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## VI. CONCLUSIONS 

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We considered the problem of maximizing the weighted sum rate and the proportional fairness utility over dual connectivity enabled HetNets by exploiting load balancing. We constructed efficient algorithms to solve the resulting mixed optimization problems and proved that they yield approximately optimal solutions. 

## APPENDIX 

Definition 1. (Ω , I ) , where Ω is a ground set and I is collection of some subsets of Ω , is said to be a matroid if 

• I is downward closed, i.e., A ∈ I & B ⊆ A ⇒ B ∈ I 

• For any two members F$_{1}$ ∈ I and F$_{2}$ ∈ I such that |F$_{1}$| < |F$_{2}$| , there exists e ∈ F$_{2}$ \ F$_{1}$ such that F$_{1}$ ∪ { e } ∈ I . This property is referred to as the exchange property. 

Definition 2. Let J be a collection of some subsets of Ω and h : J → I R be a real-valued set function defined on the members of J . The set function h ( . ) is a submodular set function over J if it satisfies, h ( B ∪ a ) − h ( B ) ≤ h ( A ∪ a ) − h ( A ) for all A ⊆ B ∈ J & a ∈ Ω \ B : B ∪ a ∈ J. 

The following lemma is invoked in the proof of Proposition 4 and its proof follows from the arguments used in that of Proposition 2. 

