Approximation algorithms for general packing problems with modified logarithmic potential function

In this paper we present an approximation algorithm based on a Lagrangian decomposition via a logarithmic potential reduction to solve a general packing or min-max resource sharing problem with M nonnegative convex constraints on a convex set B. We generalize a method by Grigoriadis et al to the case with weak approximate block solvers (i.e. with only constant, logarithmic or even worse approximation ratios). We show that the algorithm needs at most O(M (epsilon(-2) ln epsilon(-1) + ln M)) calls to the block solver, a bound independent of the data and the approximation ratio of the block solver. For small approximation ratios the algorithm needs at most O(M(epsilon(-2) + lnM)) calls to the block solver.