Approximability of Integer Programming with Generalised Constraints

TL;DR

This paper classifies how well the maximum solution problem can be approximated for various constraint languages over small domains, extending known results from Boolean to larger domains using algebraic methods.

Contribution

It provides a complete classification of approximability for maximal constraint languages over domains of size up to 4, and when all permutation constraints are allowed, using algebraic techniques.

Findings

Complete classification for domains of size up to 4.

Classification when all permutation constraints are included.

Use of clone theory to determine approximability.

Abstract

We study a family of problems, called \prob{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. When the domain is Boolean (i.e. restricted to $\{0,1\}$), the maximum solution problem is identical to the well-studied \prob{Max Ones} problem, and the approximability is completely understood for all restrictions on the underlying constraints [Khanna et al., SIAM J. Comput., 30 (2001), pp. 1863-1920]. 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 over domains of cardinality at most 4, and a complete classification of the approximability of the problem when the set of allowed constraints contains all permutation constraints. Under the assumption that a conjecture due to Szczepara holds, we give a complete classification for all maximal constraint languages. These classes of languages are well-studied in universal algebra and computer science; they have, for instance, been considered in connection with machine learning and constraint satisfaction. Our results are proved by using algebraic results from clone theory and the results indicates that this approach is very powerful for classifying the approximability of certain optimisation problems.