Online Covering with Sum of $\ell_q$-Norm Objectives

We consider fractional online covering problems with $\ellq$-norm objectives. The problem of interest is of the form $\min{ f(x) \,:\, Ax\ge 1, x\ge 0}$ where $f(x)=\sum{e} ce |x(Se)|{qe} $ is the weighted sum of $\ellq$-norms and $A$ is a non-negative matrix. The rows of $A$ (i.e. covering constraints) arrive online over time. We provide an online $O(\log d+\log \rho)$-competitive algorithm where $\rho = \frac{\max a{ij}}{\min a{ij}}$ and $d$ is the maximum of the row sparsity of $A$ and $\max |Se|$. This is based on the online primal-dual framework where we use the dual of the above convex program. Our result expands the class of convex objectives that admit good online algorithms: prior results required a monotonicity condition on the objective $f$ which is not satisfied here. This result is nearly tight even for the linear special case. As direct applications we obtain (i) improved online algorithms for non-uniform buy-at-bulk network design and (ii) the first online algorithm for throughput maximization under $\ell_p$-norm edge capacities.