We give a lower bound on the iteration complexity of a natural class of
Lagrangean-relaxation 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(ρlog(m)/ϵ2) iterations to compute a
(1+ϵ)-approximate solution, where ρ is the width of the input.
The bound is tight for a range of the parameters (m,n,ρ,ϵ).
The algorithms in the class include Dantzig-Wolfe decomposition, Benders'
decomposition, Lagrangean relaxation as developed by Held and Karp [1971] for
lower-bounding TSP, and many others (e.g. by Plotkin, Shmoys, and Tardos [1988]
and Grigoriadis and Khachiyan [1996]). To prove the bound, we use a discrepancy
argument to show an analogous lower bound on the support size of
(1+ϵ)-approximate mixed strategies for random two-player zero-sum
0/1-matrix games