On Lagrangian Relaxation and Reoptimization Problems

TL;DR

This paper demonstrates how Lagrangian relaxation can be used to develop approximation algorithms for a broad class of subset selection problems with linear constraints, achieving near-optimal solutions efficiently.

Contribution

It introduces a unified framework leveraging Lagrangian relaxation for approximation and reoptimization of subset selection problems with linear constraints.

Findings

Achieves a ratio of r/(r+1) - ε for problems with an r-approximation of the relaxation.

The algorithms use a linear number of calls to the r-approximation algorithm.

Provides reapproximation algorithms for problems like real-time scheduling, GAP, and independent set.

Abstract

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 {\em subset selection} problems with linear constraints. Given a problem in this class and some small $\eps \in (0,1)$, we show that if there exists an $r$-approximation algorithm for the Lagrangian relaxation of the problem, for some $r \in (0,1)$, then our technique achieves a ratio of $\frac{r}{r+1} -\! \eps$ to the optimal, and this ratio is tight. The number of calls to the $r$-approximation algorithm, used by our algorithms, is {\em linear} in the input size and in $\log (1 / \eps)$ for inputs with cardinality constraint, and polynomial in the input size and in $\log (1 / \eps)$ for inputs with arbitrary linear constraint. Using the technique we obtain (re)approximation algorithms for natural (reoptimization) variants of classic subset selection problems, including real-time scheduling, the {\em maximum generalized assignment problem (GAP)} and maximum weight independent set.