  We give new approximation algorithms for the submodular joint replenishment
problem and the inventory routing problem, using an iterative rounding
approach. In both problems, we are given a set of $N$ items and a discrete time
horizon of $T$ days in which given demands for the items must be satisfied.
Ordering a set of items incurs a cost according to a set function, with
properties depending on the problem under consideration. Demand for an item at
time $t$ can be satisfied by an order on any day prior to $t$, but a holding
cost is charged for storing the items during the intermediate period; the goal
is to minimize the sum of the ordering and holding cost.
  Our approximation factor for both problems is $O(\log \log \min(N,T))$; this
improves exponentially on the previous best results.
