PrefaceHistorically, there is a close connection between geometry and optImization. This is illustrated by methods like the gradient method and the simplex method, which are associated with clear geometric pictures. In combinatorial optimization, however, many of the strongest and most frequently used algorithms are based on the discrete structure of the problems: the greedy algorithm, shortest path and alternating path methods, branch-and-bound, etc. In the last several years geometric methods, in particular polyhedral combinatorics, have played a more and more profound role in combinatorial optimization as well.Our book discusses two recent geometric algorithms that have turned out to have particularly interesting consequences in combinatorial optimization, at least from a theoretical point of view. These algorithms are able to utilize the rich body of results in polyhedral combinatorics.The first of these algorithms is the ellipsoid method, developed for nonlinear programming by N. Z. Shor, D. B. Yudin, and A. S. NemirovskiI. It was a great surprise when L. G. Khachiyan showed that this method can be adapted to solve linear programs in polynomial time, thus solving an important open theoretical problem. While the ellipsoid method has not proved to be competitive with the simplex method in practice, it does have some features which make it particularly suited for the purposes of combinatorial optimization.The second algorithm we discuss finds its roots in the classical "geometry of numbers", developed by Minkowski. This method has had traditionally deep applications in number theory, in particular in diophantine approximation. Methods from the geometry of numbers were introduced in integer programming by H. W. Lenstra. An important element of his technique, called basis reduction, goes in fact back to Hermite. An efficient version of basis reduction yields a polynomial time algorithm useful not only in combinatorial optimization, but also in fields like number theory, algebra, and cryptography.A combination of these two methods results in a powerful tool for combinatorial optimization. It yields a theoretical framework in which the polynomial time solvability of a large number of combinatorial optimization problems can be shown quite easily. It establishes the algorithmic equivalence of problems which are "dual" in various senses.Being this general, this method cannot be expected to give running times comparable with special-purpose algorithms. Our policy in this book is, therefore, not to attempt to obtain the best possible running times; rather, it is to derive just the polynomial time solvability of the problems as quickly and painlessly as VI Preface possible. Thus, our results are best conceived as "almost pure" existence results for polynomial time algorithms for certain problems and classes of problems.Nevertheless, we could not get around quite a number of tedious technical details. We did try to outline the essential ideas in certain sections, which should give an outline of the underlying geometric an...

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In this paper we consider a clustering problem that arises in qualitative data analysis. This problem can be transformed to a combinatorial optimization problem, the clique partitioning problem. We have studied the latter problem from a polyhedral point of view and determined large classes of facets of the associated polytope. These theoretical results are utilized in this paper. We describe a cutting plane algorithm that is based on the simplex method and uses exact and heuristic separation routines for some of the classes of facets mentioned before. We discuss some details of the implementation of our code and present our computational results. We mention applications from, e.g., zoology, economics, and the political sciences. IntroductionThe need of analysing data that arise from the measurement of a number of characteristics (or attributes) associated with each object of a given set, occurs very frequently in sociology, zoology, economics, and many other sciences. The areas of study concerned with this type of problem are known as data analysis, multivariate analysis, and taxonomy.We consider here a problem occurring in qualitative data analysis, of the following type. Given a data set consisting of the description of a set of objects with respect to a number of characteristics, find a best partition of the object set into "homogeneous" disjoint classes (or clusters).In this paper we give a precise formulation of this clustering problem and show how it can be reduced to a graph optimization problem which we call clique partitioning problem (CPP). This is done in Section 2. In Section 3 we summarize some results of Gr&schel and Wakabayashi (1987) on the polyhedron associated with the CPP. In Section 4 we describe a cutting plane algorithm for CPP which is based on these theoretical results. Finally, in Section 5 we report on the computational results with our code. Many applications from zoology, marketing, and the 6 0 M. Gr6tschel and Y. Wakabayashi / On a clustering problempolitical sciences are given and the optimization process for each of these applications is illustrated. I. Definitions and notationWe assume that the reader is familiar with the basic concepts of graph theory. The definitions not given here can be found in Bondy and Murty (1976). All graphs we consider are simple. We denote a graph G with node set V and edge set E by G = (V, E). An edge e with endnodes u and v is denoted by uv. If S is a node set of G = (V, E) then we denote the set of edges in G with both endnodes in S by E(S), that is, E ( S ) = { u v c E l u , v6 S}.Moreover, if $ 1 , . . . , Sk are subsets of V then k E (Sl,.. ., Sk):--CJ E(S,). i--1 IfS, T_c V a n d S~T = O t h e n [S: T]:={uvlucS, vc T}denotes the set of edges with one endnode in S and the other in T.A graph is called complete if every pair of its nodes is linked by an edge. A clique is a subgraph of a graph that is complete (a clique is not necessarily a maximal complete subgraph). We will denote the (up to isomorphism unique) complete graph with n nodes by Kn = ...