On Lagrangian Relaxation and Subset Selection Problems

We prove a general result demonstrating the power of Lagrangian relaxation in solving constrained maximization problems with arbitrary objective functions. This yields a unified approach for solving a wide class of subset selection problems with linear constraints. Given a problem in this class and some small e ∈ (0,1), we show that if there exists a ρ-approximation algorithm for the Lagrangian relaxation of the problem, for some ρ ∈ (0,1), then our technique achieves a ratio of $\frac{\rho}{\rho+1} -\! \varepsilon$ to the optimal, and this ratio is tight. The number of calls to the ρ-approximation algorithm, used by our algorithms, is linear in the input size and in log(1 / e) for inputs with cardinality constraint, and polynomial in the input size and in log(1 / e) for inputs with arbitrary linear constraint. Using the technique we obtain approximation algorithms for natural variants of classic subset selection problems, including real-time scheduling, the maximum generalized assignment problem (GAP) and maximum weight independent set.