On Lagrangian relaxation for constrained maximization and reoptimization 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 ε∈(0,1), we show that if there exists an r-approximation algorithm for the Lagrangian relaxation of the problem, for some r∈(0,1), then our technique achieves a ratio of rr+1−ε to the optimal, and this ratio is tight. Using the technique we obtain (re)approximation algorithms for natural (reoptimization) variants of classic subset selection problems, including real-time scheduling, the maximum generalized assignment problem (GAP) and maximum weight independent set. For all of the problems studied in this paper, the number of calls to the r-approximation algorithm, used by our algorithms, is linear in the input size and in log(1∕ε) for inputs with a cardinality constraint, and polynomial in the input size and in log(1∕ε) for inputs with an arbitrary linear constraint.