The traveling salesman problem (TSP) is the problem of finding the shortest tour through all the nodes that a salesman has to visit. The TSP is probably the most famous and extensively studied problem in the field of combinatorial optimization. Because this problem is an NP-hard problem, practical large-scale instances cannot be solved by exact algorithms within acceptable computational times. ...

Time considerations have been largely ignored in the study of vehicle routing problems with stochastic demands, even though they are crucial in practice. We show that tour duration limits can effectively and efficiently be incorporated in solution approaches that build fixed, or a priori, tours for such problems. We do so by assuming that each tour must be duration-feasible for all demand reali...

The Travelling Salesman Problem (TSP) is a classical problem in the field of combinatorial optimization. Main objective of TSP is to find an optimal tour which starts from an arbitrary city (vertex), visits remaining cities exactly once and returns back to the city at which tour commenced. TSP belongs to the class of NP Complete problems, has been studied for many years and is still being studi...

In this work, we propose a hybrid heuristic in order to solve the Team Orienteering Problem (TOP). Given a set of points (or customers), each with associated score (profit or benefit), and a team that has a fixed number of members, the problem to solve is to visit a subset of points in order to maximize the total collected score. Each member performs a tour starting at the start point, visiting...

We motivate, derive and implement a multilevel approach to the travelling salesman problem. The resulting algorithm progressively coarsens the problem, initialises a tour and then employs either the LinKernighan (LK) or the Chained Lin-Kernighan (CLK) algorithm to refine the solution on each of the coarsened problems in reverse order. In experiments on a well established test suite of 79 proble...

We study the problem of finding a tour of n points in which every edge is long. More precisely, we wish to find a tour that visits every point exactly once, maximizing the length of the shortest edge in the tour. The problem is known as Maximum Scatter TSP, and was introduced by Arkin et al. (SODA 1997), motivated by applications in manufacturing and medical imaging. Arkin et al. gave a 0.5 -ap...

We study the problem of finding a tour of n points in R in which every edge is long. More precisely, we wish to find a tour that maximizes the length of the shortest edge in the tour. The problem is known as Maximum Scatter TSP, and it was introduced by Arkin et al. (SODA 1997), motivated by applications in manufacturing and medical imaging. Arkin et al. gave a 2-approximation for the metric ve...

We study the problem of finding a shortest tour visiting a given sequence of convex bodies in R. To our knowledge, this is the first attempt to attack the problem in its full generality: we investigate high-dimensional cases (d ≥ 2); we consider convex bodies bounded by (hyper)planes and/or (hyper)spheres; we do not restrict the start and the goal positions of the tour to be single points, we m...

We study the online version of the Prize-Collecting Traveling Salesman Problem (PCTSP), a generalization of the Traveling Salesman Problem (TSP). In the TSP, the salesman has to visit a set of cities while minimizing the length of the overall tour. In the PCTSP, each city has a given weight and penalty, and the goal is to collect a given quota of the weights of the cities while minimizing the l...

Branch-and-bound intelligently searches the set of feasible solutions to a combinatorial optimization problem: It, in effect, proves that the optimal solution is found without necessarily examining all feasible solutions. The feasible solutions are not given. They can be generated from the problem description. However, doing so usually is computationally infeasible: The number of feasible solut...