This paper presents fast algorithms that find approximate solutions for a general class of problems, which we call fractional packing and covering problems. The only previously known algorithms for solving these problems are based on general linear programming techniques. The techniques developed in this paper greatly outperform the general methods in many applications, and are extensions of a method previously applied to find approximate solutions to multicommodity flow problems. Our algorithm is a Lagrangean relaxation technique; an important aspect of our results is that we obtain a theoretical analysis of the running time of a Lagrangean relaxation-based algorithm. We give several applications of our algorithms. The new approach yields several orders of magnitude of improvement over the best previously known running times for algorithms for the scheduling of unrelated parallel machines in both the preemptive and the non-preemptive models, for the job shop problem, for th...

All previously known algorithms for solving the multicommodity flow problem with capacities are based on linear programming. The best of these algorithms [15] uses a fast matrix multiplication algorithm and takes O(k 3:5 n 3 m :5 log(nDU )) time for the multicommodity flow problem with integer demands and at least O(k 2:5 n 2 m :5 log(nffl \Gamma1 DU )) time to find an approximate solution, where k is the number of commodities, n and m denote the number of nodes and edges in the network, D is the largest demand, and U is the largest edge capacity. Substantially more time is needed to find an exact solution. As a consequence, even multicommodity flow problems with just a few commodities are believed to be much harder than single-commodity maximum-flow or minimum-cost flow problems. In this paper, we describe the first polynomial-time combinatorial algorithms for approximately solving the multicommodity flow problem. The running time of our randomized algorithm i...

Abstract. This paper describes new algorithms for approximately solving the concurrent multicommodity flow problem with uniform capacities. These algorithms are much faster than algorithms discovered previously. Besides being an important problem in its own right, the uniform-capacity concurrent flow problem has many interesting applications. Leighton and Rao used uniform-capacity concurrent flow to find an approximately "sparsest cut " in a graph and thereby approximately solve a wide variety of graph problems, including minimum feedback arc set, minimum cut linear arrangement, and minimum area layout. However, their method appeared to be impractical as it required solving a large linear program. This paper shows that their method might be practical by giving an O(m log m) expectedtime randomized algorithm for their concurrent flow problem on an m-edge graph. Raghavan and Thompson used uniform-capacity concurrent flow to solve approximately a channel width minimization problem in very large scale integration. An O (k 3/2 (m + n log n)) expected-time randomized algorithm and an O (k min {n, k} (m + n log n) log k) deterministic algorithm is given for this problem when the channel width is f2 (log n), where k denotes the number of wires to be routed in an n-node, m-edge network. Key words, multicommodity flow, approximation, concurrent flow, graph separators, VLSI routing AMS subject classification. 68Q25, 90C08, 90C27 1. Introduction. The

Minimum cost multicommodity flow is an instance of a simpler problem (multicommodity flow) to which a cost constraint has been added. In this paper we present a general scheme for solving a large class of such "cost-added" problems---even if more than one cost is added. One of the main applications of this method is a new deterministic algorithm for approximately solving the minimumcost multicommodity flow problem. Our algorithm finds a (1 + ffl) approximation to the minimum cost flow in ~ O(ffl \Gamma3 kmn) time, where k is the number of commodities, m is the number of edges, and n is the number vertices in the input problem. This improves the previous best deterministic bounds of O(ffl \Gamma4 kmn 2 ) [9] and ~ O(ffl \Gamma2 k 2 m 2 ) [15] by factors of n=ffl and fflkm=n respectively. In fact, it even dominates the best randomized bound of ~ O(ffl \Gamma2 km 2 ) [15]. The algorithm presented in this paper efficiently solves several other interesting generali...

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We consider a generalization of the maximum flow problem in which the amounts of flow entering and leaving an arc are linearly related. More precisely, if x(e) units of flow enter an arc e, x(e)fl(e) units arrive at the other end. For instance, nodes of the graph can correspond to different currencies, with the multipliers being the exchange rates. We require conservation of flow at every node except a given source node. The goal is to maximize the amount of flow excess at the source. This problem is a special case of linear programming, and therefore can be solved in polynomial time. In this paper we present the first polynomial time combinatorial algorithms for this problem. The algorithms are simple and intuitive.

The minimum-cost 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.

The multicommodity flow problem involves simultaneously shipping multiple commodities through a single network so that the total amount of flow on each edge is no more than the capacity of the edge. This problem can be expressed as a large linear program, and most known algorithms for it, both theoretical and practical, are linear programming algorithms designed to take advantage of the structure of multicommodity flow problems. The size of the linear programs, however, makes it prohibitively difficult to solve large multicommodity flow problems. In this paper, we describe and examine a multicommodity flow implementation based on the recent combinatorial approximation algorithm of Leighton et al. [13]. The theory predicts that the running time of the algorithm increases linearly with the number of commodities. Our experiments verify this behavior. The theory also predicts that the running time increases as the square of the desired precision. Our experiments show that the running time ...

In this paper, we present a new interior-point based polynomial algorithm for the multicommodity flow problem and its variants. Unlike all previously known interior point algorithms for multicommodity flow that have the same complexity for approximate and exact solutions, our algorithm improves running time in the approximate case by a polynomial factor. For many cases, the exact bounds are better as well. Instead of using the conventional linear programming formulation for the multicommodity flow problem, we model it as a quadratic optimization problem which is solved using interior-point techniques. This formulation allows us to exploit the underlying structure of the problem and to solve it efficiently. The algorithm is also shown to have improved stability properties. The improved complexity results extend to minimum cost multicommodity flow, concurrent flow and generalized flow problems. 1 Introduction The multicommodity flow problem is the problem of finding several network flo...

Minimum-cost multicommodity flow problem is one of the classical optimization problems that arises in a variety of contexts. Applications range from finding optimal ways to route information through communication networks to VLSI layout. In this paper, we describe an efficient deterministic approximation algorithm, which given that there exists a multicommodity flow of cost B that satisfies all the demands, produces a flow of cost at most (1 + ffi )B that satisfies (1 \Gamma ffl)-fraction of each demand. For constant ffi and ffl, our algorithm runs in O (kmn 2 ) time, which is an improvement over the previously fastest (deterministic) approximation algorithm for this problem due to Plotkin, Shmoys, and Tardos, that runs in O (k 2 m 2 ) time. The presented algorithm is inherently parallel and can be implemented to run in O (mn) time on PRAM with linear number of processors, instead of O (kmn) time with O(n 3 ) processors of the previously known approximation algorit...