Approximability of integer programming with generalised constraints

Given a set of variables and a set of linear inequalities over those variables, the objective in the Integer Linear Programming problem is to find an integer assignment to the variables such that the inequalities are satisfied and a linear goal function is maximised. We study a family of problems, called Maximum Solution, which are related to Integer Linear Programming. In a Maximum Solution problem, the constraints are drawn from a set of allowed relations, hence arbitrary constraints are studied instead of just linear inequalities. 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 [Khanna et al., SIAM J. Comput., 30 (2000), pp. 1863-1920]. We continue this line of research by considering domains containing more than two elements. Our main results are two new large tractable fragments for the maximum solution problem and a complete classification for the approximability of all maximal constraint languages. Moreover, we give 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