is neither likely to be fixed-parameter tractable nor efficiently approximable unless FPT = W [2].

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Optimalitymentioning

confidence: 99%

See 1 more Smart Citation

Shortcutting directed and undirected networks with a degree constraint

Tan

Leeuwen

2017

Discrete Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…The more general problem and many variations have also been studied before, e.g., see [1,3,5,6,9,12,13,15] and the references therein. Consider a general graph G in which edges have non-negative lengths.…”

Section: Related Workmentioning

confidence: 99%

An Improved Algorithm for Diameter-Optimally Augmenting Paths in a Metric Space

Wang

2017

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…The problem of adding edges to a graph in order to modify some general properties has been widely studied. To the best of our knowledge, the problems that aim at optimizing some property by adding a limited number of edges are: minimizing the average shortest-path distance between all pair of nodes [Meyerson and Tagiku 2009;Papagelis et al 2011;Parotsidis et al 2015], minimizing the average number of hops in shortest paths of weighted graphs [Bauer et al 2012], maximizing the leading eigenvalue of the adjacency matrix [Saha et al 2015;Tong et al 2012], minimizing the diameter [Bilò et al 2012;Frati et al 2015], maximizing or minimizing the number of triangles [Dehghani et al 2015;Li and Yu 2015], minimizing the eccentricity [Perumal et al 2013], and minimizing the characteristic path length [Papagelis 2015].…”

Section: Introductionmentioning

confidence: 99%

Greedily Improving Our Own Closeness Centrality in a Network

Crescenzi

D’Angelo

Severini

et al. 2016

ACM Trans. Knowl. Discov. Data

View full text Add to dashboard Cite

International audienceThe closeness centrality is a well-known measure of importance of a vertex within a given complex network. Having high closeness centrality can have positive impact on the vertex itself: hence, in this paper we consider the optimization problem of determining how much a vertex can increase its centrality by creating a limited amount of new edges incident to it. We will consider both the undirected and the directed graph cases. In both cases, we first prove that the optimization problem does not admit a polynomial-time approximation scheme (unless P = NP), and then propose a greedy approximation algorithm (with an almost tight approximation ratio), whose performance is then tested on synthetic graphs and real-world networks

show abstract

Augmenting Graphs to Minimize the Diameter

Abstract: Abstract. We study the problem of augmenting a weighted graph by inserting edges of bounded total cost while minimizing the diameter of the augmented graph. Our main result is an FPT 4-approximation algorithm for the problem.

Cited by 25 publications

References 19 publications

Shortcutting directed and undirected networks with a degree constraint

Shortcutting directed and undirected networks with a degree constraint

An Improved Algorithm for Diameter-Optimally Augmenting Paths in a Metric Space

Greedily Improving Our Own Closeness Centrality in a Network

Contact Info

Product

Resources

About