Gerhard Reinelt

TSPLIB—A Traveling Salesman Problem Library

This paper contains the description of a traveling salesman problem library (TSPLIB) which is meant to provide researchers with a broad set of test problems from various sources and with various properties. For every problem a short description is given along with known lower and upper bounds. Several references to computational tests on some of the problems are given. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A Cutting Plane Algorithm for the Linear Ordering Problem

The linear ordering problem is an NP-hard combinatorial optimization problem with a large number of applications including triangulation of input-output matrices, archaeological senation, minimizing total weighted completion time in one-machine scheduling, and aggregation of individual preferences. In a former paper, we have investigated the facet structure of the 0/1-polytope associated with the linear ordering problem. Here we report on a new algorithm that is based on these theoretical results. The main part of the algorithm is a cutting plane procedure using facet defining inequalities. This procedure is combined with various heuristics and branch and bound techniques. Our computational results compare favorably with the results of existing codes. In particular, we could triangulate all input-output matrices, of size up to 60 × 60, available to us within acceptable time bounds.

An application of combinatorial optimization to statistical physics and circuit layout design

We study the problem of finding ground states of spin glasses with exterior magnetic field, and the problem of minimizing the number of vias holes on a printed circuit board, or contacts on a chip subject to pin preassignments and layer preferences. The former problem comes up in solid-state physics, and the latter in very-large-scale-integrated VLSI circuit design and in printed circuit board design. Both problems can be reduced to the max-cut problem in graphs. Based on a partial characterization of the cut polytope, we design a cutting plane algorithm and report on computational experience with it. Our method has been used to solve max-cut problems on graphs with up to 1,600 nodes.

Experiments in quadratic 0-1 programming

We present computational experience with a cutting plane algorithm for 0–1 quadratic programming without constraints. Our approach is based on a reduction of this problem to a max-cut problem in a graph and on a partial linear description of the cut polytope.

Facets of the linear ordering polytope

LetDn be the complete digraph onn nodes, and letPLOn denote the convex hull of all incidence vectors of arc sets of linear orderings of the nodes ofDn (i.e. these are exactly the acyclic tournaments ofDn). We show that various classes of inequalities define facets ofPLOn, e.g. the 3-dicycle inequalities, the simplek-fence inequalities and various Möbius ladder inequalities, and we discuss the use of these inequalities in cutting plane approaches to the triangulation problem of input-output matrices, i.e. the solution of permutation resp. linear ordering problems.

50 Years of Integer Programming 1958-2008

I The Early Years.- Solution of a Large-Scale Traveling-Salesman Problem.- The Hungarian Method for the Assignment Problem.- Integral Boundary Points of Convex Polyhedra.- Outline of an Algorithm for Integer Solutions to Linear Programs An Algorithm for the Mixed Integer Problem.- An Automatic Method for Solving Discrete Programming Problems.- Integer Programming: Methods, Uses, Computation.- Matroid Partition.- Reducibility Among Combinatorial Problems.- Lagrangian Relaxation for Integer Programming.- Disjunctive Programming.- II From the Beginnings to the State-of-the-Art.- Polyhedral Approaches to Mixed Integer Linear Programming.- Fifty-Plus Years of Combinatorial Integer Programming.- Reformulation and Decomposition of Integer Programs.- III Current Topics.- Integer Programming and Algorithmic Geometry of Numbers.- Nonlinear Integer Programming.- Mixed Integer Programming Computation.- Symmetry in Integer Linear Programming.- Semidefinite Relaxations for Integer Programming.- The Group-Theoretic Approach in Mixed Integer Programming.

Exact Ground States of Ising Spin Glasses: New Experimental Results With a Branch and Cut Algorithm

In this paper we study two-dimensional Ising spin glasses on a grid with nearest neighbor and periodic boundary interactions, based on a Gaussian bond distribution, and an exterior magnetic field. We show how using a technique called branch and cut, the exact ground states of grids of sizes up to 100×100 can be determined in a moderate amount of computation time, and we report on extensive computational tests. With our method we produce results based on more than 20,000 experiments on the properties of spin glasses whose errors depend only on the assumptions on the model and not on the computational process. This feature is a clear advantage of the method over other, more popular ways to compute the ground state, like Monte Carlo simulation including simulated annealing, evolutionary, and genetic algorithms, that provide only approximate ground states with a degree of accuracy that cannot be determineda priori. Our ground-state energy estimation at zero field is −1.317.

