Tight LP-Relaxations of Overlapping Global Cardinality Constraints

Motivation

The global cardinality constraint (gcc) [7], written as

$$ cardinality(x,J;l,u),x_{j}\in D_{j},j\in J, $$

states that each value d is received by at least l_d and by at most u_d of the variables {x_j:j ∈ J}, where $d\in D={\mathop{\textstyle \bigcup }}_{j\in J}D_{j}=\{0,\ldots ,|D|-1\};$ also, 0 ≤ l _d ≤ u_d and u_d ≥ 1 for all d ∈ D. The gcc has several applications [2, 10], thus having been studied from the Constraint Programming community mainly for accomplishing various forms of consistency [4, 6–8] or for examining the tractability of a natural generalization [9].

This research has been co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: Thales. Investing in knowledge society through the European Social Fund.