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A polyhedral approach to the alldifferent system

Magos, Dimitris, Mourtos, Ioannis and Appa, Gautam (2012) A polyhedral approach to the alldifferent system. Mathematical Programming, 132 (1-2). pp. 209-260. ISSN 0025-5610

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Identification Number: 10.1007/s10107-010-0390-6


This paper examines the facial structure of the convex hull of integer vectors satisfying a system of alldifferent predicates, also called an alldifferent system. The underlying analysis is based on a property, called inclusion, pertinent to such a system. For the alldifferent systems for which this property holds, we present two families of facet-defining inequalities, establish that they completely describe the convex hull and show that they can be separated in polynomial time. Consequently, the inclusion property characterises a group of alldifferent systems for which the linear optimization problem (i.e. the problem of optimizing a linear function over that system) can be solved in polynomial time. Furthermore, we establish that, for systems with three predicates, the inclusion property is also a necessary condition for the convex hull to be described by those two families of inequalities. For the alldifferent systems that do not possess that property, we establish another family of facet-defining inequalities and an accompanied polynomial-time separation algorithm. All the separation algorithms are incorporated within a cutting-plane scheme and computational experience on a set of randomly generated instances is reported. In concluding, we show that the pertinence of the inclusion property can be decided in polynomial time.

Item Type: Article
Official URL:
Additional Information: © 2010 Springer and Mathematical Optimization Society
Divisions: Management
Subjects: H Social Sciences > HD Industries. Land use. Labor > HD28 Management. Industrial Management
H Social Sciences > HM Sociology
Q Science > QA Mathematics
Date Deposited: 04 Apr 2012 15:52
Last Modified: 20 Oct 2021 00:45

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