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In heterogeneous networks (HetNets), the load between macrocell base stations (MBSs) and small-cell base stations (SBSs) is imbalanced due to their different transmission powers and locations. This load imbalance significantly impacts system performance and affects the experience of mobile users (MUs) with different priorities. In this paper, we aim to distributively optimize the user association in HetNets with various user priorities to solve the load balancing problem. Since the user association is a binary matching problem, which is NP-hard, we propose a distributed belief propagation (BP) algorithm to approach the optimal solution. We first develop a factor graph model, using the network topology, to represent this user association problem. With this factor graph, we propose a novel distributed BP algorithm by adopting the proportional fairness as the objective. Next, we theoretically prove the existence of the fixed point in our BP algorithm. To be more practical, we develop an approximation method to significantly reduce the computational and communication complexity of the BP algorithm. Furthermore, we analyze some properties of the factor graph relevant to the performance of the BP algorithm using the stochastic geometry. Simulation results show that 1) the proposed BP algorithm well approaches the optimal system performance and achieves a much better performance compared with other association schemes and that 2) the analytical results on the average degree distribution and sparsity of the factor graph match with those obtained from the Monte Carlo simulations. [ABSTRACT FROM PUBLISHER]

In an interference limited network, joint power and admission control (JPAC) aims at supporting a maximum number of links at their specified signal-to-interference-plus-noise ratio (SINR) targets while using minimum total transmission power. Various convex approximation deflation approaches have been developed for the JPAC problem. In this paper, we propose an effective polynomial time non-convex approximation deflation approach for solving the problem. The approach is based on the non-convex \ell q~(0< q< 1) approximation of an equivalent sparse \ell 0 reformulation of the JPAC problem. We show that, for any instance of the JPAC problem, there exists a \bar q\in (0,1) such that it can be exactly solved by solving its \ell q approximation problem with any q\in (0, \bar q]. We also show that finding the global solution of the \ell q approximation problem is NP-hard. Then, we propose a potential reduction interior-point algorithm, which can return an \epsilon -KKT solution of the NP-hard \ell q approximation problem in polynomial time. The returned solution can be used to check the simultaneous supportability of all links in the network and to guide an iterative link removal procedure, resulting in the polynomial time non-convex approximation deflation approach for the JPAC problem. Numerical simulations show that the proposed approach outperforms the existing convex approximation approaches in terms of the number of supported links and the total transmission power, particularly exhibiting a quite good performance in selecting which subset of links to support. [ABSTRACT FROM PUBLISHER]