Combinatorial Search. by Martin AignerReview by: E. Rodney CanfieldSIAM Review, Vol. 33, No. 1 (Mar., 1991), pp. 132-133Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2030672 .

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132 BOOK REVIEWS

maximum entropy pnnciple in spectral esti- mation. There is only one chapter on multi- dimensional problems.

In keeping with the author's intention of making the book accessible to a broad spec- trum of readers, the mathematical level is not high. The author says that the reader needs a familiarity with calculus, complex numbers, and matrix algebra. This seems to be accurate.

L. L. CAMPBELL

Queen's University

Combinatorial Search. By Martin Aigner. Teubner, Stuttgart; John Wiley, Chichester, UK, 1988. iv + 368 pp. $44.95. ISBN 0-471- 92142-4 (Wiley). A volume in the Wiley- Teubner Series in Computer Science.

This highly readable book contains a wealth of material from diverse areas of combinato- rics. It is remarkable that so many subjects can be viewed from the perspective of "combina- torial search." A search process is a pair (S, aY) in which S is a nonempty set, called the search domain, and Y is a family of functions on S, called the test family. Given a target x* in S, a search algorithm consists of a choice of functionsf1 ,f2, . in Y such that the se- quence of valuesf1 (x* ),f2(x*), - uniquely determines x*. Two search problems which many readers will be familiar with are (1) de- tecting the counterfeit coin, and (2) inserting a new element into its proper position in a sorted sequence.

After an introductory chapter explaining the general model, along with some illustrative ex- amples, the book consists of five additional chapters on weighing problems, graph prob- lems, sorting problems, poset problems, and other problems.

Weighing problems are probably the oldest search problems. In this treatment, the prob- lems are classified according to single coun- terfeit versus many, and balance scale versus spring scale. Several algorithms are analyzed carefully, and their asymptotic efficiency is worked out, either in the text or in the exercises.

Chapter 3 was particularly interesting, and I will review it in greater detail than the other chapters. First, we have the problem of iden- tifying an unknown edge e* in a given graph G = (V, E). The legal tests are sets A c V of vertices. We distinguish the binary variant,

where we are told whether e* is incident with at least one vertex of A or none; and the ternary variant, where we are told that both endpoints of e*, one endpoint of e*, or no endpoint of e*, lie in A. These two variants are worked out thoroughly for complete graphs, complete bi- partite graphs, and trees. Also worked out is the case in which the test sets A are restricted to one-vertex sets, in which case the binary and ternary variants are the same.

Another graphical problem involves an un- known graph G*, for which membership in a class 4X of graphs on a fixed set V of n vertices must be decided. We are permitted to make probes of the form: is the edge e in G*? Ex- amples of 4, which is typically determined by a familiar graphical property, are acyclic graphs and regular graphs. Exact results are obtained for the classes of matchings, trees, and con- nected bipartite graphs. Some interesting con- jectures are stated at the end of this section.

In full generality, the above graph recogni- tion problem may be stated thus: we are given ( 1 ) a set T of t elements el, * - , e,, (2) a family of subsets w c 2T and (3) an unknown X c T. We want to decide if X X 4. The process may be seen as a game, in which player A makes the decision by asking questions of the form "is e, in X?" and player B must answer consistently but in such a way as to prolong A's decision. The length of the game, with both players playing optimally, is called the recog- nition complexity C( 4). For example, C( 4) = t when W consists of all k-element subsets of T, I T I = t. Such families, of max- imum complexity, are termed elusive. Con- ditions for elusiveness are established. Any graph property for n -< is elusive, but for n = 6, an example of a nonelusive property is given.

The reader begins to wonder if all complex- ities are asymptotically of order n 2, but the ex- ample of the "scorpion graph property" is shown to have complexity less than or equal to 6n - 10. By the method of packing, Bollobas and Eldridge have shown that C(P) ? 2n - 4 for any nontrivial property P. The truth about the range between 2n and 6n is not known. Nontrivial monotone graph properties are shown to have complexity at least n2/ 16. Many of the references in this chapter and the next two chapters are recent and reflect current re- search.

