On the Optimality of Rate Balancing for Max-Min Fair Multicasting
Sadaf Syed, Wolfgang Utschick, Michael Joham
TL;DR
This work tackles the NP-hard MMF multicasting problem in downlink MISO systems and shows that, under certain conditions, rate balancing achieves the MMF optimum. By applying fractional programming and Lagrangian duality, it derives an optimal closed-form dual-variable solution and proves that balanced rates are optimal when the user channels are independent and $K \le M$; otherwise an active-set approach is needed. A low-complexity algorithm with closed-form expressions is proposed, with convergence guaranteed via block coordinate descent and complexity dominated by inverting a $K\times K$ matrix. Simulations under 3GPP channel models demonstrate the proposed method matches or outperforms SDR-based CVX methods while offering substantial computational savings, particularly in overloaded regimes.
Abstract
The max-min fair (MMF) multicasting problem is known to be NP-hard. In this work, we analytically derive the optimal solution to this NP-hard problem and establish the equivalence between rate balancing and the optimal MMF multicasting solution under certain conditions. Based on this theoretical insight, we propose a low-complexity algorithm for MMF multicasting that yields closed-form solutions. Simulation results validate our analysis and demonstrate that the proposed algorithm outperforms the state-of-the-art methods while being computationally more efficient.