Network verification via routing table queries

Bampas, Evangelos; Bilò, Davide; Drovandi, Guido; Gualà, Luciano; Klasing, Ralf; Proietti, Guido

doi:10.1016/j.jcss.2014.06.003

Cited by 16 publications

(9 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Related notions. A weaker model is studied in [2,4] as a network discovery problem, where we seek a set S of vertices such that for each edge e, there exists a vertex x of S and a vertex y of G such that e belongs to some shortest path from x to y. A different (also weaker) model is the Link Monitoring problem studied in [5], in which one seeks to monitor the edges of a graph network by selecting vertices to act as probes.…”

Section: Distance-edge-monitoring Setmentioning

confidence: 99%

“…Probes that measure distances in graphs are present in real-life networks, for instance this is useful in the fundamental task of routing [8,14]. They are also frequently used for problems concerning network verification [2,4,6].…”

Section: Introductionmentioning

confidence: 99%

“…Even for graphs G that are (a) of diameter 4, (b) bipartite and of diameter 6, or (c) bipartite and of maximum degree 3, Min Distance-Edge-Monitoring Set is not approximable within a factor of (1 − ǫ) ln |E(G)| in polynomial time, unless P = N P . Moreover, for such instances, Distance-Edge-Monitoring Set cannot be solved in time |G| o(k) , unless FPT=W[1], and it is W[2]-hard for parameter k.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Monitoring the Edges of a Graph Using Distances

Foucaud

Klasing

Miller

et al. 2020

Algorithms and Discrete Applied Mathematics

Self Cite

View full text Add to dashboard Cite

We introduce a new graph-theoretic concept in the area of network monitoring. A set M of vertices of a graph G is a distance-edge-monitoring set if for every edge e of G, there is a vertex x of M and a vertex y of G such that e belongs to all shortest paths between x and y. We denote by dem(G) the smallest size of such a set in G. The vertices of M represent distance probes in a network modeled by G; when the edge e fails, the distance from x to y increases, and thus we are able to detect the failure. It turns out that not only we can detect it, but we can even correctly locate the failing edge. In this paper, we initiate the study of this new concept. We show that for a nontrivial connected graph G of order n, 1 ≤ dem(G) ≤ n − 1 with dem(G) = 1 if and only if G is a tree, and dem(G) = n − 1 if and only if it is a complete graph. We compute the exact value of dem for grids, hypercubes, and complete bipartite graphs. Then, we relate dem to other standard graph parameters. We show that dem(G) is lowerbounded by the arboricity of the graph, and upper-bounded by its vertex cover number. It is also upper-bounded by twice its feedback edge set number. Moreover, we characterize connected graphs G with dem(G) = 2. Then, we show that determining dem(G) for an input graph G is an NP-complete problem, even for apex graphs. There exists a polynomial-time logarithmic-factor approximation algorithm, however it is NP-hard to compute an asymptotically better approximation, even for bipartite graphs of small diameter and for bipartite subcubic graphs. For such instances, the problem is also unlikey to be fixed parameter tractable when parameterized by the solution size. * This paper is dedicated to the memory of Mirka Miller, who sadly passed away in January 2016. This research was started when Mirka Miller and Joe Ryan visited Ralf Klasing at the LaBRI, University of Bordeaux, in April 2015. † A preliminary version of this paper appeared as [12]. ‡ The authors acknowledge the financial support from the ANR project HOSIGRA (ANR-17-CE40-0022), the IFCAM project "Applications of graph homomorphisms" (MA/IFCAM/18/39), and the Programme IdEx Bordeaux-SysNum (ANR-10-IDEX-03-02).

show abstract

Section: Distance-edge-monitoring Setmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Monitoring the Edges of a Graph Using Distances

Foucaud

Klasing

Miller

et al. 2020

Algorithms and Discrete Applied Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…These concepts have many applications and connections to other areas. For example, the metric dimension can be applied to network discovery [4,3], robot navigation [21], coin-weighing problems [26], T -joins [26], the Mastermind game [13], or chemistry [12].…”

Section: Introductionmentioning

confidence: 99%

Bounding the Order of a Graph Using Its Diameter and Metric Dimension: A Study Through Tree Decompositions and VC Dimension

Beaudou¹,

Dankelmann²,

Foucaud³

et al. 2018

SIAM J. Discrete Math.

View full text Add to dashboard Cite

The metric dimension of a graph is the minimum size of a set of vertices such that each vertex is uniquely determined by the distances to the vertices of that set. Our aim is to upper-bound the order n of a graph in terms of its diameter d and metric dimension k. In general, the bound n ≤ d k + k is known to hold. We prove a bound of the form n = O(kd 2 ) for trees and outerplanar graphs (for trees we determine the best possible bound and the corresponding extremal examples). More generally, for graphs having a tree decomposition of width w and length ℓ, we obtain a bound of the form n = O(kd 2 (2ℓ + 1) 3w+1 ). This implies in particular that n = O(kd O(1) ) for graphs of constant treewidth and n = O(f (k)d 2 ) for chordal graphs, where f is a doubly-exponential function. Using the notion of distance-VC dimension (introduced in 2014 by Bousquet and Thomassé) as a tool, we prove the bounds n ≤ (dk + 1) t−1 + 1 for Kt-minor-free graphs, and n ≤ (dk + 1) d(3·2 r +2) + 1 for graphs of rankwidth at most r. *

show abstract

“…Roughly speaking, instead of the distances to S, we ask for the vertices to be distinguished by their neighbourhood within S. These problems have been widely studied since their introduction in the 1970s and 1980s. They have been applied in various areas such as network verification [4,5], fault-detection in networks [41,56], graph isomorphism testing [3] or the logical definability of graphs [43]. We note that the similar problem of finding a test cover of a hypergraph (where hyperedges distinguish the vertices) has been studied under several names by various authors, see e.g.…”

Section: Introductionmentioning

confidence: 99%

Identification, Location-Domination and Metric Dimension on Interval and Permutation Graphs. II. Algorithms and Complexity

et al. 2016

View full text Add to dashboard Cite

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted Identifying Code, (Open) Locating-Dominating Set and Metric Dimension) of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph by a subset of the vertices, using either the neighbourhood within the solution set or the distances to the solution vertices. Using a general reduction for this class of problems, we prove that the decision problems associated to these four notions are NP-complete, even for interval graphs of diameter 2 and permutation graphs of diameter 2. While Identifying Code and (Open) Locating-Dominating Set are trivially fixed-parameter-tractable when parameterized by solution size, it is known that in the same setting Metric Dimension is W [2]-hard. We show that for interval graphs, this parameterization of Metric Dimension is fixed-parameter-tractable. * A short version of this paper, containing only the results about location-domination and metric dimension, appeared in the proceedings of the WG 2015 conference [28].pair in X has a vertex in S (totally) separating it. We have the three key definitions, that merge the concepts of (total) domination and (total) separation:Definition 1 (Slater [52,53], Babai [3]). A set S of vertices of a graph G is a locating-dominating set if it is a dominating set and it separates the vertices of V (G) \ S.

show abstract

Network verification via routing table queries

Cited by 16 publications

References 12 publications

Monitoring the Edges of a Graph Using Distances

Monitoring the Edges of a Graph Using Distances

Bounding the Order of a Graph Using Its Diameter and Metric Dimension: A Study Through Tree Decompositions and VC Dimension

Identification, Location-Domination and Metric Dimension on Interval and Permutation Graphs. II. Algorithms and Complexity

Contact Info

Product

Resources

About