Variable neighborhood search (VNS) is a recent metaheuristic for solving combinatorial and global optimization problems whose basic idea is systematic change of neighborhood within a local search. In this survey paper we present basic rules of VNS and some of its extensions. Moreover, applications are briefly summarized. They comprise heuristic solution of a variety of optimization problems, ways to accelerate exact algorithms and to analyze heuristic solution processes, as well as computer-assisted discovery of conjectures in graph theory.

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.

Systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization, called variable neighborhood search (VNS). We present a basic scheme for this purpose, which can easily be implemented using any local search algorithm as a subroutine. Its effectiveness is illustrated by solving several classical combinatorial or global optimization problems. Moreover, several extensions are proposed for solving large problem instances: using VNS within the successive approximation method yields a two-level VNS, called variable neighborhood decomposition search (VNDS); modifying the basic scheme to explore easily valleys far from the incumbent solution yields an efficient skewed VNS (SVNS) heuristic. Finally, we show how to stabilize column generation algorithms with help of VNS and discuss various ways to use VNS in graph theory, i.e., to suggest, disprove or give hints on how to prove conjectures, an area where metaheuristics do not appear

This paper is the result of a literature study carried out by the authors. It is a review of the dierent attempts made to solve the Travelling Salesman Problem with Genetic Algorithms. We present crossover and mutation operators, developed to tackle the Travelling Salesman Problem with Genetic Algorithms with dierent representations such as: binary representation, path representation, adjacency representation, ordinal representation and matrix representation. Likewise, we show the experimental results obtained with dierent standard examples using combination of crossover and mutation operators in relation with path representation. Keywords: Travelling Salesman Problem; Genetic Algorithms; Binary representation; Path representation; Adjacency representation; Ordinal representation; Matrix representation; Hybridation. 1 1 Introduction In nature, there exist many processes which seek a stable state. These processes can be seen as natural optimization processes. Over the last...

A new local search heuristic, called J-Means, is proposed for solving the minimum sum-of-squares clustering problem. The neighborhood of the current solution is defined by all possible centroid-to-entity relocations followed by corresponding changes of assignments. Moves are made in such neighborhoods until a local optimum is reached. The new heuristic is compared with two other well-known local search heuristics, KMeans and H-Means as well as with H-Means+, an improved version of the latter in which degeneracy is removed. Moreover, another heuristic, which fits into the Variable Neighborhood Search metaheuristic framework and uses J-Means in its local search step, is proposed too. Results on standard test problems from the literature are reported. It appears that J-Means outperforms the other local search methods, quite substantially when many entities and clusters are considered. 1 Introduction Consider a set X = fx 1 ; : : : ; xN g, x j = (x 1j ; : : : ; x qj ) 2 R q of N entiti...

We discuss several issues that arise in the implementation of Martin, Otto, and Felten's Chained Lin-Kernighan heuristic for large-scale traveling salesman problems. Computational results are presented for TSPLIB instances ranging in size from 11,849 cities up to 85,900 cities; for each of these instances, solutions within 1% of the optimal value can be found in under 1 CPU minute on a 300 Mhz Pentium II workstation, and solutions within 0.5% of optimal can be found in under 10 CPU minutes. We also demonstrate the scalability of the heuristic, presenting results for randomly generated Euclidean instances having up to 25,000,000 cities. For the largest of these random instances, a tour within 1% of an estimate of the optimal value can be obtained in under 1 CPU day on a 64-bit IBM RS6000 workstation.

The traveling salesman problem, or TSP for short, is easy to state: given a finite number of "cities" along with the cost of travel between each pair of them, find the cheapest way of visiting all the cities and returning to your starting point. The travel costs are symmetric in the sense that traveling from city X to city Y costs just as much as traveling from Y to X; the "way of visiting all the cities" is simply the order in which the cities are visited. In this report we consider the relaxed version of the TSP where we ask only for a tour of low cost. This is a preliminary version of a chapter of planned monograph on the TSP.

This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectivity invariants, branchwidth and treewidth, were first introduced to aid in proving the Graph Minors Theorem, a wellknown conjecture (Wagner’s conjecture) in graph theory. The algorithmic importance of branch decompositions and tree decompositions for solving NP-hard problems modelled on graphs was first realized by computer scientists in relation to formulating graph problems in monadic second order logic. The dynamic programming techniques utilizing branch decompositions and tree decompositions, called branch decomposition and tree decomposition based algorithms, fall into a class of algorithms known as fixed-parameter tractable algorithms and have been shown to be effective in a practical setting for NP-hard problems such as minimum domination, the travelling salesman problem, general minor containment, and frequency assignment problems.

The p-Center problem consists in locating p facilities and assigning clients to them in order to minimize the maximum distance between a client and the facility to which he is allocated. In this paper we present a basic Variable Neighborhood Search and two Tabu Search heuristics for the p-Center problem without triangle inequality. Both proposed methods use the 1-interchange (or vertex substitution) neighborhood structure.

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