PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS

VOLUME X

COMBINATORIAL ANALYSIS

AMERICAN MATHEMATICAL SOCIETY

PROVIDENCE, RHODE ISLAND

1960

http://dx.doi.org/10.1090/psapm/010

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 . . . . . . 1 5 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 . . . . . . 9 5 By A. L. WHITEMAN

Some Recent Applications of the Theory of Linear Inequalities to Extremal Combinatorial Analysis. . . . . . 1 1 3

By A. J . HOFFMAN

A Combinatorial Equivalence of Matrices . . . . . 1 2 9 By A. W. TUCKER

Linear Inequalities and the Pauli Principle . . . . . 1 4 1 By H. W. Kuhn

Compound and Induced Matrices in Combinatorial Analysis . . 1 4 9 By H. J . RYSER

Permanents of Doubly Stochastic Matrices . . . . . 1 6 9 By MARVIN MARCUS and MORRIS NEWMAN

iii

iv CONTENTS

A Search Problem in the n-cube . . . . . . . 1 7 5 By A. M. GLEASON

Teaching Combinatorial Tricks to a Computer. . . . . 179 D. H. LEHMER

Isomorph Rejection in Exhaustive Search Techniques . . . 1 9 5 By J. D. SWIFT

Some Discrete Variable Computations . . . . . . 2 0 1 By OLGA TAUSSKY and J O H N TODD

Solving Linear Programming Problems in Integers . . . . 2 1 1 By R. E. GOMORY

Combinatorial Processes and Dynamic Programming . . . 2 1 7 By RICHARD BELLMAN

Solution of Large Scale Transportation Problems . . . . 2 5 1 By MURRAY GERSTENHABER

On Some Communication Network Problems . . . . . 261 By ROBERT KALABA

Directed Graphs and Assembly Schedules . . . . . 2 8 1 By J. D. FOULKES

A Problem in Binary Encoding . . . . . . . 2 9 1 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 com­munication 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 in­equalities. 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. I I . Combinatorial analysis of discrete extremal problems.

I I I . 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. I t 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, geo­metric, 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 Algorithm, 299, 305 Allocation problem, 218 Alternating graph, 124 Approximation

Cebycev, 231 functional, 229 in policy space, 233, 235 monotonicity of, 232 successive, 232, 236

Arcs, 117, 123 Assembly schedules, 281 Assignment problem, 299 Autotopism, 68

Back-track, 92 Baker-Campbell-Hausdorff coefficients,

203 Bernoulli number, 206 Block designs, 2-5 Blocking, 269 BoldyrefT, A., 235 Book-binding problem, 237 Bottleneck problems, 238 Burst noise, 291

Caterer problem, 124, 237 Cebycev approximation, 231 Central collineation, 50 Chinese remainder theorem, 296 Circulation theorem, 117, 118, 119 Classes of quadratic forms, 206 Collineation, 15, 49, 62

central, 50 elementary, 64, 65 group, 9-13, 62

Combinations, 184 Combinatorial equivalence, 129ff. Communication network problems,

261ff. Commutators, 203 Complete graph, 162 Compound, 150 Configuration, v, fc, A, 165 Congruence, 201 Constraints, 221, 239

Continuous version of the simplex tech­nique, 239

Convex hull, 142 set, 113, 124 spaces, 6

Coordinated, 62, 63 Covering theorems, 204 Curse of dimensionality, 227 Cyclic, 15 Cyclotomic numbers, 95 Cyclotomy, theory of, 95

Decomposable, 169 Desarguesian plane, 10, 11, 63 Design of experiments, 246 Dickson-Hurwitz sums, 97 Difference set, 16, 109 Dimensionality

curse of, 227 reduction in, 241

Directed graphs, 281 Discreteness, 222 Distribution problem, Hitchcock, 299 Distributive lattice, 85 Division algebra, 54, 61 Doubly stochastic matrix, 166, 169ff. Dreyfus, S., 217 Dual, 253

representation, 93 Duality theorem of linear programming,

115, 117, 256 Dynamic programming, 217ff.

Elementary collineations, 64, 65 operations, 139

Encoding, 291 full, 294

Endpoints, 162 Equivalence relation on matrices, 129 Error-correcting, 291 Exhaustive searches, 195ff. Extreme

point, 142 ray, 145

310

Fermat's last theorem, 205 Finite

Fourier series, 97 graph, 162 Parseval relation, 97 planes, 53ff. projective plane, 2, 3, 62 homomorphisms, 49

Flood, M. M., 257 Flooding technique, 235 Flow problems, multi-commodity, Flow theorems, 118 Flows in networks, 117 Football pool, 204 Ford, L. R., Jr., 224, 234, 257 Forecast, 204 Fourier series, finite, 97 Free plane, 45, 52 Fulkerson, D. R., 224, 235, 257 Functional

approximations, 226, 229 equations, 221, 225

Gauss forms, 207 Graphs, 117, 118, 123, 162

alternating, 124 directed, 281

Harris, T. E., 234 Harris transportation problem, 233 Hitchcock distribution problem, 299 Hitchcock-Koopmans transportation

problem, 224, 232, 251 Homomorphisms

of loops, 48 of projective planes, 45ff.

