On the simplex algorithm for networks and generalized networks

Bibliography: p. 21-22.by James B. Orlin

A genuinely polynomial primal simplex algorithm for the assignment problem

Cataloged from PDF version of article.Akgil, M., A genuinely polynomial primal simplex algorithm for the assignment problem, Discrete Applied Mathematics 45 (1993) 93-l 15. We present a primal simplex algorithm that solves the assignment problem in :n(n+3)-4 pivots. Starting with a problem of size 1, we sequentially solve problems of size 2,3,4,. ..,lt. The algorithm utilizes degeneracy by working with strongly feasible trees and employs Dantdg’s rule for entering edges for the subproblem. The number of nondegenerate simplex pivots is bounded by n-l. The number of consecutive degenerate simplex pivots is bounded by : (n-2)(n+ 1). All three bounds are sharp. The algorithm can be implemented to run in O(ni) time for dense graphs. For sparse graphs, using state of the art data structures, it runs in O(n2 log n+nm) time, where the bipartite graph has 2n nodes and m edges

A polynomial time primal network simplex algorithm for minimum cost flows

Cover title.Includes bibliographical references (p. 25-27).Supported by ONR. N00014-94-1-0099 Supported in part by a grant from the UPS foundation.by James B. Orlin

Constrained shortest paths for QoS routing and path protection in communication networks.

The CSDP (k) problem requires the selection of a set of k > 1 link-disjoint paths with minimum total cost and with total delay bounded by a given upper bound. This problem arises in the context of provisioning paths in a network that could be used to provide resilience to link failures. Again we studied the LP relaxation of the ILP formulation of the problem from the primal perspective and proposed an approximation algorithm.We have studied certain combinatorial optimization problems that arise in the context of two important problems in computer communication networks: end-to-end Quality of Service (QoS) and fault tolerance. These problems can be modeled as constrained shortest path(s) selection problems on networks with each of their links associated with additive weights representing the cost, delay etc.The problems considered above assume that the network status is known and accurate. However, in real networks, this assumption is not realistic. So we considered the QoS route selection problem under inaccurate state information. Here the goal is to find a path with the highest probability that satisfies a given delay upper bound. We proposed a pseudo-polynomial time approximation algorithm, a fully polynomial time approximation scheme, and a strongly polynomial time heuristic for this problem.Finally we studied the constrained shortest path problem with multiple additive constraints. Using the LARAC algorithm as a building block and combining ideas from mathematical programming, we proposed a new approximation algorithm.First we studied the QoS single route selection problem, i.e., the constrained shortest path (CSP) problem. The goal of the CSP problem is to identify a minimum cost route which incurs a delay less than a specified bound. It can be formulated as an integer linear programming (ILP) problem which is computationally intractable. The LARAC algorithm reported in the literature is based on the dual of the linear programming relaxation of the ILP formulation and gives an approximate solution. We proposed two new approximation algorithms solving the dual problem. Next, we studied the CSP problem using the primal simplex method and exploiting certain structural properties of networks. This led to a novel approximation algorithm

Network Flows

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Combinatorics and Geometry of Transportation Polytopes: An Update

A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have interest for discrete mathematics because permutation matrices, latin squares, and magic squares appear naturally as lattice points of these polytopes. In this paper we survey advances on the understanding of the combinatorics and geometry of these polyhedra and include some recent unpublished results on the diameter of graphs of these polytopes. In particular, this is a thirty-year update on the status of a list of open questions last visited in the 1984 book by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure

An Efficient Extension of Network Simplex Algorithm

In this paper, an efficient extension of network simplex algorithm is presented. In static scheduling problem, where there is no change in situation, the challenge is that the large problems can be solved in a short time. In this paper, the Static Scheduling problem of Automated Guided Vehicles in container terminal is solved by Network Simplex Algorithm (NSA) and NSA+, which extended the standard NSA. The algorithms are based on graph model and their performances are at least 100 times faster than traditional simplex algorithm for Linear Programs. Many random data are generated and fed to the model for 50 vehicles. We compared results of NSA and NSA+ for the static automated vehicle scheduling problem. The results show that NSA+ is significantly more efficient than NSA. It is found that, in practice, NSA and NSA+ take polynomial time to solve problems in this application