Abstract
We study a family of problems, called Maximum Solution, where the objective is to maximise a linear goal function over the feasible integer assignments to a set of variables subject to a set of constraints. This problem is closely related to Integer Linear Programming.When the domain is Boolean (i.e. restricted to {0,1}), the maximum solution problem is identical to the well-studied Max Ones problem, and the approximability is completely understood for all restrictions on the underlying constraints. We continue this line of research by considering domains containing more than two elements. We present two main results: a complete classification for the approximability of all maximal constraint languages, and a complete classification of the approximability of the problem when the set of allowed constraints contains all permutation constraints.Our results are proved by using algebraic results from clone theory and the results indicates that this approach is very useful for classifying the approximability of certain optimisation problems.
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