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173
How bad is selfish routing?
 JOURNAL OF THE ACM
, 2002
"... We consider the problem of routing traffic to optimize the performance of a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion; the objective is to route t ..."
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Cited by 513 (28 self)
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We consider the problem of routing traffic to optimize the performance of a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion; the objective is to route traffic such that the sum of all travel times—the total latency—is minimized. In many settings, it may be expensive or impossible to regulate network traffic so as to implement an optimal assignment of routes. In the absence of regulation by some central authority, we assume that each network user routes its traffic on the minimumlatency path available to it, given the network congestion caused by the other users. In general such a “selfishly motivated ” assignment of traffic to paths will not minimize the total latency; hence, this lack of regulation carries the cost of decreased network performance. In this article, we quantify the degradation in network performance due to unregulated traffic. We prove that if the latency of each edge is a linear function of its congestion, then the total latency of the routes chosen by selfish network users is at most 4/3 times the minimum possible total latency (subject to the condition that all traffic must be routed). We also consider the more general setting in which edge latency functions are assumed only to be continuous and nondecreasing in the edge congestion. Here, the total
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & ..."
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Cited by 190 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Selfish Routing and the Price of Anarchy
, 2005
"... Abstract Selfish routing is a classical mathematical model of how selfinterested users might route traffic through a congested network. The outcome of selfish routing is generally inefficient, in that it fails to optimize natural objective functions. The price of anarchy is a quantitative measure o ..."
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Cited by 175 (11 self)
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Abstract Selfish routing is a classical mathematical model of how selfinterested users might route traffic through a congested network. The outcome of selfish routing is generally inefficient, in that it fails to optimize natural objective functions. The price of anarchy is a quantitative measure of this inefficiency. We survey recent work that analyzes the price of anarchy of selfish routing. We also describe related results on bounding the worstpossible severity of a phenomenon called Braess's Paradox, and on three techniques for reducing the price of anarchy of selfish routing. This survey concentrates on the contributions of the author's PhD thesis, but also discusses several more recent results in the area.
Approximation Algorithms for Disjoint Paths Problems
, 1996
"... The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for w ..."
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Cited by 140 (0 self)
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The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in highspeed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.
Finding Regular Simple Paths In Graph Databases
, 1989
"... We consider the following problem: given a labelled directed graph G and a regular expression R, find all pairs of nodes connected by a simple path such that the concatenation of the labels along the path satisfies R. The problem is motivated by the observation that many recursive queries in relatio ..."
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Cited by 115 (5 self)
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We consider the following problem: given a labelled directed graph G and a regular expression R, find all pairs of nodes connected by a simple path such that the concatenation of the labels along the path satisfies R. The problem is motivated by the observation that many recursive queries in relational databases can be expressed in this form, and by the implementation of a query language, G+ , based on this observation. We show that the problem is in general intractable, but present an algorithm than runs in polynomial time in the size of the graph when the regular expression and the graph are free of conflicts. We also present a class of languages whose expressions can always be evaluated in time polynomial in the size of both the graph and the expression, and characterize syntactically the expressions for such languages. Key words. Labelled directed graphs, NPcompleteness, polynomialtime algorithms, regular expressions, simple paths AMS(MOS) subject classifications. 68P, 6...
Nearoptimal hardness results and approximation algorithms for edgedisjoint paths and related problems
 Journal of Computer and System Sciences
, 1999
"... We study the approximability of edgedisjoint paths and related problems. In the edgedisjoint paths problem (EDP), we are given a network G with sourcesink pairs (si, ti), 1 ≤ i ≤ k, and the goal is to find a largest subset of sourcesink pairs that can be simultaneously connected in an edgedisjo ..."
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Cited by 107 (10 self)
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We study the approximability of edgedisjoint paths and related problems. In the edgedisjoint paths problem (EDP), we are given a network G with sourcesink pairs (si, ti), 1 ≤ i ≤ k, and the goal is to find a largest subset of sourcesink pairs that can be simultaneously connected in an edgedisjoint manner. We show that in directed networks, for any ɛ> 0, EDP is NPhard to approximate within m 1/2−ɛ. We also design simple approximation algorithms that achieve essentially matching approximation guarantees for some generalizations of EDP. Another related class of routing problems that we study concerns EDP with the additional constraint that the routing paths be of bounded length. We show that, for any ɛ> 0, bounded length EDP is hard to approximate within m 1/2−ɛ even in undirected networks, and give an O ( √ m)approximation algorithm for it. For directed networks, we show that even the single sourcesink pair case (i.e. find the maximum number of paths of bounded length between a given sourcesink pair) is hard to approximate within m 1/2−ɛ, for any ɛ> 0.
