Skip to main content
Open Access Publications from the University of California

UC Riverside

UC Riverside Previously Published Works bannerUC Riverside

On the Number of Iterations for Dantzig--Wolfe Optimization and Packing-Covering Approximation Algorithms

Published Web Location

We give a lower bound on the iteration complexity of a natural class of Lagrangianrelaxation algorithms for approximately solving packing/covering linear programs. We show that, given an input with m random 0/1-constraints on n variables, with high probability, any such algorithm requires Ω(ρ log(m)/∈2) iterations to compute a (1 + ∈)-approximate solution, where ρ is the width of the input. The bound is tight for a range of the parameters (m, n, ρ, ∈). The algorithms in the class include Dantzig-Wolfe decomposition, Benders' decomposition, Lagrangian relaxation as developed by Held and Karp for lower-bounding TSP, and many others (e.g., those by Plotkin, Shmoys, and Tardos and Grigoriadis and Khachiyan). To prove the bound, we use a discrepancy argument to show an analogous lower bound on the support size of (1 + ∈)-approximate mixed strategies for random two-player zero-sum 0/1-matrix games.

Many UC-authored scholarly publications are freely available on this site because of the UC's open access policies. Let us know how this access is important for you.

Main Content
For improved accessibility of PDF content, download the file to your device.
Current View