COMBINATORIAL ANALYSIS

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2 PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS VOLUME X COMBINATORIAL ANALYSIS AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND 1960

3 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 ORD-4545 with the Ordnance Corps, U.S. Army. International Standard Serial Number International Standard Book Number Library of Congress Catalog Card Number 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.

4 CONTENTS PREFACE V Current Studies on Combinatorial Designs By MARSHALL HALL, Jr. Quadratic Extensions of Cyclic Planes By R. H. BRUCK On Homomorphisms of Projective Planes By D. R. HUGHES Finite Division Algebras and Finite Planes 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 By R. P. DILWORTH An Enumerative Technique for a Class of Combinatorial Problems. 91 By R. J. WALKER The Cyclotomic Numbers of Order Ten By A. L. WHITEMAN Some Recent Applications of the Theory of Linear Inequalities to Extremal Combinatorial Analysis By A. J. HOFFMAN A Combinatorial Equivalence of Matrices By A. W. TUCKER Linear Inequalities and the Pauli Principle By H. W. Kuhn Compound and Induced Matrices in Combinatorial Analysis..149 By H. J. RYSER Permanents of Doubly Stochastic Matrices By MARVIN MARCUS and MORRIS NEWMAN iii

5 iv CONTENTS A Search Problem in the n-cube By A. M. GLEASON Teaching Combinatorial Tricks to a Computer D. H. LEHMER Isomorph Rejection in Exhaustive Search Techniques By J. D. SWIFT Some Discrete Variable Computations By OLGA TAUSSKY and JOHN TODD Solving Linear Programming Problems in Integers By R. E. GOMORY Combinatorial Processes and Dynamic Programming By RICHARD BELLMAN Solution of Large Scale Transportation Problems By MURRAY GERSTENHABER On Some Communication Network Problems By ROBERT KALABA Directed Graphs and Assembly Schedules By J. D. FOULKES A Problem in Binary Encoding By E. N. GILBERT An Alternative Proof of a Theorem of Konig as an Algorithm for the Hitchcock Distribution Problem 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.

6 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 Fulkerson. 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

7 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

8 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

9 310 INDEX 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, 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, 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 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

10 Ordinary point, 62 Oriented graph, 162 Orthogonal latin squares, 5-9, 91, 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, 206 Quasigroup, 54, 62 Queens problem, 91, 93 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 INDEX 311 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

A Course in Combinatorics

A Course in Combinatorics J. H. van Lint Technical Universüy of Eindhoven and R. M. Wilson California Institute of Technology H CAMBRIDGE UNIVERSITY PRESS CONTENTS Preface xi 1. Graphs 1 Terminology of