Chapter 4 deals with sorting in all its many aspects. Topics as familiar as selecting the minimum, and as intricate as the Ford-John-

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BOOK REVIEWS 133

son sorting algorithm, are discussed. Merging, selection, networks, and parallel sorting all make an appearance. Professor Aigner himself has made numerous original contributions to this subject. The chapter is very much in the spirit of Volume III of Knuth's Art of Computer Programming, but the many recent references and results make Chapter 4 a valuable update of Knuth's classic.

The subject matter of Chapter 5 is partially ordered sets (posets). In this context general sorting problems are considered: (1) finding linear extensions of a poset, (2) sorting ele- ments about which certain relations are already known. The very beautiful " 13, 8 theorem" of Kahn and Saks is discussed, although not proved. Finally, in Chapter 6 several additional topics are treated: the polyhedra membership problem, graph colouring, and longest in- creasing subsequences.

Thorough bibliographic notes and references are given on a chapter by chapter basis. The writing style is very pleasant, not too formal, but with a well-chosen level of detail in proofs. Aigner is a good expositor. Each chapter con- cludes with a set of exercises, and selected so- lutions are provided at the end of the book. All in all, the book is a highly recommended reference for combinatorialists whose interests include posets, sorting, and searching, as well as for computer scientists interested in the same topics or other theoretical matters such as in- formation theoretic lower bounds and entropy. Thanks to the exercises, the book would make an excellent text for a topics course, and nat- urally would be the ideal starting point for graduate students contemplating research in related areas.

E. RODNEY CANFIELD

University of Georgia

Numerical Solution of Optimal Control Prob- lems with State Constraints by Sequential Quadratic Programming in Function Spaces. by K. C. P. Machielsen. Stichting Mathema- tisch Centrum, Amsterdam, the Netherlands, 1988. vi + 214 pp. Dfl., 33.00, paper. No ISBN. Centre for Mathematics and Computer Sci- ence, CWI Tract No. 53.

It must have been experienced by both mathematicians and control engineers that finding explicit closed-form solutions for non- linear optimal control problems with state

(and/or control) constraints is impossible in general, and hence numerical solutions are necessary, in particular, for practical purposes. Not many texts and monographs have been written, however, to provide general techniques for solving such control problems numerically. Some have appeared in the literature, even re- cently, such as those written by Boltyanskii (1971), Gruver and Sachs (1981), and Toe and Wu ( 1983 ), as well as some others written for the general purpose of solving nonlinear optimization problems. Therefore, the book under review should be welcome as a new ad- dition to the control and optimization litera- ture.

The context of the present book is outlined as follows.

In the first chapter, the Introduction, a gen- eral form of optimal control problems in state- variable description with different types of constraints is first briefly described. An ex- ample of a state-constrained optimal control problem in robotics is then given to motivate the investigation. Optimal conditions for state- constrained optimal control problems in the finite-dimensional case is finally stated without derivation by converting it to a mathematical programming problem.

The second chapter of the book is entitled "Nonlinear programming in Banach spaces." In order to explain why nonlinear program- ming is studied and why it should be formu- lated on a general Banach space setting, the author states in the Summary, which is in the very beginning of the book, "Because state- constrained optimal control problems can be identified as special cases of the abstract opti- mization problems, the theory reviewed for abstract optimization problems [in Chap. 2] can be applied directly." Furthermore, "the method, which is proposed for the numerical solution of the optimal control problems, is presented first in terms of the abstract opti- mization problems." Under these considera- tions, Chapter 2 reviews a number of theoret- ical results from the theory of functional anal- ysis concerned with optimization. A very general description of the constrained opti- mization is stated therein as follows.

Given a Banach space X, an objective func- tional J: X -- R, and a constraint set U c X, find a uc c U such that

J(uz)-J(u) forallucU.

Here, of course, some necessary conditions on

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