Algorithms For Pure And Generalized Network Flow Problems

In the past few decades, there has been a large amount of work on algorithms for linear network flow problems, special classes of network problems such as assignment problems (linear and quadratic), Steiner tree problem, topology network design and nonconvex cost network flow problems.Network optimization problems find numerous applications in transportation, in communication network design, in production and inventory planning, in facilities location and allocation, and in VLSI design.The purpose of this book is to cover a spectrum of recent developments in network optimization problems, from linear networks to general nonconvex network flow problems./a

Linear programming; the simplex method for network program; the out-of-kilter algorithm for the network program; the simplex method for the generalized network problem; the multicommodity network flow problem; the simplex method for the network with side constraints model; appendixes: characterization of a tree; data structures for network programs; convergence of subgradient optimization algorithm; projection operation for subgradient algorithm; a product form representation of the inverse of a multicommodity cycle matrix; NETFLO; references; index.

Michael Holzhauser discusses generalizations of well-known network flow and packing problems by additional or modified side constraints. By exploiting the inherent connection between the two problem classes, the author investigates the complexity and approximability of several novel network flow and packing problems and presents combinatorial solution and approximation algorithms.

Network flow theory has been used across a number of disciplines, including theoretical computer science, operations research, and discrete math, to model not only problems in the transportation of goods and information, but also a wide range of applications from image segmentation problems in computer vision to deciding when a baseball team has been eliminated from contention. This graduate text and reference presents a succinct, unified view of a wide variety of efficient combinatorial algorithms for network flow problems, including many results not found in other books. It covers maximum flows, minimum-cost flows, generalized flows, multicommodity flows, and global minimum cuts and also presents recent work on computing electrical flows along with recent applications of these flows to classical problems in network flow theory.

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The specialization of the simplex algorithm for the solution of generalized network flow problems rests on the fact that a basis for the problem may be represented graphically as a spanning forest in which each component is either a one-tree or a rooted tree. The design of a specialized algorithm for efficient solution of generalized network problems necessarily depends on data structures chosen to represent the basis. This paper presents the design and detailed algorithmic specification of the primal simplex algorithm for such problems. Computational testing to determine the overhead required by generalized network data structures over pure network data structures indicates that generalized network algorithms are on the order of 2.5 to 3.5 times slower than pure network algorithms. Computational testing with generalized network problems with up to 1000 nodes and 7000 arcs establishes the suitability of the data-structures for efficient implementation of primal simplex calculations. Keywords: Linear programming. (Author).