Optimality of Dual Methods for Discrete Multiuser Multicarrier Resource Allocation Problems

Dual methods based on Lagrangian relaxation are the state of the art to solve multiuser multicarrier resource allocation problems. This applies to concave utility functions as well as to practical systems employing adaptive modulation, in which users' data rates can be described by step functions. We show that this discrete resource allocation problem can be formulated as an integer linear program belonging to the class of multiple-choice knapsack problems. As a knapsack problem with additional constraints, this problem is NP-hard, but facilitates approximation algorithms based on Lagrangian relaxation. We show that these dual methods can be described as rounding methods. As an immediate result, we conclude that prior claims of optimality, based on a vanishing duality gap, are insufficient. To answer the question of optimality of dual methods for discrete multicarrier resource allocation problems, we present bounds on the absolute integrality gap for three exemplary downlink resource allocation problems with different objectives when employing rounding methods. The obtained bounds are asymptotically optimal in the sense that the relative performance loss vanishes as the number of subcarriers tends to infinity. The exemplary problems considered in this work are sum rate maximization, sum power minimization and max-min fairness.