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+ | ====== Variable complexity problems ====== | ||

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+ | Variable complexity problems are problems in which the main problem is to find a suitable set of values for a set of parameters, but we don't know how many. However we already know the relationships between the parameters. Problems falling into this class include | ||

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+ | * A few optimisation problems, where the aim is to find real-number coefficients x1,...,xn where we don't know how large n needs to be beforehand - this includes cases such as power or fourier series approximations | ||

+ | * Some discrete optimisation problems, we may not know ahead of time how many parameters x1,..., are required | ||

+ | * For example knapsack problems, stock cutting problems | ||

+ | * Polynomial regression, where we don't know ahead of time how high degree polynomial is required to get an acceptable fit | ||

+ | * Most interesting attribute learning problems fall into this class - decision trees, decision rules etc | ||

+ | * While solutions aren't completely unstructured, there are many transformation symmetries (transitivity, commutativity,...) so discovering the structure is generally not the hard part of the problem | ||

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+ | Methods for solving such problems include | ||

+ | * Gradient descent and similar deterministic eager search methods | ||

+ | * Stochastic search (simulated annealing, tabu search, genetic algorithms) | ||

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+ | While this isn't the primary focus of research in the structural complexity lab, we do some research in these areas from time to time; we tend to be more interested in | ||

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+ | * Dynamic problems, where the optimal solution changes with time | ||

+ | * Problems belonging to a family of problems, so tough versions can be solved incrementally from first solving simpler ones | ||