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 non-negative. 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 non-decreasing solution $(y,\mu)$. We provide an online primal-dual 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 problem-dependent 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 non-separable production costs. Some of these results are new and others provide a unified view of previous results, with matching or slightly worse competitive ratios.