Following the theoretical studies of J.B. Robinson and H.W. Kuhn in the late 1940s and the early 1950s, G.B. Dantzig, R. Fulkerson, and S.M. Johnson demonstrated in 1954 that large instances of the TSP could be solved by linear programming. Their approach remains the only known tool for solving TSP instances with more than several hundred cities; over the years, it has evolved further through the work of M. Grötschel, S. Hong, M. Junger, P. Miliotis, D. Naddef, M. Padberg, W.R. Pulleyblank, G. Reinelt, G. Rinaldi, and others. We enumerate some of its refinements that led to the solution of a 13,509-city instance.

Abstract not found

Dantzig, Fulkerson, and Johnson (1954) introduced the cutting-plane method as a means of attacking the traveling salesman problem; this method has been applied to broad classes of problems in combinatorial optimization and integer programming. In this paper we discuss an implementation of Dantzig et al.'s method that is suitable for TSP instances having 1,000,000 or more cities. Our aim is to use the study of the TSP as a step towards understanding the applicability and limits of the general cutting-plane method in large-scale applications.

ABSTRACT In the MINIMUM BOUNDED DEGREE SPANNING TREE problem,we are given an undirected graph with a degree upper bound Bv oneach vertex v, and the task is to find a spanning tree of minimumcost which satisfies all the degree bounds. Let OPT be the costof an optimal solution to this problem. In this paper, we present a polynomial time algorithm which returns a spanning tree T ofcost at most OPT and dT (v) ^ Bv + 1 for all v, where dT (v)denotes the degree of v in T. This generalizes a result of Furerand Raghavachari [8] to weighted graphs, and settles a 15-year-old conjecture of Goemans [10] affirmatively. The algorithm general-izes when each vertex v has a degree lower bound Av and a degreeupper bound Bv, and returns a spanning tree with cost at most OPTand Av \Gamma 1 ^ dT (v) ^ Bv + 1 for all v. This is essentially thebest possible. The main technique used is an extension of the iterative rounding method introduced by Jain [12] for the design ofapproximation algorithms.

We consider the vertex-weighted version of the undirected Steiner tree problem. In this problem, a cost is incurred both for the vertices and the edges present in the Steiner tree. We completely describe the associated polytope by linear inequalities when the underlying graph is series-parallel. For general graphs, this formulation can be interpreted as a (partial) extended formulation for the Steiner tree problem. By projecting this formulation, we obtain some very large classes of facet-defining valid inequalities for the Steiner tree polytope.

The first computer implementation of the Dantzig-Fulkerson-Johnson 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

We consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worst-case improvement resulting from their addition to the subtour polyhedron. For example, we show that the comb inequalities cannot improve the subtour bound by a factor greater than ~. The corresponding factor for the class of clique tree inequalities is 8, while it is 4 for the path configuration inequalities. Keywords: Polyhedral combinatorics; Valid inequalities; Travelling salesman; Worst-case analysis 1.

We consider the minimum cost spanning tree problem under the restriction that all degrees must be at most a given value k. We show that we can efficiently find a spanning tree of maximum degree at most k + 2 whose cost is at most the cost of the optimum spanning tree of maximum degree at most k. This is almost best possible. The approach uses a sequence of simple algebraic, polyhedral and combinatorial arguments. It illustrates many techniques and ideas in combinatorial optimization as it involves polyhedral characterizations, uncrossing, matroid intersection, and graph orientations (or packing of spanning trees). The result generalizes to the setting where every vertex has both upper and lower bounds and gives then a spanning tree which violates the bounds by at most two units and whose cost is at most the cost of the optimum tree. It also gives a better understanding of the subtour relaxation for both the symmetric and asymmetric traveling salesman problems. The generalization to l-edge-connected subgraphs is briefly discussed.

Given an undirected graph G = (V; E) and a cost vector c 2 IR E , the weighted girth problem is to find a circuit in G having minimum total cost. This problem is in general NP-hard since the traveling salesman problem can be reduced to it. A promising approach to hard combinatorial optimization problems is given by the so-called cutting plane methods. These involve linear programming techniques based on a partial description of the convex hull of the incidence vectors of possible solutions. We consider the weighted girth problem in the case where G is the complete graph K n and study the facial structure of the circuit polytope P n C and some related polyhedra. In the appendix we give complete characterizations of P n C for n up to 6.

The successful application of Branch and Cut methods to the TSP has drawn attention also to the polyhedral properties of the symmetric capacitated vehicle routing problem, which is the capacitated counterpart of the TSP. We investigate three classes of valid inequalities for the CVRP, multistars, pathbin inequalities and hypotours and give computational results we obtained with a Branch and Cut implementation.