Theory of linear and integer programming
Theory of linear and integer programming
Simple and Fast Algorithms for Linear and Integer Programs with Two Variables Per Inequality
SIAM Journal on Computing
Tractable constraints on ordered domains
Artificial Intelligence
Closure properties of constraints
Journal of the ACM (JACM)
Near-optimal nonapproximability results for some NPO PB-complete problems
Information Processing Letters
Complexity classifications of boolean constraint satisfaction problems
Complexity classifications of boolean constraint satisfaction problems
The complexity of maximal constraint languages
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The Approximability of Constraint Satisfaction Problems
SIAM Journal on Computing
A Dichotomy Theorem for Constraints on a Three-Element Set
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
A Graph of a Relational Structure and Constraint Satisfaction Problems
LICS '04 Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science
A new tractable class of constraint satisfaction problems
Annals of Mathematics and Artificial Intelligence
Classifying the Complexity of Constraints Using Finite Algebras
SIAM Journal on Computing
A dichotomy theorem for constraint satisfaction problems on a 3-element set
Journal of the ACM (JACM)
The Approximability of Three-valued MAX CSP
SIAM Journal on Computing
Tight approximability results for the maximum solution equation problem over Zp
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
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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.