Network Coding in Undirected Networks
, 2004
"... Recent work in network coding shows that, it is necessary to consider both the routing and coding strategies to achieve optimal throughput of information transmission in data networks. So far, most research on network coding has focused on the model of directed networks, where each communication li ..."
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Cited by 70 (14 self)
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Recent work in network coding shows that, it is necessary to consider both the routing and coding strategies to achieve optimal throughput of information transmission in data networks. So far, most research on network coding has focused on the model of directed networks, where each communication link has a fixed direction. In this paper, we study the benefits of network coding in undirected networks, where each communication link is bidirectional. Our theoretical results show that, for a single unicast or broadcast session, there are no improvements with respect to throughput due to network coding. In the case of a single multicast session, such an improvement is bounded by a factor of two, as long as half integer routing is permitted. This is dramatically different from previous results obtained in directed networks. We also show that multicast throughput in an undirected network is independent of the selection of the sender within the multicast group. We finally show that, rather than improving the optimal achievable throughput, the benefit of network coding is to significantly facilitate the design of efficient algorithms to compute and achieve such optimal throughput. I.
How Much Can Taxes Help Selfish Routing?
 EC'03
, 2003
"... ... in networks. We consider a model of selfish routing in which the latency experienced by network tra#c on an edge of the network is a function of the edge congestion, and network users are assumed to selfishly route tra#c on minimumlatency paths. The quality of a routing of tra#c is historically ..."
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Cited by 62 (5 self)
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... in networks. We consider a model of selfish routing in which the latency experienced by network tra#c on an edge of the network is a function of the edge congestion, and network users are assumed to selfishly route tra#c on minimumlatency paths. The quality of a routing of tra#c is historically measured by the sum of all travel times, also called the total latency. It is well known
Designing networks for selfish users is hard
 In Proceedings of the 42nd Annual Symposium on Foundations of Computer Science
, 2001
"... Abstract We consider a directed network in which every edge possesses a latency function specifying the time needed to traverse the edge given its congestion. Selfish, noncooperative agents constitute the network traffic and wish to travel from a source s to a sink t as quickly as possible. Since th ..."
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Cited by 61 (8 self)
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Abstract We consider a directed network in which every edge possesses a latency function specifying the time needed to traverse the edge given its congestion. Selfish, noncooperative agents constitute the network traffic and wish to travel from a source s to a sink t as quickly as possible. Since the route chosen by one network user affects the congestion (and hence the latency) experienced by others, we model the problem as a noncooperative game. Assuming each agent controls only a negligible portion of the overall traffic, Nash equilibria in this noncooperative game correspond to st flows in which all flow paths have equal latency. A natural measure for the performance of a network used by selfish agents is the common latency experienced by each user in a Nash equilibrium. It is a counterintuitive but wellknown fact that removing edges from a network may improve its performance; the most famous example of this phenomenon is the socalled Braess's Paradox. This fact motivates the following network design problem: given such a network, which edges should be removed to obtain the best possible flow at Nash equilibrium? Equivalently, given a large network of candidate edges to be built, which subnetwork will exhibit the best performance when used selfishly? We give optimal inapproximability results and approximation algorithms for several network design problems of this type. For example, we prove that for networks with n vertices and continuous, nondecreasing latency functions, there is no approximation algorithm for this problem with approximation ratio less than n/2 (unless P = N P). We also prove this hardness result to be best possible by exhibiting an n/2approximation algorithm. For networks in which the latency of each edge is a linear function of the congestion, we prove that there is no ( 43 ffl)approximation algorithm for the problem (for any ffl> 0, unless P = N P); the existence of a 43approximation algorithm follows easily from existing work, proving this hardness result sharp. Moreover, we prove that an optimal approximation algorithm for these problems is what we call the trivial algorithm: given a network of candidate edges, build the entire network. A consequence of this result is that Braess's Paradox (even in its worstpossible manifestation) is impossible to detect efficiently.