Minimum-Cost Multiple Paths Subject to Minimum Link and Node Sharing in a Network

Zheng, S. Q.; Wang, Jianping; Yang, Bing; Yang, Mei

doi:10.1109/tnet.2010.2044514

Cited by 9 publications

(4 citation statements)

References 18 publications

(19 reference statements)

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“…We will use the notation OPT Uumv (G, s, t, r, κ) to denote the number of shared edges in an optimal solution of an instance of Uumv. 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%

“…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. When r = 1, the Uumv problem is called the "minimum shared edges" (Mse) problem.…”

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%

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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“…The network failure was due to fibre cuts, hardware and software failures or power outages occurred in the networks. The heuristic algorithm, exhaustive search algorithm helps to finding the diversely routed path and Integer Linear Programming (ILP) algorithm guaranteed on minimum cost SRLG diverse routing problem [6]. The Mixed Integer Linear Programming (MILP) had an integer optimal solution which implemented on shared risks and reduced running time.…”

Section: Related Workmentioning

confidence: 99%

Optimization of Disjoints Using WDM-PON in an Optical Network

Rani¹,

Devi²,

Suganthi³

2016

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An optical network is a type of data communication network built with optical fibre technology. It utilizes optical fibre cables as the primary communication medium for converting data and passing data as light pulses between sender and receiver nodes. The major issue in optical networking is disjoints that occur in the network. A polynomial time algorithm Wavelength Division Multiplexing-Passive Optical Networking (WDM-PON) computes disjoints of an optical network and reduces the count of disjoints that occur in the network by separating Optical Network Units (ONU) into several virtual point-to-point connections. The Arrayed Waveguide Grating (AWG) filter is included in WDM-PON to avoid the traffic in the network thereby increasing the bandwidth capacity. In case of a failure or disjoint Ant Colony Optimization (ACO) algorithm is used to find the optimized shortest path for re-routing. For enhanced security, modified Rivert Shamir Adleman (RSA) algorithm encrypts the message during communication between the nodes. The efficiency is found to be improved in terms of delay in packet delivery, longer optical reach, optimized shortest path, packet error rate.

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“…Their algorithm converts the problem into an instance of the minimum cost flow problem. Using the same approach, in [153,149], the authors present a polynomial time solution for finding minimum sum of shared nodes/edges on both directed and undirected networks.…”

Section: Disjoint Paths Problemmentioning

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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Minimum-Cost Multiple Paths Subject to Minimum Link and Node Sharing in a Network

Cited by 9 publications

References 18 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

Optimization of Disjoints Using WDM-PON in an Optical Network

Path Problems in Geographic Information Systems

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