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, 195ff. paths through networks, 264 Isomorphic planes, 68 routing of messages, 262 Isotope, 54, 57, 60 routing problems, 270 Isotopic algebras, 54, 55, 66, 68 trajectory, 236 Optimality, principle of, 220, 235, 240 Jacobi sum, 96 Order, 45 Join, 162 of a pivotal transformation, 136 INDEX 311

Ordinary point, 62 Right Oriented graph, 162 characteristic function, 58 Orthogonal latin squares, 5-9, 91, 93, powers, 58 195 Routing problem, 235 10 x 10, 71fT., 197 Orthogonality relation, 99 SEAC—the National Bureau Standards Eastern Automatic Computer, 201 Parseval relation, 97 SWAC, 91, 93 Partially ordered sets, 85 Scheduling problems, 237 Partitioned, 20 Semigroups of order five, 199 Paths, 117, 123, 124 Sequential Pauli exclusion principle, 141 search, 245 Permanent, 166, 169ff. testing, 246 Permutation matrix, 113, 124 Shadow prices, 272 Permutations, 184ff. Shears, 65 Pivot, 133 Sieve, 182 Pivotal transform, 135 Simplex method, 129 Planar ternary rings, 45, 62 continuous version of, 239 Poincare, H., 163 Slightly intertwined Point at infinity, 62 matrices, 239 Points, 62, 162 symmetric matrices, 241 Polar cone, 146 Solvability problem, 144 Polyhedral cone, 146 Steiner triple systems, 1, 2, 9, 197 Polyhedron problem, 142, 147 of order 15, 199 Power matrix, 151 Stochastic matrix, doubly, 166, 169fT. Primitive roots, 206 Successive approximations, 226, 232, 236 Principle of optimality, 220, 235, 240 Systems of Problem of the queens, 91, 93 distinct representatives, 113,114,124,166 Programming representatives, 114, 122 dynamic, 217 restricted representatives, 115, 116 linear, 211 Ternary ring, 45, 46, 62 Projective plane, 45, 197 Threshold methods, 252 finite, 2, 3, 62 Tinsley, M. F., 166 homomorphisms, 45ff. Translate, 17 of order eight, 94, 198 Translation group, 64 order, 15 Transportation problem, 124, 251, 299 Harris, 233 Quadratic field units, 206 Hitchcock-Koopmans, 224, 232, 251 Quadratic forms, classes of, 206 nonlinear, 225, 233 Quasigroup, 54, 62 Traveling salesman problem, 235, 299 Queens problem, 91, 93 Twisted fields, 69 Rado-Hall theorem, 87 Unimodular property, 114, 115, 117, 118, Recurrence relations, 221 123, 124, 125 Reduction in dimensionality, 241 Urn problem, 141 Redundancy, 294 v, k, A configuration, 165 Reliability, 243 van der Waerden, B. L., 166 Remainder theorem, Chinese, 296 Vandermonde matrix, 160 Representatives, systems of, 114, 122 Veblen, O., 163 distinct, 113, 114, 124, 166 Veblen-Wedderburn systems of order 16, restricted, 115, 116, Residue difference sets, 109 198 Resolvent of Lagrange, 98 Warehousing problems, 124, 237