Combinatorial optimization is a lively eld of applied mathematics, combining techniques from combinatorics, linear programming, and the theory of algorithms, to solve optimization problems over discrete structures. There are a number of classic texts in this eld, but we felt that there is a place for a new treatment of the subject, covering some of the advances that have been made in the past decade. We set out to describe the material in an elementary text, suitable for a one semester course. The urge to include advanced topics proved to be irresistible, however, and the manuscript, in time, grew beyond the bounds of what one could reasonably expect to cover in a single course. We hope that this is a plus for the book, allowing the instructor to pick and choose among the topics that are treated. In this way, the book may b e s u i table for both graduate and undergraduate courses, given in departments of mathematics, operations research, and computer science. An advanced theoretical course might spend a lecture or two o n c hapter 2 and sections 3.1 and 3.2, then concentrate on 3.3, 3.4, 4.1, most of chapters 5 and 6 and some of chapters 8 and 9. An introductory course might c o ver chapter 2, sections 3.1 to 3.3, section 4.1 and one of 4.2 or 4.3, and sections 5.1 through 5.3. A course oriented more towards integer linear programming and polyhedral methods could be based mainly on chapters 6 and 7 and would include section 3.6. The most challenging exercises have been marked in boldface. These should probably only be used in advanced courses. The only real prerequisite for reading our text is a certain mathematical maturity. W e d o m a k e frequent use of linear programming duality, so a reader unfamiliar with this subject matter should be prepared to study the linear programming appendix before proceeding with the main part of the text. We bene tted greatly from thoughtful comments given by m a n y o f o u r colleagues who read early drafts of the book. In particular, we w ould like to thank

Chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. The basic ingredient in these cutting-plane proofs is that for a polyhedron P and integral ve.:tor w, if max( wx Ix E P, wx integer}= I, then wx"' t is valid for all integral vectors in P. We consider the variant of this step where the requirement that wx be integer may be replaced by the requirement that wx be integer for some other integral vector w. The cutting-plane proofs thus obtained ma) be seen either as an abstraction of Gomory's mixed integer cutting-plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by Balas and Jeroslow. Our main result is that for a given polyhedron P, the set of vectors that satisfy every cutting plane for P with respect to a specified subset of integer variables is again a polyhedron. This allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron, analogous to the process of repeatedly taking Chvatal closures in the integer programming case. These results are illustrated with a number of examples from combinatorial optimization. Our work can be seen as a continuation of that of Nemhauser and Wolsey on mixed integer cutting planes.

We discuss several issues that arise in the implementation of Martin, Otto, and Felten's Chained Lin-Kernighan heuristic for large-scale traveling salesman problems. Computational results are presented for TSPLIB instances ranging in size from 11,849 cities up to 85,900 cities; for each of these instances, solutions within 1% of the optimal value can be found in under 1 CPU minute on a 300 Mhz Pentium II workstation, and solutions within 0.5% of optimal can be found in under 10 CPU minutes. We also demonstrate the scalability of the heuristic, presenting results for randomly generated Euclidean instances having up to 25,000,000 cities. For the largest of these random instances, a tour within 1% of an estimate of the optimal value can be obtained in under 1 CPU day on a 64-bit IBM RS6000 workstation.

Abstract. The first computer implementation of the Dantzig-FulkersonJohnson cutting-plane method for solving the traveling salesman problem, written by Martin, used subtour inequalities as well as cutting planes of Gomory's type. The practice of looking for and using cuts that match prescribed templates in conjunction with Gomory cuts was continued in computer codes of Miliotis, Land, and Fleischmann. Grötschel, Padberg, and Hong advocated a different policy, where the template paradigm is the only source of cuts; furthermore, they argued for drawing the templates exclusively from the set of linear inequalities that induce facets of the TSP polytope. These policies were adopted in the work of Crowder and Padberg, in the work of Grötschel and Holland, and in the work of Padberg and Rinaldi; their computer codes produced the most impressive computational TSP successes of the nineteen eighties. Eventually, the template paradigm became the standard frame of reference for cutting planes in the TSP. The purpose of this paper is to describe a technique for finding cuts that disdains all understanding of the TSP polytope and bashes on regardless of all prescribed templates. Combining this technique with the traditional template approach was a crucial step in our solutions of a 13,509-city TSP instance and a 15,112-city TSP instance.

We make several observations on the implementation of Edmonds' blossom algorithm for solving minimum-weight perfectmatching problems and we present computational results for geometric problem instances ranging in size from 1,000 nodes up to 5,000,000 nodes. A key feature in our implementation is the use of multiple search trees with an individual dual-change ⑀ for each tree. As a benchmark of the algorithm's performance, solving a 100,000-node geometric instance on a 200 Mhz Pentium-Pro computer takes approximately 3 minutes.