Polynomial-time highest-gain augmenting path algorithms for the generalized circulation problem

Includes bibliographical references (p. 15-16).Supported in part by NSF. DMS 94-14438 DMS 95-27124 Supported in part by DOE. DE-FG02-92ER25126 Supported as well by grants from UPS and ONR. N00014-96-1-0051by Donald Goldfarb, Zhiying Jin, James B. Orlin

A faster strongly polynomial minimum cost flow algorithm

Bibliography: p. 11.Supported in part by a grant from the National Science Foundation Presidential Young Investigator. 8451517-ECS Supported in part by a grant from the Air Force Office of Scientific Research. AFOSR-88-0088 Supported in part by grants from Analog Devices, Apple Computer Inc., and Prime Computer.James B. Orlin

On Combinatorial Network Flows Algorithms and Circuit Augmentation for Pseudoflows

There is a wealth of combinatorial algorithms for classical min-cost flow problems and their simpler variants like max flow or shortest path problems. It is well-known that many of these algorithms are related to the Simplex method and the more general circuit augmentation schemes: prime examples are the network Simplex method, a refinement of the primal Simplex method, and min-mean cycle canceling, which corresponds to a steepest-descent circuit augmentation scheme. We are interested in a deeper understanding of the relationship between circuit augmentation and combinatorial network flows algorithms. To this end, we generalize from the consideration of primal or dual flows to so-called pseudoflows, which adhere to arc capacities but allow for a violation of flow balance. We introduce `pseudoflow polyhedra,' in which slack variables are used to quantify this violation, and characterize their circuits. This enables the study of combinatorial network flows algorithms in view of the walks that they trace in these polyhedra, and in view of the pivot rules for the steps. In doing so, we provide an `umbrella,' a general framework, that captures several algorithms. We show that the Successive Shortest Path Algorithm for min-cost flow problems, the Shortest Augmenting Path Algorithm for max flow problems, and the Preflow-Push algorithm for max flow problems lead to (non-edge) circuit walks in these polyhedra. The former two are replicated by circuit augmentation schemes for simple pivot rules. Further, we show that the Hungarian Method leads to an edge walk and is replicated, equivalently, as a circuit augmentation scheme or a primal Simplex run for a simple pivot rule

A faster strongly polynomial minimum cost flow algorithm

Also issued as: Working Paper (Sloan School of Management) ; WP 2042-88Includes bibliographical references (leaf 11).Supported by a National Science Foundation Presidential Young Investigator Grant. 8451517-ECS Supported by the Air Force Office of Scientific Research, Analog Devices, Apple Computer, and Prime Computer.by James B. Orlin

Concave Generalized Flows with Applications to Market Equilibria

We consider a nonlinear extension of the generalized network flow model, with the flow leaving an arc being an increasing concave function of the flow entering it, as proposed by Truemper and Shigeno. We give a polynomial time combinatorial algorithm for solving corresponding flow maximization problems, finding an epsilon-approximate solution in O(m(m+log n)log(MUm/epsilon)) arithmetic operations and value oracle queries, where M and U are upper bounds on simple parameters. This also gives a new algorithm for linear generalized flows, an efficient, purely scaling variant of the Fat-Path algorithm by Goldberg, Plotkin and Tardos, not using any cycle cancellations. We show that this general convex programming model serves as a common framework for several market equilibrium problems, including the linear Fisher market model and its various extensions. Our result immediately extends these market models to more general settings. We also obtain a combinatorial algorithm for nonsymmetric Arrow-Debreu Nash bargaining, settling an open question by Vazirani.Comment: Major revision. Instead of highest gain augmenting paths, we employ the Fat-Path framework. Many parts simplified, running time for the linear case improve

An Algorithmic Theory of Dependent Regularizers, Part 1: Submodular Structure

We present an exploration of the rich theoretical connections between several classes of regularized models, network flows, and recent results in submodular function theory. This work unifies key aspects of these problems under a common theory, leading to novel methods for working with several important models of interest in statistics, machine learning and computer vision. In Part 1, we review the concepts of network flows and submodular function optimization theory foundational to our results. We then examine the connections between network flows and the minimum-norm algorithm from submodular optimization, extending and improving several current results. This leads to a concise representation of the structure of a large class of pairwise regularized models important in machine learning, statistics and computer vision. In Part 2, we describe the full regularization path of a class of penalized regression problems with dependent variables that includes the graph-guided LASSO and total variation constrained models. This description also motivates a practical algorithm. This allows us to efficiently find the regularization path of the discretized version of TV penalized models. Ultimately, our new algorithms scale up to high-dimensional problems with millions of variables

Network Flows

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