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)).

. The scaling push-relabel method is an important theoretical development in the area of minimum-cost flow algorithms. We study practical implementations of this method. We are especially interested in heuristics which improve real-life performance of the method. Our implementation works very well over a wide range of problem classes. In our experiments, it was always competitive with the established codes, and usually outperformed these codes by a wide margin. Some heuristics we develop may apply to other network algorithms. Our experimental work on the minimum-cost flow problem motivated theoretical work on related problems. Supported in part by ONR Young Investigator Award N00014-91-J-1855, NSF Presidential Young Investigator Grant CCR-8858097 with matching funds from AT&T and DEC, Stanford University Office of Technology Licensing, and a grant form the Powell Foundation. 1 1. Introduction. Significant theoretical progress has been made recently in the area of minimum-cost flow ...

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We review a class of recently-proposed linear-cost network flow methods which are amenable to distributed implementation. All the methods in the class use the notion of e-complementary slackness, and most do not explicitly manipulate any "global " objects such as paths, trees, or cuts. Interestingly, these methods have stimulated a large number of new serial computational complexity results. We develop the basic theory of these methods and present two specific methods, the e-relaxation algorithm for the minimum-cost flow problem, and the auction algorithm for the assignment problem. We show how to implement these methods with serial complexities of O(N 3 log NC) and O(NA log NC), respectively. We also discuss practical implementation issues and computational experience to date. Finally, we show how to implement e-relaxation in a completely asynchronous, "chaotic" environment in which some processors compute faster than others, some processors communicate faster than others, and there can be arbitrarily large communication delays.

We present a simple sequential algorithm for the maximum flow problem on a network with n nodes, m arcs, and integer arc capacities bounded by U. Under the practical assumption that U is polynomially bounded in n, our algorithm runs in time O(nm + n 2 log n). This result improves the previous best bound of O(nm log(n 2 /m)), obtained by Goldberg and Taran, by a factor of log n for networks that are both nonsparse and nondense without using any complex data structures. We also describe a parallel implementation of the algorithm that runs in O(n'log U log p) time in the PRAM model with EREW and uses only p processors where p = [m/n

For every family of sets F ` f0; 1g n the following problems are strongly polynomial time equivalent: given a feasible point x 0 2 F and a linear objective function c 2 ZZ n , ffl find a feasible point x 2 F that maximizes c x (Optimization), ffl find a feasible point x new 2 F with c x new ? c x 0 (Augmentation), and ffl find a feasible point x new 2 F with c x new ? c x 0 such that x new \Gamma x 0 is "irreducible" (Irreducible Augmentation). This generalizes results and techniques that are well known for 0=1-- integer programming problems that arise from various classes of combinatorial optimization problems.

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We propose two new methods for the solution of the single commodity, separable convex cost network flow problem: the #-relaxation method and the auction/sequential shortest path method. Both methods were originally developed for linear cost problems and reduce to their linear conterparts when applied to such problems. We show that both methods stem from a common algorithmic framework, that they terminate with a near optimal solution, and we provide an associated complexity analysis. We also present computational results showing that these methods are much faster than earlier relaxation methods, particularly for ill-conditioned problems.

The minimum cost flow problem is perhaps the most useful problem in the traditional network literature. It combines two well known special cases — the maximum flow and the shortest route problem. The problem statement is given below: 1.1 Problem Given a directed network G =[N; A] andrealnumbersqifor each i ∈ N and li,j, ui,j,andci,j for each (i, j) ∈ A, find arc flows fi,j satisfying the relations: X (fi,j − fj,i) =qi ∀ i ∈ N (1.1)

Contents Preface 7 Lecture I. The Stable Marriage and Stable Roommates Problems 8 1. The Stable Marriage Problem 8 1.1. Definition of the Stable Marriage Problem 8 1.2. The Gale-Shapley Algorithm 9 1.3. Correctness 10 1.4. Male Optimality 10 2. Introduction to Network Stability 11 2.1. The Stable Roommates Problem 11 2.2. Network Definitions 11 2.3. The Stable Roommate Problem and Network Stability 13 Lecture II. The Stable Roommates Problem and Network Stability 15 1. The Stable Roommates Problem and X-Network Stability 15 2. Adjacency-Preserving Circuits and X-Networks 16 3. Network Stability and Simplification 16 Lecture III. The Maximum Flow Problem 20 1. Network Flows 20 1.1. Flows and Pseudo-flows 20 1.2. Cuts 20 1.3. Residual Networks 21 1.4. Augmenting Paths 22 2. The Augmenting Path Algorithm 22 CONTENTS 3 3. The Decomposition Theorem 24 4. The Capacity Scaling Algorithm 26 Lecture IV. Th