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Faster and simpler algorithms for multicommodity flow and other fractional packing problems
"... This paper considers the problem of designing fast, approximate, combinatorial algorithms for multicommodity flows and other fractional packing problems. We present new faster and much simpler algorithms for these problems. ..."
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Cited by 279 (5 self)
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This paper considers the problem of designing fast, approximate, combinatorial algorithms for multicommodity flows and other fractional packing problems. We present new faster and much simpler algorithms for these problems.
Potential Function Methods for Approximately Solving Linear Programming Problems: Theory and Practice
, 2001
"... After several decades of sustained research and testing, linear programming has evolved into a remarkably reliable, accurate and useful tool for handling industrial optimization problems. Yet, large problems arising from several concrete applications routinely defeat the very best linear programming ..."
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Cited by 123 (4 self)
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After several decades of sustained research and testing, linear programming has evolved into a remarkably reliable, accurate and useful tool for handling industrial optimization problems. Yet, large problems arising from several concrete applications routinely defeat the very best linear programming codes, running on the fastest computing hardware. Moreover, this is a trend that may well continue and intensify, as problem sizes escalate and the need for fast algorithms becomes more stringent. Traditionally, the focus in optimization algorithms, and in particular, in algorithms for linear programming, has been to solve problems "to optimality." In concrete implementations, this has always meant the solution ofproblems to some finite accuracy (for example, eight digits). An alternative approach would be to explicitly, and rigorously, trade o# accuracy for speed. One motivating factor is that in many practical applications, quickly obtaining a partially accurate solution is much preferable to obtaining a very accurate solution very slowly. A secondary (and independent) consideration is that the input data in many practical applications has limited accuracy to begin with. During the last ten years, a new body ofresearch has emerged, which seeks to develop provably good approximation algorithms for classes of linear programming problems. This work both has roots in fundamental areas of mathematical programming and is also framed in the context ofthe modern theory ofalgorithms. The result ofthis work has been a family ofalgorithms with solid theoretical foundations and with growing experimental success. In this manuscript we will study these algorithms, starting with some ofthe very earliest examples, and through the latest theoretical and computational developments.
Approximating Fractional Multicommodity Flow Independent of the Number of Commodities
, 1999
"... We describe fully polynomial time approximation schemes for various multicommodity flow problems in graphs with m edges and n vertices. We present the first approximation scheme for maximum multicommodity flow that is independent of the number of commodities k, and our algorithm improves upon the ru ..."
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Cited by 95 (6 self)
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We describe fully polynomial time approximation schemes for various multicommodity flow problems in graphs with m edges and n vertices. We present the first approximation scheme for maximum multicommodity flow that is independent of the number of commodities k, and our algorithm improves upon the runtime of previous algorithms by this factor of k, performing in O (ffl \Gamma2 m 2 ) time. For maximum concurrent flow, and minimum cost concurrent flow, we present algorithms that are faster than the current known algorithms when the graph is sparse or the number of commodities k is large, i.e. k ? m=n. Our algorithms build on the framework proposed by Garg and Konemann [4]. They are simple, deterministic, and for the versions without costs, they are strongly polynomial. Our maximum multicommodity flow algorithm extends to an approximation scheme for the maximum weighted multicommodity flow, which is faster than those implied by previous algorithms by a factor of k= log W where W is ...
OnLine Routing of Virtual Circuits with Applications to Load Balancing and Machine Scheduling
, 1993
"... In this paper we study the problem of online allocation of routes to virtual circuits (both pointtopoint and multicast) where the goal is to minimize the required bandwidth. We concentrate on the case of permanent virtual circuits (i.e., once a circuit is established, it exists forever), and descr ..."
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Cited by 77 (7 self)
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In this paper we study the problem of online allocation of routes to virtual circuits (both pointtopoint and multicast) where the goal is to minimize the required bandwidth. We concentrate on the case of permanent virtual circuits (i.e., once a circuit is established, it exists forever), and describe an algorithm that achieves an O(log n) competitive ratio with respect to maximum congestion, where n is the number of nodes in the network. Informally, our results show that instead of knowing all of the future requests, it is sufficient to increase the bandwidth of the communication links by an O(log n) factor. We also show that this result is tight, i.e. for any online algorithm there exists a scenario in which O(log n) increase in bandwidth is necessary. We view virtual circuit routing as a generalization of an online load balancing problem, defined as follows: jobs arrive on line and each job must be assigned to one of the machines immediately upon arrival. Assigning a job to a machine increases this machine’s load by an amount that depends both on the job and on the machine. The goal is to minimize the maximum load. For the related machines case, we describe the first algorithm that achieves constant competitive ratio. For the unrelated case (with n machines), we describe a new method that yields O(log n)competitive
Sequential and parallel algorithms for mixed packing and covering
 IN 42ND ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 2001
"... We describe sequential and parallel algorithms that approximately solve linear programs with no negative coefficients (a.k.a. mixed packing and covering problems). For explicitly given problems, our fastest sequential algorithm returns a solution satisfying all constraints within a ¦ ¯ factor in Ç ..."
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Cited by 46 (2 self)
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We describe sequential and parallel algorithms that approximately solve linear programs with no negative coefficients (a.k.a. mixed packing and covering problems). For explicitly given problems, our fastest sequential algorithm returns a solution satisfying all constraints within a ¦ ¯ factor in Ç Ñ � ÐÓ � Ñ � ¯ time, where Ñ is the number of constraints and � is the maximum number of constraints any variable appears in. Our parallel algorithm runs in time polylogarithmic in the input size times ¯ � and uses a total number of operations comparable to the sequential algorithm. The main contribution is that the algorithms solve mixed packing and covering problems (in contrast to pure packing or pure covering problems, which have only “� ” or only “� ” inequalities, but not both) and run in time independent of the socalled width of the problem.
