Polynomially searchable exponential neighbourhoods for sequencing problems in combinatorial optimisation

Abstract

In this thesis, we study neighbourhoods of exponential size that can be searched in polynomial time. Such neighbourhoods are used in local search algorithms for classes of combinatorial optimisation problems. We introduce a method, called dynasearch, of constructing new neighbourhoods, and of viewing some previously derived exponentially sized neighbourhoods which are searchable in polynomial time. We produce new neighbourhoods by combining simple well-known neighbourhood moves (such as swap, insert, and k-opt) so that the moves can be performed together as a single move. In dynasearch neighbourhoods, the moves are combined in such a way that the effect of the combined move on the objective function is equal to the sum of the effects of the individual moves from the underlying neighbourhood. Dynasearch neighbourhoods can be formed using dynamic programing from underlying moves:
• nested within each other;
• disjoint from each other;
• and in the case of the TSP overlapping one another.
Our dynasearch neighbourhoods made from underlying disjoint moves are successfully implemented within well-known local search methods to form competitive algorithms for the travelling salesman problem and state-of-the-art algorithms for the total weighted tardiness problem and linear ordering problem. By viewing moves from some known travelling salesman problem neighbourhoods as a combination of underlying moves, each reversing a section of the tour, greater insight into the structure of the neighbourhoods may be obtained. This insight has both enabled us to calculate the size of a number of neighbourhoods and demonstrate how some neighbourhoods are contained within others.

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