In this paper, we present a new strongly polynomial time algorithm for the minimum cost flow problem, based on a refinement of the Edmonds-Karp scaling technique. Our algorithm solves the uncapacitated minimum cost flow problem as a sequence of O(n log n) shortest path problems on networks with n nodes and m arcs and runs in O(n log n (m + n log n)) time. Using a standard transformation, thjis approach yields an O(m log n (m + n log n)) algorithm for the capacitated minimum cost flow problem. This algorithm improves the best previous strongly polynomial time algorithm, due to Z. Galil and E. Tardos, by a factor of n 2 /m. Our algorithm for the capacitated minimum cost flow problem is even more efficient if the number of arcs with finite upper bounds, say n', is much less than m. In this case, the running time of the algorithm is O((m ' + n)log n(m + n log n)).

In this paper we suggest new scaling algorithms for the assignment and minimum mean cycle problems. Our assignment algorithm is based on applying scaling to a hybrid version of the recent auction algorithm of Bertsekas and the successive shortest path algorithm. The algorithm proceeds by relaxing the optimality conditions, and the amount of relaxation is successively reduced to zero. On a network with 2n nodes, m arcs, and integer arc costs bounded by C, the algorithm runs in O(,/-n m log(nC)) time and uses very simple data structures. This time bound is comparable to the time taken by Gabow and Tarjan's scaling algorithm, and is better than all other time bounds under the similarity assumption, i.e., C = O(n k) for some k. We next consider the minimum mean cycle problem. The mean cost of a cycle is defined as the cost of the cycle divided by the number of arcs it contains. The minimum mean cycle problem is to identify a cycle whose mean cost is minimum. We show that by using ideas of the assignment algorithm in an approximate binary search procedure, the minimum mean cycle problem can also be solved in O(~/n m log nC) time. Under the similarity assumption, this is the best available time bound to solve the minimum mean cycle problem.

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This paper aims at describing the state of the art on linear assignment problems (LAPs). Besides sum LAPs it discusses also problems with other objective functions like the bottleneck LAP, the lexicographic LAP, and the more general algebraic LAP. We consider different aspects of assignment problems, starting with the assignment polytope and the relationship between assignment and matching problems, and focusing then on deterministic and randomized algorithms, parallel approaches, and the asymptotic behaviour. Further, we describe different applications of assignment problems, ranging from the well know personnel assignment or assignment of jobs to parallel machines, to less known applications, e.g. tracking of moving objects in the space. Finally, planar and axial three-dimensional assignment problems are considered, and polyhedral results, as well as algorithms for these problems or their special cases are discussed. The paper will appear in the Handbook of Combinatorial Optimization to be published

In this paper, we study network flow algorithms for bipartite networks. A network G = (V; E) is called bipartite if its vertex set V can be partitioned into two subsets V 1 and V 2 such that all edges have one endpoint in V 1 and the other in V 2 . Let n = jV j, n 1 = jV 1 j, n 2 = jV 2 j, m = jEj and assume without loss of generality that n 1 n 2 . We call a bipartite network unbalanced if n 1 ø n 2 and balanced otherwise. (This notion is necessarily imprecise.) We show that several maximum flow algorithms can be substantially sped up when applied to unbalanced networks. The basic idea in these improvements is a two-edge push rule that allows us to "charge" most computation to vertices in V 1 , and hence develop algorithms whose running times depend on n 1 rather than n. For example, we show that the two-edge push version of Goldberg and Tarjan's FIFO preflow push algorithm runs in O(n 1 m + n 3 1 ) time and that the analogous version of Ahuja and Orlin's excess scaling algori...

In the last two decades, a number of algorithms for the linear single-commodity Min Cost Flow problem (MCF) have been proposed, and several efficient codes are available that implement different variants of the algorithms. The practical significance of the algorithms has been tested by comparing the time required by their implementations for solving "from scratch" instances of (MCF), of different classes, as the size of the problem (number of nodes and arcs) increases. However, in many applications several closely related instances of (MCF) have to be sequentially solved, so that reoptimization techniques can be used to speed up computations, and the most attractive algorithm is the one which minimizes the total time required to solve all the instances in the sequence. In this paper we compare the performances of four different efficient implementations of algorithms for (MCF) under cost reoptimization in the context of decomposition algorithms for the Multicommodity Min Cost Flow problem (MMCF), showing that for some classes of instances the relative performances of the codes doing "from scratch" optimization do not accurately predict the relative performances when reoptimization is used. Since the best solver depends both on the class and on the size of the instance, this work also shows the usefulness of a standard interface for (MCF) problem solvers that we have proposed and implemented.

In this paper we consider the asymmetric assignment problem and we propose a new auction algorithm for its solution. The algorithm uses in a novel way the recently proposed idea of reverse auction, where in addition to persons bidding for objects by raising their prices, we also have objects competing for persons by essentially o#ering discounts. In practice, the new algorithm apparently deals better with price wars than the currently existing auction algorithms. As a result it frequently does not require #-scaling for good practical performance, and tends to terminate substantially (and often dramatically) faster than its competitors. 1 This work was supported in part by NSF under Grant CCR-9108058, and in part by the BM/C3 Technology branch of the United States Army Strategic Defense Command. It will be published also by the Journal of Computational Optimization and its Applications. 2 Department of Electrical Engineering and Computer Science, M. I. T., Cambridge, Mass., 02139. 3 Department of Electrical Engineering, Boston University, and ALPHATECH, Inc., Burlington, Mass., 01803. 1 1.

. This paper presents a unified analysis of decomposition algorithms for continuously differentiable optimization problems defined on Cartesian products of convex feasible sets. The decomposition algorithms are analyzed using the framework of cost approximation algorithms. A convergence analysis is made for three decomposition algorithms: a sequential algorithm which extends the classical Gauss--Seidel scheme, a synchronized parallel algorithm which extends the Jacobi method, and a partially asynchronous parallel algorithm. The analysis validates inexact computations in both the subproblem and line search phases, and includes convergence rate results. The range of feasible step lengths within each algorithm is shown to have a direct correspondence to the increasing degree of parallelism and asynchronism, and the resulting usage of more outdated information in the algorithms. Keywords: Cartesian product sets, decomposition, cost approximation, sequential algorithm, parallel processing,...

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This paper considers the polynomially solvable case in which there is a single depot and one type of vehicle. Several assignment types of formulations are discussed and a new quasi-assignment algorithm is provided. This algorithm is based on an auction algorithm for the linear assignment problem. Furthermore, several ways for dealing with a large number of feasible connections between trips, a bottleneck for vehicle scheduling, are discussed. An important contribution of this paper is a computational study that compares our algorithms with the most successful algorithms for the vehicle scheduling problem, using both randomly generated and real life data. This can serve as a guide for both scientists and practioners. The new quasi-assignment algorithm is shown to be considerably faster for most of the test problems. This improvement in computation time is also very important when this particular vehicle scheduling problem appears as a subproblem in more complex vehicle and crew scheduling problems.