Routing and Admission Control in General Topology Networks with Poisson Arrivals
 7th ACMSIAM Symposium on Discrete Algorithms
, 1996
"... Emerging high speed networks will carry traffic for services such as videoondemand and video teleconferencing  that require resource reservation along the path on which the traffic is sent. High bandwidthdelay product of these networks prevents circuit rerouting, i.e. once a circuit is routed o ..."
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Cited by 40 (3 self)
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Emerging high speed networks will carry traffic for services such as videoondemand and video teleconferencing  that require resource reservation along the path on which the traffic is sent. High bandwidthdelay product of these networks prevents circuit rerouting, i.e. once a circuit is routed on a certain path, the bandwidth taken by this circuit remains unavailable for the duration (holding time) of this circuit. As a result, such networks will need effective routing and admission control strategies. Recently developed online routing and admission control strategies have logarithmic competitive ratios with respect to the admission ratio (the fraction of admitted circuits). Such guarantees on performance are rather weak in the most interesting case where the rejection ratio of the optimum algorithm is very small or even 0. Unfortunately, these guarantees can not be improved in the context of the considered models, making it impossible to use these models to identify algorithms th...
Approximate MinimumCost Multicommodity Flows In
, 1995
"... We show that an \epsilonapproximate solution of the costconstrained Kcommodity flow problem on an Nnode Marc network G can be computed by sequentially solving O(K(\epsilon^{2} log K) log M log(\epsilon^{1}K) singlecommodity minimumcost flow problems on the same network. In particular, an ap ..."
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Cited by 26 (0 self)
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We show that an \epsilonapproximate solution of the costconstrained Kcommodity flow problem on an Nnode Marc network G can be computed by sequentially solving O(K(\epsilon^{2} log K) log M log(\epsilon^{1}K) singlecommodity minimumcost flow problems on the same network. In particular, an approximate minimumcost multicommodity flow can be computed in O^~(\epsilon^{2}KNM) running time, where the notation O^~(.) means "up to logarithmic factors". This result improves the time bound mentioned in Grigoriadis and Khachiyan (1994) by a factor of M/N and that developed recently in Karger and Plotkin(1995) by a factor of \epsilon^{1}. We also provide a simple O^~(NM)time algorithm for singlecommodity budgetconstrained minimumcost flows which is O^~(\epsilon^{3}) times faster than the algorithm of Karger and Plotkin (1995).
An Implementation of a Combinatorial Approximation Algorithm for MinimumCost Multicommodity Flow
, 1997
"... The minimumcost multicommodity flow problem involves simultaneously shipping multiple commodities through a single network so that the total flow obeys arc capacity constraints and has minimum cost. ..."
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Cited by 26 (3 self)
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The minimumcost multicommodity flow problem involves simultaneously shipping multiple commodities through a single network so that the total flow obeys arc capacity constraints and has minimum cost.
On the number of iterations for dantzigwolfe optimization and packingcovering approximation algorithms
 In Proceedings of the 7th International IPCO Conference
, 1999
"... We start with definitions given by Plotkin, Shmoys, and Tardos [16]. Given A ∈ IR m×n, b ∈ IR m and a polytope P ⊆ IR n,thefractional packing problem is to find an x ∈ P such that Ax ≤ b if such an x exists. An ɛapproximate solution to this problem is an x ∈ P such that Ax ≤ (1 + ɛ)b. Anɛrelaxed d ..."
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Cited by 21 (2 self)
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We start with definitions given by Plotkin, Shmoys, and Tardos [16]. Given A ∈ IR m×n, b ∈ IR m and a polytope P ⊆ IR n,thefractional packing problem is to find an x ∈ P such that Ax ≤ b if such an x exists. An ɛapproximate solution to this problem is an x ∈ P such that Ax ≤ (1 + ɛ)b. Anɛrelaxed decision
Faster Approximation Schemes for Fractional Multicommodity Flow Problems
"... We present fully polynomial approximation schemes for concurrent multicommodity flow problems that run in time of minimum possible dependency on the number of commodities k. We showthat by modifying the algorithms by Garg & K"onemann [7] and Fleischer [5] we can reduce their running ..."
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Cited by 20 (0 self)
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We present fully polynomial approximation schemes for concurrent multicommodity flow problems that run in time of minimum possible dependency on the number of commodities k. We showthat by modifying the algorithms by Garg & K&quot;onemann [7] and Fleischer [5] we can reduce their running time on a graph with n vertices and m edges from ~O(&quot;2(m2 + km)) to ~O(&quot;2m2) foran implicit representation of the output, or ~ O(&quot;2(m2 + kn)) for an explicit representation, where ~ O(f) denotes a quantity that is O(f logO(1) m). The implicit representation consists of a set oftrees rooted at sources (there can be more than one tree per source), and with sinks as their leaves, together with flow values for the flow directed from the source to the sinks in a particular tree.Given this implicit representation, the approximate value of the concurrent flow is known, but if we want the explicit flow per commodity per edge, we would have to combine all these trees together,and the cost of doing so may be prohibitive. In case we want to calculate explicitly the solution flow, we modify our schemes so that they run in time polylogarithmic in nk (n is the numberof nodes in the network). This is within a polylogarithmic factor of the trivial lower bound of time \Omega (nk) needed to explicitly write down a multicommodity flow of k commodities in a network of n nodes. Therefore our schemes are within a polylogarithmic factor of the minimum possible dependency of the running time on the number of commodities k.