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In this paper, the authors study the complexity of spectrum assignment problems in Cognitive Radio Networks (CRNs) in the presence of several constraints. Although optimal spectrum assignment for secondary transmissions in CRNs is generally believed to be NP complete, the impact of fairness and link quality constraints on the hardness of the problem is not well studied. In this paper, they show that when a minimum quality constraint is imposed on secondary transmissions, the spectrum assignment problem can be solved in polynomial time. However, such assignments may not guarantee fairness. They also show that when fairness is desired, even in the presence of quality constraints spectrum assignment problems remain NP complete.
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