Baseline Cluster size 1 Cluster size 3 Cluster size 57 0 5 10 15 Macro−cell Spectral Efficiency (bps/Hz) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 600 700 800 900 1000 1100 1200 Macro Resource Weighted sum rate Decreasing min. rates Fig. 3. Cell SE for different cluster sizes Fig. 2. Optimized weighted sum rate vs macro budget for different min. rates VI. CONCLUSIONS APPENDIX 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. 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 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. • 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.