Optimal Random Matchings, Tours, and Spanning Trees in Hierarchically Separated Trees

We derive tight bounds on the expected weights of several combinatorial optimization problems for random point sets of size $n$ distributed among the leaves of a balanced hierarchically separated tree. We consider {\it monochromatic} and {\it bichromatic} versions of the minimum matching, minimum spanning tree, and traveling salesman problems. We also present tight concentration results for the monochromatic problems.Comment: 24 pages, to appear in TC

The Traveling Salesman Problem

This paper presents a self-contained introduction into algorithmic and computational aspects of the traveling salesman problem and of related problems, along with their theoretical prerequisites as seen from the point of view of an operations researcher who wants to solve practical problem instances. Extensive computational results are reported on most of the algorithms described. Optimal solutions are reported for instances with sizes up to several thousand nodes as well as heuristic solutions with provably very high quality for larger instances

On the heterogeneous vehicle routing problem under demand uncertainty

In this paper we study the heterogeneous vehicle routing problem under demand uncertainty, on which there has been little research to our knowledge. The focus of the paper is to provide a strong formulation that also easily allows tractable robust and chance-constrained counterparts. To this end, we propose a basic Miller-Tucker-Zemlin (MTZ) formulation with the main advantage that uncertainty is restricted to the right-hand side of the constraints. This leads to compact and tractable counterparts of demand uncertainty. On the other hand, since the MTZ formulation is well known to provide a rather weak linear programming relaxation, we propose to strengthen the initial formulation with valid inequalities and lifting techniques and, furthermore, to dynamically add cutting planes that successively reduce the polyhedral region using a branch-and-cut algorithm. We complete our study with extensive computational analysis with different performance measures on different classes of instances taken from the literature. In addition, using simulation, we conduct a scenario-based risk level analysis for both cases where either unmet demand is allowed or not

Lifting and Separation Procedures for the Cut Polytope

The max-cut problem and the associated cut polytope on complete graphs have been extensively studied over the last 25 years. However, little research has been conducted for the cut polytope on arbitrary graphs. In this study we describe new separation and lifting procedures for the cut polytope on such graphs. These procedures exploit algorithmic and structural results known for the cut polytope on complete graphs to generate valid, and sometimes facet defining, inequalities for the cut polytope on arbitrary graphs in a cutting plane framework. We report computational results on a set of well-established benchmark problems

Polyhedral techniques in combinatorial optimization II: computations

Combinatorial optimization problems appear in many disciplines ranging from management and logistics to mathematics, physics, and chemistry. These problems are usually relatively easy to formulate mathematically, but most of them are computationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable progress in techniques based on the polyhedral description of combinatorial problems. leading to a large increase in the size of several problem types that can be solved. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. Ideally we can then solve the problem as a linear programming problem, which can be done efficiently. The purpose of this manuscript is to give an overview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also present some modern applications, and computational experience

The traveling salesman problem with pickups, deliveries, and draft limits

open3siResearch supported by Air Force Office of Scientific Research (Grants FA9550-17-1-0025 and FA9550-17-1-0067 ) and by MIUR- Italy (Grant PRIN 2015 ).We introduce a new generalization of the traveling salesman problem with pickup and delivery, that stems from applications in maritime logistics, in which each node represents a port and has a known draft limit. Each customer has a demand, characterized by a weight, and pickups and deliveries are performed by a single ship of given weight capacity. The ship is able to visit a port only if the amount of cargo it carries is compatible with the draft limit of the port. We present an integer linear programming formulation and we show how classical valid inequalities from the literature can be adapted to the considered problem. We introduce heuristic procedures and a branch-and-cut exact algorithm. We examine, through extensive computational experiments, the impact of the various cuts and the performance of the proposed algorithms.openMalaguti, Enrico; Martello, Silvano*; Santini, AlbertoMalaguti, Enrico; Martello, Silvano*; Santini, Albert