Hungarian method of Kuhn, 305

Incidence matrix, 2, 113, 117, 123, 124 164

of G, 163 of the v, k, A configuration, 165

Induced matrix, 151 Invariants, 196, 197 Inverse of a matrix, 140 Isomorph rejection, 195ff. Isomorphic planes, 68 Isotope, 54, 57, 60 Isotopic algebras, 54, 55, 66, 68

Jacobi sum, 96 Join, 162

INDEX

Kantorovitch, L. V., 120, 251 Konig's theorem, 86, 299, 300 Konigsberg bridge problem, 261 Kronecker forms, 207 Kruskal tree, 263 Kuhn, H. W., 257, 299, 305

Lagrange multipliers, 218, 226, 227, 244 resolvent, 98

2 7 0 Latin squares, 5-8, 7Iff. Left vector space, 61 Legendre-Sophie St. Germain criterion, 205 Line at infinity, 63 Linear inequalities, 113ff., 141

solvability of, 144 Linear programming, 129, 270

duality theorem of, 115, 117 integer solutions in, 21 Iff.

Lines, 62, 162 Loop, 62

Majorized, 152 Marcus, M., 156, 166 Matrices with non-negative elements, 170 Matrix of relations, 208 Min-cut max-flow theorem, 115, 118, 119 Minimal cost connecting networks, 261 Monotonicity of approximation, 232 Multi-commodity flow problems, 270 Multiplier, 16, 17 Mutually exclusive activities, 222

nth shortest paths, 267 n-tuple decomposition, 201 Network flow, 122 Network problems, 234

communication, 261 Newman, M., 166 Nodes, 117, 123, 124 Nonlinear transportation problem, 225,233 Norm, 155

Operations, elementary, 139 Optimal

paths through networks, 264 routing of messages, 262 routing problems, 270 trajectory, 236

Optimality, principle of, 220, 235, 240 Order, 45

of a pivotal transformation, 136

INDEX 311

Ordinary point, 62 Oriented graph, 162 Orthogonal latin squares, 5-9, 91,

195 10 x 10, 71fT., 197

Orthogonality relation, 99

Parseval relation, 97 Partially ordered sets, 85 Partitioned, 20 Paths, 117, 123, 124 Pauli exclusion principle, 141 Permanent, 166, 169ff. Permutation matrix, 113, 124 Permutations, 184ff. Pivot, 133 Pivotal transform, 135 Planar ternary rings, 45, 62 Poincare, H., 163 Point at infinity, 62 Points, 62, 162 Polar cone, 146 Polyhedral cone, 146 Polyhedron problem, 142, 147 Power matrix, 151 Primitive roots, 206 Principle of optimality, 220, 235, 240 Problem of the queens, 91, 93 Programming

dynamic, 217 linear, 211

Projective plane, 45, 197 finite, 2, 3, 62 homomorphisms, 45ff. of order eight, 94, 198 order, 15

Quadratic field units, 206 Quadratic forms, classes of, Quasigroup, 54, 62 Queens problem, 91, 93

206

Rado-Hall theorem, 87 Recurrence relations, 221 Reduction in dimensionality, 241 Redundancy, 294 Reliability, 243 Remainder theorem, Chinese, 296 Representatives, systems of, 114, 122

distinct, 113, 114, 124, 166 restricted, 115, 116,

Residue difference sets, 109 Resolvent of Lagrange, 98

Right characteristic function, 58

93, powers, 58 Routing problem, 235

SEAC—the National Bureau Standards Eastern Automatic Computer, 201

SWAC, 91, 93 Scheduling problems, 237 Semigroups of order five, 199 Sequential

search, 245 testing, 246

Shadow prices, 272 Shears, 65 Sieve, 182 Simplex method, 129

continuous version of, 239 Slightly intertwined

matrices, 239 symmetric matrices, 241

Solvability problem, 144 Steiner triple systems, 1, 2, 9, 197

of order 15, 199 Stochastic matrix, doubly, 166, 169fT. Successive approximations, 226, 232, 236 Systems of

distinct representatives, 113,114,124,166 representatives, 114, 122 restricted representatives, 115, 116

Ternary ring, 45, 46, 62 Threshold methods, 252 Tinsley, M. F., 166 Translate, 17 Translation group, 64 Transportation problem, 124, 251, 299

Harris, 233 Hitchcock-Koopmans, 224, 232, 251 nonlinear, 225, 233

Traveling salesman problem, 235, 299 Twisted fields, 69

Unimodular property, 114, 115, 117, 118, 123, 124, 125

Urn problem, 141

v, k, A configuration, 165 van der Waerden, B. L., 166 Vandermonde matrix, 160 Veblen, O., 163 Veblen-Wedderburn systems of order 16,

198

Warehousing problems, 124, 237