Online Packing and Covering Framework with Convex Objectives
Abstract
We consider online fractional covering problems with a convex objective, where the covering constraints arrive over time. Formally, we want to solve $\min\,\{f(x) \mid Ax\ge \mathbf{1},\, x\ge 0\},$ where the objective function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is convex, and the constraint matrix $A_{m\times n}$ is nonnegative. The rows of $A$ arrive online over time, and we wish to maintain a feasible solution $x$ at all times while only increasing coordinates of $x$. We also consider "dual" packing problems of the form $\max\,\{c^\intercal y  g(\mu) \mid A^\intercal y \le \mu,\, y\ge 0\}$, where $g$ is a convex function. In the online setting, variables $y$ and columns of $A^\intercal$ arrive over time, and we wish to maintain a nondecreasing solution $(y,\mu)$. We provide an online primaldual framework for both classes of problems with competitive ratio depending on certain "monotonicity" and "smoothness" parameters of $f$; our results match or improve on guarantees for some special classes of functions $f$ considered previously. Using this fractional solver with problemdependent randomized rounding procedures, we obtain competitive algorithms for the following problems: online covering LPs minimizing $\ell_p$norms of arbitrary packing constraints, set cover with multiple cost functions, capacity constrained facility location, capacitated multicast problem, set cover with set requests, and profit maximization with nonseparable production costs. Some of these results are new and others provide a unified view of previous results, with matching or slightly worse competitive ratios.
 Publication:

arXiv eprints
 Pub Date:
 December 2014
 arXiv:
 arXiv:1412.8347
 Bibcode:
 2014arXiv1412.8347B
 Keywords:

 Computer Science  Data Structures and Algorithms