Finding good approximate vertex and edge partitions is NP-hard

Bui, Tuan Anh; Jones, Curt

doi:10.1016/0020-0190(92)90140-q

Cited by 277 publications

(138 citation statements)

References 7 publications

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“…The problem of finding the exact optimal routing is mathematically tied to the problem of finding the minimal sparsity vertex separator [10], which has been shown to be an NPhard problem [27]. This means that the number of flops necessary for the computation of an exact solution increases with the number of nodes N faster than any polynomial.…”

Section: Introductionmentioning

confidence: 99%

Transport optimization on complex networks

Danila

Marsh

et al. 2007

Chaos: An Interdisciplinary Journal of Nonlinear Science

101

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We present a comparative study of the application of a recently introduced heuristic algorithm to the optimization of transport on three major types of complex networks. The algorithm balances network traffic iteratively by minimizing the maximum node betweenness with as little path lengthening as possible. We show that by using this optimal routing, a network can sustain significantly higher traffic without jamming than in the case of shortest path routing. A formula is proved that allows quick computation of the average number of hops along the path and of the average travel times once the betweennesses of the nodes are computed. Using this formula, we show that routing optimization preserves the small-world character exhibited by networks under shortest path routing, and that it significantly reduces the average travel time on congested networks with only a negligible increase in the average travel time at low loads. Finally, we study the correlation between the weights of the links in the case of optimal routing and the betweennesses of the nodes connected by them.Keywords: complex networks, scaling laws, transport * dbogdan@mail.uh.edu † marshj@ainfosec.com ‡ bassler@uh.edu 1 One of the most important problems in the study of complex networks is how to best route transport on the networks. This problem is important because transport is the main function of many natural and human-made networks. Often, the transport routes used on networks are the so-called shortest-path routes, which are the routes with the minimum number of hops between any two nodes. However, this approach, which is currently used to route transport of information packets on the Internet, typically leads to congestion and eventually jamming of highly connected nodes of the networks called hubs. For this reason and in light of recent research, interest has developed in finding the routing rules that allow a given network to bear the maximum possible traffic. Specifically, the problem can be stated as follows. Given a complex network and a set of processing power and traffic demand constraints for its nodes, find the set of routing rules which allow the network to bear the highest possible amount of traffic without jamming. This problem is known to be NP -hard, meaning that the time required for the computation of an exact solution increases with the number of nodes faster than any polynomial. In this paper we argue that a heuristic transport routing optimization algorithm recently published by us achieves near-optimal transport routing in polynomial time and show this to be true for three important types of complex networks. Of course, any optimized routing when compared to shortest-path routing occurs at the expense of increasing the average number of hops between the nodes. We show that with our algorithm the average number of hops after optimization increases with the number of nodes no faster than logarithmically and that optimization significantly decreases the average travel time on congested networks.

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Section: Introductionmentioning

confidence: 99%

Transport optimization on complex networks

Danila

Marsh

et al. 2007

Chaos: An Interdisciplinary Journal of Nonlinear Science

101

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“…An important point is that, even though the minimization procedure pertains to a single scalar quantity, such an optimization algorithm will implicitly reshape the betweenness landscape across the whole network, lowering traffic through the initially busy nodes at the expense of increased traffic through the initially idle nodes until the traffic spreads out and an as narrow as possible betweenness distribution is achieved. The problem of finding the exact optimal routing is mathematically tied to the problem of finding the minimal sparsity vertex separator [7], which has been shown [17] to be an N P -hard problem. This means that the number of flops necessary for the computation of an exact solution increases with the number of nodes N faster than any polynomial.…”

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confidence: 99%

Optimal transport on complex networks

Danila

Marsh³

et al. 2006

Phys. Rev. E

222

187

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We present a novel heuristic algorithm for routing optimization on complex networks. Previously proposed routing optimization algorithms aim at avoiding or reducing link overload. Our algorithm balances traffic on a network by minimizing the maximum node betweenness with as little path lengthening as possible, thus being useful in cases when networks are jamming due to queuing overload. By using the resulting routing table, a network can sustain significantly higher traffic without jamming than in the case of traditional shortest path routing.

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“…As a consequence of the above, the b-balanced oneway cut problem is NP-hard for every 1 2 − 1 n < b ≤ 1 2 , since in this case, for an even n, every b-balanced cut is a bisection. We now turn to show the hardness of the b-balanced oneway cut problem for b = 1 2 − , where 1 n ≤ = o 1 log n .…”

Section: Balanced Oneway Cutsmentioning

confidence: 95%

“…Minimum b-vertex separator is the problem of finding a minimum set of vertices whose removal separates the vertices of the graph into two disconnected sets, each of size at least bn. This problem is known to be NP-hard [1]. One approach for approximating this problem is by a reduction to the directed b-balanced cut problem.…”

Section: Related Workmentioning

confidence: 99%