Introduction to the Maximum Solution Problem
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Approximability of the Maximum Solution Problem for Certain Families of Algebras
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We study a family of problems, called Maximum Solution, where the objective is to maximize 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 Max Ones problem, and the approximability is completely understood for all restrictions on the underlying constraints [S. Khanna, M. Sudan, L. Trevisan, and D. P. Williamson, 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 [Minimal Clones Generated by Groupoids, Ph.D. thesis, Université de Móntreal, Montreal, QC, 1996] 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 indicate that this approach is very powerful for classifying the approximability of certain optimization problems.