Travelling salesman problem in the context of Hamiltonian path problem

The vehicle routing problem (VRP) is a combinatorial optimization and integer programming problem which asks "What is the optimal set of routes for a fleet of vehicles to traverse in order to deliver to a given set of customers?" The problem first appeared, as the truck dispatching problem, in a paper by George Dantzig and John Ramser in 1959, in which it was applied to petrol deliveries. Often, the context is that of delivering goods located at a central depot to customers who have placed orders for such goods. However, variants of the problem consider, e.g, collection of solid waste and the transport of the elderly and the sick to and from health-care facilities. The standard objective of the VRP is to minimise the total route cost. Other objectives, such as minimising the number of vehicles used or travelled distance are also considered.

The VRP generalises the travelling salesman problem (TSP), which is equivalent to requiring a single route to visit all locations. As the TSP is NP-hard, the VRP is also NP-hard.

Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead.

Combinatorial optimization is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, VLSI, applied mathematics and theoretical computer science.

Jan Karel Lenstra (born 19 December 1947, in Zaandam) is a Dutch mathematician and operations researcher, known for his work on scheduling algorithms, local search, and the travelling salesman problem.

Lenstra received his Ph.D. from the University of Amsterdam in 1976, advised by Gijsbert de Leve. He then became a researcher at the Centrum Wiskunde & Informatica, where he remained until 1989. After taking positions at the Eindhoven University of Technology (where he became Dean of the Faculty of Mathematics and Computer Science) and the Georgia Institute of Technology, he returned to CWI as its director in 2003. He stepped down in 2011, and at that time became a CWI Fellow. He was editor-in-chief of Mathematics of Operations Research from 1993 to 1998, and is editor-in-chief of Operations Research Letters since 2002.

Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. For large numbers of local optima, SA can find the global optimum. It is often used when the search space is discrete (for example the traveling salesman problem, the boolean satisfiability problem, protein structure prediction, and job-shop scheduling). For problems where a fixed amount of computing resource is available, finding an approximate global optimum may be more relevant than attempting to find a precise local optimum. In such cases, SA may be preferable to exact algorithms such as gradient descent or branch and bound. The problems solved by SA are currently formulated by an objective function of many variables, subject to several mathematical constraints. In practice, a constraint violation can be penalized as part of the objective function.

Similar techniques have been independently introduced on several occasions, including Pincus (1970), Khachaturyan et al. (1979, 1981), Kirkpatrick, Gelatt and Vecchi (1983), and Cerny (1985). In 1983, this approach was used by Kirkpatrick, Gelatt Jr., and Vecchi for a solution of the traveling salesman problem. They also proposed its current name, simulated annealing.