Finding Paths with Minimum Shared Edges

Omran, Masoud T.; Sack, Jörg-Rüdiger; Zarrabi-Zadeh, Hamid

doi:10.1007/978-3-642-22685-4_49

Cited by 5 publications

(6 citation statements)

References 10 publications

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“…Problem 1 (Unweighted Uncapacitated Minimum Vulnerability problem (Uumv) [4,29,38]) The input to this problem a graph G = (V, E), two nodes s, t ∈ V , and two positive integers 0 < r < κ. The goal is to find a set of κ paths between s and t that minimizes the number of "shared edges", where an edge is called shared if it is in more than r of these κ paths between s and t.…”

Section: Network Design Application: Minimizing Bottleneck Edgesmentioning

confidence: 99%

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Effect of Gromov-Hyperbolicity Parameter on Cuts and Expansions in Graphs and Some Algorithmic Implications

et al. 2017

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δ-hyperbolic graphs, originally conceived by Gromov in 1987, occur often in many network applications; for fixed δ, such graphs are simply called hyperbolic graphs and include non-trivial interesting classes of "non-expander" graphs. The main motivation of this paper is to investigate the effect of the hyperbolicity measure δ on expansion and cut-size bounds on graphs (here δ need not be a constant), and the asymptotic ranges of δ for which these results may provide improved approximation algorithms for related combinatorial problems. To this effect, we provide constructive bounds on node expansions for δ-hyperbolic graphs as a function of δ, and show that many witnesses (subsets of nodes) for such expansions can be computed efficiently even if the witnesses are required to be nested or sufficiently distinct from each other. To the best of our knowledge, these are the first such constructive bounds proven. We also show how to find a large family of s-t cuts with relatively small number of cut-edges when s and t are sufficiently far apart. We then provide algorithmic consequences of these bounds and their related proof techniques for two problems for δ-hyperbolic graphs (where δ is a function f of the number of nodes, 2 Bhaskar DasGupta et al.the exact nature of growth of f being dependent on the particular problem considered). Mathematics Subject Classification1 IntroductionUseful insights for many complex systems such as the world-wide web, social networks, metabolic networks, and protein-protein interaction networks can often be obtained by representing them as parameterized networks and analyzing them using graph-theoretic tools. Some standard measures used for such investigations include degree based measures (e.g., maximum/minimum/average degree or degree distribution) connectivity based measures (e.g., clustering coefficient, claw-free property, largest cliques or densest sub-graphs), and geodesic based measures (e.g., diameter or betweenness centrality). It is a standard practice in theoretical computer science to investigate and categorize the computational complexities of combinatorial problems in terms of ranges of these parameters. For example:◮ Bounded-degree graphs are known to admit improved approximation as opposed to their arbitrary-degree counter-parts for many graph-theoretic problems. ◮ Claw-free graphs are known to admit improved approximation as opposed to general graphs for graph-theoretic problems such as the maximum independent set problem.In this paper we consider a topological measure called Gromov-hyperbolicity (or, simply hyperbolicity for short) for undirected unweighted graphs that has recently received significant attention from researchers in both the graph theory and the network science community. This hyperbolicity measure δ was originally conceived in a somewhat different group-theoretic context by Gromov [20]. The measure was first defined for infinite continuous metric space via properties of geodesics [10], but was later also adopted for finite graphs. Lately, there have been...

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Section: Network Design Application: Minimizing Bottleneck Edgesmentioning

confidence: 99%

“…Uumv has applications in several communication network design problems (see [36][37][38] for further details). The following computational complexity results are known regarding Uumv and Mse for a graph with n nodes and m edges (see [4,29]):…”

Section: Network Design Application: Minimizing Bottleneck Edgesmentioning

confidence: 99%

Effect of Gromov-Hyperbolicity Parameter on Cuts and Expansions in Graphs and Some Algorithmic Implications

et al. 2017

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“…We wish to find n paths from s to t with minimum number of critical edges. One motivation for this problem, adapted from [1,3], is as follows. We need to make a sensitive shipment from s to t. In the planning stage, n paths are chosen and prepared.…”

Section: Introductionmentioning

confidence: 99%

“…Following [1], for 1 ≤ k ≤ n, call an edge k-vulnerable if it used by at least k of the n paths. In [3] it was shown that finding n paths with minimum number of 2-vulnerable edges is NP-hard, and in [1] it was shown that for fixed n and k, finding n paths with minimum number of k-vulnerable edges can be done in polynomial time.…”

Section: Introductionmentioning

confidence: 99%

The unimodular intersection problem

Kaibel

Onn

Sarrabezolles

2015

Operations Research Letters

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“…On the positive side, we show that there exists a (k − 1)-approximation algorithm for the problem, using an adaption of a network flow algorithm. We design heuristics to improve the quality of the output, and provide empirical results [123,121].…”

Section: New Resultsmentioning

confidence: 99%

Path Problems in Geographic Information Systems

Omran¹

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Path problems are among the most widely studied problems in graph theory. Various versions of the problem with distinct real world applications have been proposed and studied (Shortest Path, Longest Path, Hamiltonian Path, Traveling Salesperson Problem, among others). The shortest path problem for a network (or directed graph) is a fundamental problem in graph theory with applications related to Navigation/Transportation Systems, Computer Networks, VLSI Design, Social Networks, and many other networks. Unlike in well-studied conventional static shortest path problems, the characteristics of underlying networks may change over time in many real-world applications. An efficient solution to the Shortest Path problem, should therefore take the dynamics of the underlying graph into consideration. In this thesis, we study a dynamic version of the Shortest Path Problem in which travel times on links of the underlying graph change over time, and present improved and easily implemented exact and approximation algorithms.With theoretical improvements in time-dependent shortest path computation and technical advancements in the provision of real-time traffic data, many travelers these days use and rely on navigation systems and time-dependent shortest path computation in their everyday lives. For example, GoogleMaps provides advice on best routes by using current traffic conditions and historical route data to keep users out of traffic jams. This may, however, compromise users' privacy since untrusted Location Based Services (LBS) are able to use inference attacks to gain access to sensitive information about users, such as their driving habits, health conditions they may have, their sexual orientation, or details about their lifestyle. In this dissertation, we propose a heuristic algorithm that protects the privacy of users who submit frequent location based queries to a Location Based Service.Although, shortest paths are preferred in most applications related to navigation and transportation, they can easily be inferred and regenerated, and should therefore be avoided in applications where user security is paramount. For example, a traveling VIP is more concerned about the security of her path rather than its length. We study the traveling VIP problem in which the goal is to randomly select a path from k candidate source-destination paths. When links are shared between many paths security is compromised and security guards may have to be placed on such links. The number of shared edges, then, should be minimized to reduce the costs associated with security guards who must be assigned to each shared link. We show that this problem is hard to approximate, and propose an approximation algorithm which has been experimentally evaluated on various networks.

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Finding Paths with Minimum Shared Edges

Cited by 5 publications

References 10 publications

Effect of Gromov-Hyperbolicity Parameter on Cuts and Expansions in Graphs and Some Algorithmic Implications

Effect of Gromov-Hyperbolicity Parameter on Cuts and Expansions in Graphs and Some Algorithmic Implications

The unimodular intersection problem

Path Problems in Geographic Information Systems

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