PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS

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2 PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS VOLUME 1 VOLUME 2 VOLUME 3 VOLUME 4 VOLUME 5 VOLUME 6 VOLUME 7 VOLUME 8 VOLUME 9 VOLUME 10 VOLUME 11 VOLUME 12 VOLUME 13 VOLUME 14 VOLUME 15 VOLUME 16 NON-LINEAR PROBLEMS IN MECHANICS OF CONTINUA Edited by E. Reissner (Brown University, August 1947) ELECTROMAGNETIC THEORY Edited by A. H. Taub (Massachusetts Institute of Technology, July 1948) ELASTICITY Edited by R. V. Churchill (University of Michigan, June 1949) FLUID DYNAMICS Edited by M. H. Martin (University of Maryland, June 1951) WAVE MOTION AND VIBRATION THEORY Edited by A. E. Heins (Carnegie Institute of Technology, June 1952) NUMERICAL ANALYSIS Edited by /. H. Curtiss (Santa Monica City College, August 1953) APPLIED PROBABILITY Edited by L. A. MacColl (Polytechnic Institute of Brooklyn, April 1955) CALCULUS OF VARIATIONS AND ITS APPLICATIONS Edited by L. M. Graves (University of Chicago, April 1956) ORBIT THEORY Edited by G. Birkhoff and R. E. Longer (New York University, April 1957) COMBINATORIAL ANALYSIS Edited by R. Bellman and M. Hall, Jr. (Columbia University, April 1958) NUCLEAR REACTOR THEORY Edited by G. Birkhoff and E. P. Wigner (New York City, April 1959) STRUCTURE OF LANGUAGE AND ITS MATHEMATICAL ASPECTS Edited by R. Jakobson (New York City, April 1960) HYDRODYNAMIC INSTABILITY Edited by R. Bellman, G. Birkhoff, C. C. Lin (New York City, April 1960) MATHEMATICAL PROBLEMS IN THE BIOLOGICAL SCIENCES Edited by R. Bellman (New York City, April 1961) EXPERIMENTAL ARITHMETIC, HIGH SPEED COMPUTING, AND MATHEMATICS Edited by N. C. Metropolis, A. H. Taub, /. Todd, C. B. Tompkins (Atlantic City and Chicago, April 1962) STOCHASTIC PROCESSES IN MATHEMATICAL PHYSICS AND ENGIN EERING Edited by R. Bellman (New York City, April 1963)

3 VOLUME 17 VOLUME 18 VOLUME 19 VOLUME 20 VOLUME 21 VOLUME 22 VOLUME 23 VOLUME 24 VOLUME 25 APPLICATIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUA TIONS IN MATHEMATICAL PHYSICS Edited by R. Finn (New York City, April 1964) MAGNETO-FLUID AND PLASMA DYNAMICS Edited by H. Grad (New York City, April 1965) MATHEMATICAL ASPECTS OF COMPUTER SCIENCE Edited by J. T. Schwartz (New York City, April 1966) THE INFLUENCE OF COMPUTING ON MATHEMATICAL RESEARCH AND EDUCATION Edited by J. P. LaSalle (University of Montana, August 1973) MATHEMATICAL ASPECTS OF PRODUCTION AND DISTRIBUTION OF ENERGY Edited by P. D. Lax (San Antonio, Texas, January 1976) NUMERICAL ANALYSIS Edited by G. H. Golub and J. Oliger (Atlanta, Georgia, January 1978) MODERN STATISTICS: METHODS AND APPLICATIONS Edited by R. V. Hogg (San Antonio, Texas, January 1980) GAME THEORY AND ITS APPLICATIONS Edited by W. F. Lucas (Biloxi, Mississippi, January 1979) OPERATIONS RESEARCH: MATHEMATICS AND MODELS Edited by S. I. Gass (Duluth, Minnesota, August 1979)

4 AMS SHORT COURSE LECTURE NOTES published as a subseries of Proceedings of Symposia in Applied Mathematics

5 PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS Volume 26 THE MATHEMATICS OF NETWORKS AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND

