Improved Primal Simplex Algorithms for Shortest Path, Assignment and Minimum

Ravindra K. Ahuja, James B. Orlin

ISBN 10: 1332265103 / ISBN 13: 9781332265107

Published by Forgotten Books, 2020

About this Item

Print on Demand. This book introduces a new and improved primal simplex pivot rule for the minimum cost flow problem, demonstrating its worst-case complexity and analyzing its implementation for the shortest path and assignment problems. The author expands on existing research and presents a detailed complexity analysis, providing valuable insights into the potential of this rule. This book positions itself as a significant contribution to the field of network optimization and is essential reading for researchers and practitioners interested in efficient algorithms for network flow problems. Its in-depth exploration of the scaling pivot rule and its applications make it an invaluable resource for advancing the theory and practice of network optimization. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book. Seller Inventory # 9781332265107_0

Bibliographic Details

Title: Improved Primal Simplex Algorithms for ...

Publisher: Forgotten Books

Publication Date: 2020

Binding: Paperback

Condition: New

Book Type: print-on-demand item

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Synopsis:

Excerpt from Improved Primal Simplex Algorithms for Shortest Path, Assignment and Minimum Cost Flow Problems: November, 1988

The network simplex algorithm maintains a feasible basis structure at each iteration and successively modifies the basis structure via pivots until it becomes an optimum basis structure. The special structure of the basis enables the simplex computations to be performed very efficiently. In the following discussion, we give a brief summary of the network simplex algorithm and the data structure required to implement the algorithm.

The basis B of the minimum cost flow problem is a spanning tree. We consider this tree as hanging from node 1. The tree arcs either are upward pointing (towards node 1) or are downward pointing (away from node We associate three indices with each node i in the tree: a predecessor index pred( i), a depth index depth( i), and a thread index, thread( i). Each node i has a unique path connecting it to node 1. The predecessor index stores the first node in that path (other than node i and the depth index stores the number of arcs in the path For node 1, these indices are zero. We say that pred(i) is the predecessor of node i, and i is the successor of node pred( i). The descendants of a node i consist of node i itself, its successors, successors of its successors, and so on. The set of descendants of a node i induce a subtree rooted at node i. We denote the nodes of the subtree by the set D(i). Node i is called an ancestor of nodes in D(i). The thread indices define a traversal of the tree, a sequence of nodes that walks or threads its way through the nodes of the tree, starting at node 1 and visiting nodes in a top to bottom and left to right order, and then finally retuming to node 1. The thread indices can be formed by performing a depth first search of the tree. The thread indices provide a means for visiting (or finding) all descendants of a node i in C(| D(i) I) time. For a detailed description of the tree indices see Kennington and Helgason In the basis there is a unique path connecting any two nodes. We refer to this path as the basis path.

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This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.