Online covering with $$\ell _q$$-norm objectives and applications to network design

Abstract

We consider fractional online covering problems with $$\ell _q$$ -norm objectives as well as its dual packing problems. The problem of interest is of the form $$\min \{ f(x) \,:\, Ax\ge 1, x\ge 0\}$$ where $$f(x)=\sum _{e} c_e \Vert x(S_e)\Vert _{q_e} $$ is the weighted sum of $$\ell _q$$ -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 $$\rho = \frac{ a_{\max }}{ a_{\min }}$$ and d is the maximum of the row sparsity of A and $$\max |S_e|$$ . This is based on the online primal-dual framework where we use the dual of the above convex program. Our result is nearly tight (even in the linear special case), and it expands the class of convex programs that admit online algorithms. We also provide two applications where such convex programs arise as relaxations of discrete optimization problems, for which our result leads to good online algorithms. In particular, we obtain an improved online algorithm (by two logarithmic factors) for non-uniform buy-at-bulk network design and a poly-logarithmic competitive ratio for throughput maximization under $$\ell _p$$ -norm capacities.