6 LECTURE NOTES PREPARED FOR THE AMERICAN MATHEMATICAL SOCIETY SHORT COURSE THE MATHEMATICS OF NETWORKS HELD IN PITTSBURGH, PENNSYLVANIA AUGUST 15-16, 1981 EDITED BY STEFAN A. BURR The AMS Short Course Series is sponsored by the Society's Committee on Employment and Education Policy (CEEP). The Series is under the direction of the Short Course Advisory Subcommittee of CEEP. library of Congress Cataloging in Publication Data The Mathematics of networks. (Proceedings of symposia in applied mathematics, ISSN ; v. 26) "Lecture notes prepared for the American Mathematical Society Short Course, The Mathematics of Networks held in Pittsburgh, Pennsylvania, August 15-16, 1981, edited by Stefan A. Burr'* T.p. verso. Includes bibliographies. 1. System analysis Congresses. 2. Electric networks Congresses. I. Burr, Stefan A. (Stefan Andrus), II. American Mathematical Society Short Course, The Mathematics of Networks (1981: Pittsburgh, Pa.) III. Series. T57.85.M ' ISBN Mathematics Subject Classification. Primary 90B99 Copyright 1982 by the American Mathematical Society. Printed in the United States of America. All rights reserved except those granted to the United States Government. This book may not be reproduced in any form without the permission of the publishers. This volume was printed directly from copy prepared by the authors. Reprinted 1984

7 CONTENTS Preface Introduction to basic network problems ix by FRANK BOESCH 1 Maximum flows in networks by FRANCES YAO 31 The computational complexity of network problems by R. M. KARP 45 Effective use of heuristic algorithms in network design by SHEN LIN 63 Some practical network problems by DANIEL J. KLEITMAN 85 Telephone switching networks by NICHOLAS PIPPENGER 101 Concluding remarks by STEFAN A. BURR 135 vii

8 PREFACE On August 15 and 16, 1981, an American Mathematical Society Short Course was held on The Mathematics of Networks at the Summer Meeting in Pittsburgh, Pennsylvania. This volume represents a considerable expansion of the lecture notes that were prepared by the speakers for the benefit of the participants. We, the speakers and the organizer, were very pleased with the substantial attendance at the Short Course and by the enthusiastic support of the AMS staff and of the Short Course subcommittee. It was also a pleasure to have contributed to such a distinguished series; this Short Course was the seventeenth in the series, which began in August, The theory of networks is a very lively one, both in terms of developments in the theory itself and in terms of the variety of its applications as the chapters in this volume should show. The range of theory and applications is so great that neither the Short Course nor this volume could hope to give a full and systematic treatment of the subject. However, it was my intention that both would introduce most of the basic ideas of network theory and develop some of these ideas to a considerable extent. In addition, my plan was to introduce a number of more specialized topics, including a number of areas of active research and quite a wide variety of applications, in order to indicate the breadth and depth of the field. Thanks to the contributors, these objectives were met. The first chapter, Introduction to Basic Network Problems, sets the stage. Here, Frank Boesch gives the basic definitions in the mathematics of networks and in the closely-related topic of graph theory. (Indeed, problems that are graph-theoretical in the mathematical sense are often thought of as being network problems when the intended application relates to a physical network.) Then, he discusses two of the most fundamental network problems, the shortest path problem and the minimum spanning tree problem, and some of their variants. Finally, he discusses the less basic, but very interesting, area of network reliability. In the second chapter, Maximum Flows in Networks, Frances Yao considers the problem most often thought of in connection with the mathematics of networks. It would not be unfair to think of network flows as the most fundamental of all problems in the field. The history of this topic is one of steady ix

9 X PREFACE improvement in the performance of the algorithms to solve this problem, and this chapter gives an account of several of the steps in this history. These problems have elegant, efficient algorithms. However, one of the most significant recent developments in discrete mathematics is that of computational complexity, and in particular the discovery that many natural problems in such areas as network theory appear to be inherently intractable. In the third chapter, The Computational Complexity of Network Problems, Richard Karp gives an account of this theory, with an emphasis on network problems. In the fourth chapter, Effective Use of Heuristic Algorithms in Network Design, Shen Lin demonstrates by means of a case study some of the ways by which, using great ingenuity, it is often possible to get plausible solutions to problems for which it is out of the question to find exact solutions. In the fifth chapter, Some Practical Network Problems, Daniel Kleitman discusses four applications of the mathematics of networks. These consist of the design of irrigation systems, the theory of electrical networks (the first area of network theory to be systematically developed), the scheduling of delivery trucks, and the physics of ice. In the sixth chapter, Telephone Switching Networks, Nicholas Pippenger presents an area of network theory that leads to difficult mathematics drawn from such apparently unrelated fields as harmonic analysis. In the final chapter, Concluding Remarks, I make a few general comments, and then present a chronology of events relating to the theory of networks and a short bibliography. In preparing these, I was heavily assisted by Frank Boesch. No talk corresponds to this chapter; it could be thought of as bearing some relation to the panel discussion that closed the Short Course. I wish to give special thanks to three people at the AMS in Providence Lincoln Durst, Peggy Reynolds, and Warren Lemoi, Jr. - and Ronald Graham, the chairman of the advisory subcommittee, for their invaluable assistance. Stefan A. Burr City College, C.U.N.Y. May, 1982

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