Online Covering with Convex Objectives and Applications

Yossi Azar; Ilan Reuven Cohen; Debmalya Panigrahi

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arXiv · 1412.3507

Online Covering with Convex Objectives and Applications

Yossi Azar·Ilan Reuven Cohen·Debmalya Panigrahi

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Abstract

We give an algorithmic framework for minimizing general convex objectives (that are differentiable and monotone non-decreasing) 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 non-linear 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 non-trivial $\ell_1$ 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 two-phase approach of obtaining a fractional solution and then rounding it online (used successfully to many linear objectives) to the non-linear objective. The fractional algorithm uses ideas from our general framework that we described above (but does not fit the framework exactly because of non-positive entries in the constraint matrix). The rounding algorithm uses ideas from offline rounding of LPs with non-linear 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.

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BibTeX RIS

Yossi Azar, Ilan Reuven Cohen, Debmalya Panigrahi. 2014-12-11. Online Covering with Convex Objectives and Applications. https://arxiv.org/abs/1412.3507

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