A Branch & Cut Algorithm for the Asymmetric Traveling Salesman Problem with Precedence Constraints

In this article we consider a variant of the classical asymmetric traveling salesman problem (ATSP), namely the ATSP in which precedence constraints require that certain nodes must precede certain other nodes in any feasible directed tour. This problem occurs as a basic model in scheduling and routing and has a wide range of applications varying from helicopter routing (Timlin, Masters Thesis, Department of Combinatorics and Optimization, University of Waterloo, 1989), sequencing in flexible manufacturing (Ascheuer et al., Integer Programming and Combinatorial Optimization, University of Waterloo, Waterloo, 1990, pp. 19–28; Idem., SIAM Journal on Optimization, vol. 3, pp. 25–42, 1993), to stacker crane routing in an automatic storage system (Ascheuer, Ph.D. Thesis, Tech. Univ. Berlin, 1995). We give an integer programming model and summarize known classes of valid inequalities. We describe in detail the implementation of a branch&cut-algorithm and give computational results on real-world instances and benchmark problems from TSPLIB. The results we achieve indicate that our implementation outperforms other implementations found in the literature. Real world instances with more than 200 nodes can be solved to optimality within a few minutes of CPU-time. As a side product we obtain a branch&cut-algorithm for the ATSP. All instances in TSPLIB can be solved to optimality in a reasonable amount of computation time.

On the acyclic subgraph polytope

The acyclic subgraph problem can be formulated as follows. Given a digraph with arc weights, find a set of arcs containing no directed cycle and having maximum total weight. We investigate this problem from a polyhedral point of view and determine several classes of facets for the associated acyclic subgraph polytope. We also show that the separation problem for the facet defining dicycle inequalities can be solved in polynomial time. This implies that the acyclic subgraph problem can be solved in polynomial time for weakly acyclic digraphs. This generalizes a result of Lucchesi for planar digraphs.

Combinatorial optimization - Eureka, you shrink!

