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Combinatorial Analysis http://dx.doi.org/10.1090/psapm/010 PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS VOLUME X COMBINATORIAL ANALYSIS AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND 1960 PROCEEDINGS OP THE TENTH SYMPOSIUM IN APPLIED MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY Held at Columbia University April 24-26, 1958 CO SPONSORED BY THE OFFICE OF ORDNANCE RESEARCH EDITED BY RICHARD BELLMAN MARSHALL HALL, JR. Prepared by the American Mathematical Society under Contract No. DA-19-020-ORD-4545 with the Ordnance Corps, U.S. Army. International Standard Serial Number 0160-7634 International Standard Book Number 0-8218-1310-2 Library of Congress Catalog Card Number 50-1183 Copyright © 1960 by the American Mathematical Society. Second printing, 1979 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. CONTENTS PREFACE ........... V Current Studies on Combinatorial Designs ..... 1 By MARSHALL HALL, Jr. Quadratic Extensions of Cyclic Planes . .15 By R. H. BRUCK On Homomorphisms of Projective Planes ..... 45 By D. R. HUGHES Finite Division Algebras and Finite Planes ..... 53 By A. A. ALBERT The Size of the 10 x 10 Orthogonal Latin Square Problem . 71 By L. J. PAIGE and C. B. TOMPKINS Some Combinatorial Problems on Partially Ordered Sets ... 85 By R. P. DILWORTH An Enumerative Technique for a Class of Combinatorial Problems . 91 By R. J. WALKER The Cyclotomic Numbers of Order Ten . .95 By A. L. WHITEMAN Some Recent Applications of the Theory of Linear Inequalities to Extremal Combinatorial Analysis. .113 By A. J. HOFFMAN A Combinatorial Equivalence of Matrices . .129 By A. W. TUCKER Linear Inequalities and the Pauli Principle . .141 By H. W. Kuhn Compound and Induced Matrices in Combinatorial Analysis . .149 By H. J. RYSER Permanents of Doubly Stochastic Matrices . .169 By MARVIN MARCUS and MORRIS NEWMAN iii iv CONTENTS A Search Problem in the n-cube . .175 By A. M. GLEASON Teaching Combinatorial Tricks to a Computer. .... 179 D. H. LEHMER Isomorph Rejection in Exhaustive Search Techniques . .195 By J. D. SWIFT Some Discrete Variable Computations . .201 By OLGA TAUSSKY and JOHN TODD Solving Linear Programming Problems in Integers . .211 By R. E. GOMORY Combinatorial Processes and Dynamic Programming . .217 By RICHARD BELLMAN Solution of Large Scale Transportation Problems . .251 By MURRAY GERSTENHABER On Some Communication Network Problems ..... 261 By ROBERT KALABA Directed Graphs and Assembly Schedules . .281 By J. D. FOULKES A Problem in Binary Encoding . .291 By E. N. GILBERT An Alternative Proof of a Theorem of Konig as an Algorithm for the Hitchcock Distribution Problem ...... 299 By M. M. FLOOD INDEX 309 PREFACE Problems in combinatorial analysis range from the study of finite geometries, through algebra and number theory, to the domains of com­ munication theory and transportation networks. Although the questions that arise are all problems of arrangement, they differ enormously in the superficial form in which they arise, and quite often intrinsically, as well. Perhaps the greatest discrepancy is between the discrete problems involving the construction of designs and the continuous problems of linear in­ equalities. Nevertheless, in a number of the papers that are presented a basic unity of the whole theory is brought to light. For example, Alan Hoffman has shown that many problems of discrete choice and arrangement may be solved in an elegant fashion by means of recent developments of the theory of linear inequalities, a continuation of work of Dantzig and Fulker- son. Similarly, Robert Kalaba and Richard Bellman have shown that a variety of combinatorial problems arising in the study of scheduling and transportation can be treated by means of functional equation techniques. Marshall Hall has observed that the solution of a problem in arrangements, in particular, the construction of pairs of orthogonal squares, is precisely equivalent to solving a certain equation for a matrix with nonnegative real entries. A very challenging area of research which is investigated in a number of the papers that follow is that of using a computer to attack combinatorial questions, both by means of theoretical algorithms and by means of sophisticated search techniques. Papers by Paige and Tompkins, Walker, Gerstenhaber, Flood, Gleason, Lehmer, Swift, Todd, and Gomory, discuss versions of this fundamental problem. Following the manner in which the Symposium was divided into four sessions, the Proceedings are divided into four sections. These are: I. Existence and construction of combinatorial designs. II. Combinatorial analysis of discrete extremal problems. III. Problems of communications, transportation and logistics. IV. Numerical analysis