Online Covering with Convex Objectives and Applications
Abstract
We give an algorithmic framework for minimizing general convex objectives (that are differentiable and monotone nondecreasing) over a set of covering constraints that arrive online. This substantially extends previous work on online covering for linear objectives (Alon {\em et al.}, STOC 2003) and online covering with offline packing constraints (Azar {\em et al.}, SODA 2013). To the best of our knowledge, this is the first result in online optimization for generic nonlinear objectives; special cases of such objectives have previously been considered, particularly for energy minimization. As a specific problem in this genre, we consider the unrelated machine scheduling problem with startup costs and arbitrary $\ell_p$ norms on machine loads (including the surprisingly nontrivial $\ell_1$ norm representing total machine load). This problem was studied earlier for the makespan norm in both the offline (Khuller~{\em et al.}, SODA 2010; Li and Khuller, SODA 2011) and online settings (Azar {\em et al.}, SODA 2013). We adapt the twophase approach of obtaining a fractional solution and then rounding it online (used successfully to many linear objectives) to the nonlinear objective. The fractional algorithm uses ideas from our general framework that we described above (but does not fit the framework exactly because of nonpositive entries in the constraint matrix). The rounding algorithm uses ideas from offline rounding of LPs with nonlinear objectives (Azar and Epstein, STOC 2005; Kumar {\em et al.}, FOCS 2005). Our competitive ratio is tight up to a logarithmic factor. Finally, for the important special case of total load ($\ell_1$ norm), we give a different rounding algorithm that obtains a better competitive ratio than the generic rounding algorithm for $\ell_p$ norms. We show that this competitive ratio is asymptotically tight.
 Publication:

arXiv eprints
 Pub Date:
 December 2014
 arXiv:
 arXiv:1412.3507
 Bibcode:
 2014arXiv1412.3507A
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Distributed;
 Parallel;
 and Cluster Computing