Congram, Richard K.
Polynomially searchable exponential neighbourhoods for sequencing problems in combinatorial optimisation
University of Southampton, Department of Mathematics,
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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