of discrete problems. What is very attractive about this field of research is that it combines both the most abstract and most nonquantitative parts of mathematics with the most arithmetic and numerical aspects. It shows very clearly that the discovery of a feasible solution of a particular problem may necessitate enormous theoretical advances. Perhaps the moral of the tale is that the division into pure and applied mathematics is certainly artificial and to the detriment of the enthusiasts on both sides. Furthermore, the way in which vi PREFACE apparently simple problems require a complex medley of algebraic, geo­ metric, analytic and numerical considerations shows that the traditional subdivisions of mathematics are themselves too rigidly labelled. There is one subject, mathematics, and one type of problem, a mathematical problem. RICHARD BELLMAN, The RAND Corporation MARSHALL HALL, Jr., California Institute of Technology INDEX Ad jugate, 150 Continuous version of the simplex tech­ Algorithm, 299, 305 nique, 239 Allocation problem, 218 Convex Alternating graph, 124 hull, 142 Approximation set, 113, 124 Cebycev, 231 spaces, 6 functional, 229 Coordinated, 62, 63 in policy space, 233, 235 Covering theorems, 204 monotonicity of, 232 Curse of dimensionality, 227 successive, 232, 236 Cyclic, 15 Arcs, 117, 123 Cyclotomic numbers, 95 Assembly schedules, 281 Cyclotomy, theory of, 95 Assignment problem, 299 Autotopism, 68 Decomposable, 169 Desarguesian plane, 10, 11, 63 Back-track, 92 Design of experiments, 246 Baker-Campbell-Hausdorff coefficients, Dickson-Hurwitz sums, 97 203 Difference set, 16, 109 Bernoulli number, 206 Dimensionality Block designs, 2-5 curse of, 227 Blocking, 269 reduction in, 241 BoldyrefT, A., 235 Directed graphs, 281 Book-binding problem, 237 Discreteness, 222 Bottleneck problems, 238 Distribution problem, Hitchcock, 299 Burst noise, 291 Distributive lattice, 85 Division algebra, 54, 61 Doubly stochastic matrix, 166, 169ff. Caterer problem, 124, 237 Dreyfus, S., 217 Cebycev approximation, 231 Dual, 253 Central collineation, 50 Chinese remainder theorem, 296 representation, 93 Circulation theorem, 117, 118, 119 Duality theorem of linear programming, Classes of quadratic forms, 206 115, 117, 256 Collineation, 15, 49, 62 Dynamic programming, 217ff. central, 50 elementary, 64, 65 Elementary group, 9-13, 62 collineations, 64, 65 Combinations, 184 operations, 139 Combinatorial equivalence, 129ff. Encoding, 291 Communication network problems, full, 294 261ff. Endpoints, 162 Commutators, 203 Equivalence relation on matrices, 129 Complete graph, 162 Error-correcting, 291 Compound, 150 Exhaustive searches, 195ff. Configuration, v, fc, A, 165 Extreme Congruence, 201 point, 142 Constraints, 221, 239 ray, 145 310 INDEX Fermat's last theorem, 205 Kantorovitch, L. V., 120, 251 Finite Konig's theorem, 86, 299, 300 Fourier series, 97 Konigsberg bridge problem, 261 graph, 162 Kronecker forms, 207 Parseval relation, 97 Kruskal tree, 263 planes, 53ff. Kuhn, H. W., 257, 299, 305 projective plane, 2, 3, 62 homomorphisms, 49 Lagrange Flood, M. M., 257 multipliers, 218, 226, 227, 244 Flooding technique, 235 resolvent, 98 Flow problems, multi-commodity, 270 Latin squares, 5-8, 7Iff. Flow theorems, 118 Left vector space, 61 Flows in networks, 117 Legendre-Sophie St. Germain criterion, 205 Football pool, 204 Line at infinity, 63 Ford, L. R., Jr., 224, 234, 257 Linear inequalities, 113ff., 141 Forecast, 204 solvability of, 144 Fourier series, finite, 97 Linear programming, 129, 270 Free plane, 45, 52 duality theorem of, 115, 117 Fulkerson, D. R., 224, 235, 257 integer solutions in, 21 Iff. Functional Lines, 62, 162 approximations, 226, 229 Loop, 62 equations, 221, 225 Majorized, 152 Gauss forms, 207 Marcus, M., 156, 166 Graphs, 117, 118, 123, 162 Matrices with non-negative elements, 170 alternating, 124 Matrix of relations, 208 directed, 281 Min-cut max-flow theorem, 115, 118, 119 Minimal cost connecting networks, 261 Harris, T. E., 234 Monotonicity of approximation, 232 Harris transportation problem, 233 Multi-commodity flow problems, 270 Hitchcock distribution problem, 299 Multiplier, 16, 17 Hitchcock-Koopmans transportation Mutually exclusive activities, 222 problem, 224, 232, 251 Homomorphisms nth shortest paths, 267 of loops, 48 n-tuple decomposition, 201 of projective planes, 45ff. Network flow, 122 Hungarian method of Kuhn, 305 Network problems, 234 communication, 261 Incidence matrix, 2, 113, 117, 123, 124 Newman, M., 166 164 Nodes, 117, 123, 124 of G, 163 Nonlinear transportation problem, 225,233 of the v, k, A configuration, 165 Norm, 155 Induced matrix, 151 Invariants, 196, 197 Operations, elementary, 139 Inverse of a matrix, 140 Optimal Isomorph rejection,
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