properties of linear independence. Beautiful paper. And various people had gone on studying the properties of these algebraic structures. That’s the year Jack was born, right? 1935? Is that true? In 1935? 1932? Two years? Jack, you were born in 1935! (laughter) Yeah, that wouldn’t make sense, would it? I think it was 1935, but here’s what was interesting. (Jack: At that time, I worked on the subject, it was independently.) Good! Let’s say in the early 30s. Now, you see the thing about matroids was that they were this abstract structure, and I guess there were actually several people who somehow had this idea of optimizing over matroids with the greedy algorithm. And, Jack, of course, knew how to do that. But he was now beginning to strut his stuff with this new paradigm he had developed, which was the idea of polyhedral combinatorics. You see, in fact when Jack first described the matching polytope, he would write the degree constraints, then he would put the blossom inequalities in. There were a lot of people at that point who said “Ah! That’s really a bad idea”, and I remember people saying “You know Jack, they don’t make computers big enough to store all those inequalities.” And Jack said “Yes, but you don’t need to store them explicitly. You can generate them, and use them when you need them”. They said “I don’t get it”, so there was a bit of a problem on that point. But it was this idea that you could add this large set of inequalities to a small set of inequalities that define an integer program, and from there go on to the algorithm and to proving optimality, and the whole NP characterization idea which was in there. Now I guess the thing that to me was particularly remarkable about this was the stunning sequence of papers that followed including matroid partition, matroid intersection – much more complex combinatorial structures which came from this very simple notion of a matroid. Then, damned if Jack didn’t go ahead and solve the corresponding optimization problems. Now there was one problem that we always ran into with the whole matroid paradigm. People would say “Now, how do you know when a set is independent in a matroid?” and again get into these questions of defining oracles. At one point Jack said, again I wasn’t there at the time, but I can just imagine how Jack finally said it, “Well, I’ll give them something simple they can understand”, so in Mathematics of the Decision Sciences, there was a paper on what Jack called “branchings”. Optimum branchings was to be an accessible example of matroid intersection, so people would actually see this stuff working. And I do remember Jack telling me at a point later that he was a little disappointed that it took so many pages to write it down, even though it seemed so simple. But the branching thing came out as an elegant concrete example of matroid intersections. Another point that I would like to stress here, this whole idea of complexity of integer calculation was discovered by Jack. There’s a paper called “Systems of distinct representatives and linear algebra” which contains the famous line: “Gaussian elimination is not a good algorithm”. And, I can remember when I first read that line, I thought “I don’t get it. You clearly only have to perform a polynomial number of steps to reduce a matrix, so what’s the deal?” And of course what Jack had observed was that the numbers can get big. And the fact that you have to pay attention to the size of the numbers was something that “Eureka – You Shrink!” 7 again became extremely influential as people began studying these areas and going on from there. Partition matroids was another class of matroids Jack invented. It was beginning to get a buzz around it. In 1969, the University of Calgary hosted a massive combinatorial meeting, two weeks long, and basically everybody who was anybody in combinatorics at that time was invited. I had just finished my Master’s degree the week before that meeting was to take place, and my Master’s thesis actually was on matroids. What my supervisor Eric Milner had done is given me stacks of papers to read on matroids. Bob, you and I are probably the only people who tried to read the Tutte paper and I think you made it further than I did. But there was all this excitement about matroids and all these matroid optimization results were coming from Jack. That was one of the reasons why they invited Jack to come and give a series of lectures at this big Calgary meeting on his research. And, in some sense, you can say this was a coming-out party. It was an endorsement of Jack’s views, it gave him a platform to present this material. It was an amazing series of lectures because I can remember that, at this point, Jack had developed the whole idea of submodular functions and polymatroids. This was matroids generalized to a much broader concept, and every so often he would throw in one of those combinatorial realizations. As someone who sat and took notes on those lectures, I remember I was exhausted at the end just trying to keep up with it. So when I think of the sort of work that Jack did in that stage, not only did he define and promote this whole concept of good algorithms and good characterizations, he solved the matching problem, he did the same for matroid intersections, he did the same for submodular functions, he laid the basis for what became arithmetic complexity. I think at one point I said to Jack: “How come you got there when all the good stuff was there?”. But I think it’s probably fairer to say that all this good stuff was there because Jack made it there. He had this vision of what could be done, he could see how it could go, and for those of you who just met him, you may not be aware of it, maybe I am the only one who has noticed this: Jack has a slightly stubborn streak. I think that served Jack incredibly well, as he created this whole agenda describing where the world of combinatorial optimization should go. I think it’s been remarkably successful. Now, what we want to do at this point is to have a few comments by some of the people who knew Jack somewhat earlier in his career. I think, Jack, the technical term for this is a “roast”. At this point, Ellis Johnson, George Nemhauser, Bob Bixby, Jean-François Maurras, Denis Naddef, and Kathie Cameron successively joined Bill Pulleyblank and all presented reminiscences and a number of entertaining stories, accompanied by lot of laughter and interaction with the audience. Finally, Jack Edmonds took the stage. We cite from his speech: I should say that I didn’t expect to give this talk tonight. These guys, I saw a session for Edmonds on the board, and then they said to me, 8 Surprise Session for Jack Edmonds Fig. 5. “I remember . . .” Fig. 6. Jack Edmonds “Hey, would you give a few words about the early days”. So here I am now. (laughter) It occurred to me that “Yeah, sure, their idea of a special session for Jack is to start him talking.” (laughter) And talk he did, sharing his personal recollections of his early career with the audience. We recommend “A Glimpse of Heaven” [6] which contains Jack’s reminiscences on this period of his life, for those readers who were not at Aussois to enjoy it live. “Eureka – You Shrink!” 9 The session closed with a standing ovation for Jack and a presentation to Jack: a poster brought to Aussois by Jean-François Maurras showing Jack in the 70s and signed by the participants of Aussois 2001. Fig. 7. Jack Edmonds in the 70s