Online Convex Covering and Packing Problems

We study the online convex covering problem and online convex packing problem. The (offline) convex covering problem is modeled by the following convex program: min ~ x∈Rn+ f(~x) s.t. A~x ≥ ~1 where f : R+ 7→ R+ is a monotone and convex cost function, and A is an m × n matrix with non-negative entries. Each row of the constraint matrix A corresponds to a covering constraint. In the online problem, each row of A comes online and the algorithm must maintain a feasible assignment ~x and may only increase ~x over time. The (offline) convex packing problem is modeled by the following convex program: max ~ y∈Rm+ ∑m j=1 yj − g(A ~y) where g : R+ 7→ R+ is a monotone and convex cost function. It is the Fenchel dual program of convex covering when g is the convex conjugate of f . In the online problem, each variable yj arrives online and the algorithm must decide the value of yj on its arrival. We propose simple online algorithms for both problems using the online primal dual technique, and obtain nearly optimal competitive ratios for both problems for the important special case of polynomial cost functions. For instance, for any convex homogeneous polynomial cost functions with non-negative coefficients and degree τ , we introduce an O(τ logn) -competitive online convex covering algorithm, and an O(τ)-competitive online convex packing algorithm, matching the known Ω(τ logn) [5] and Ω(τ) [21] lower bounds respectively. There is a large family of online resource allocation problems that can be modeled under this online convex covering and packing framework, including online covering and packing problems (with linear objectives) [14], online mixed covering and packing [4], and online combinatorial auction [7, 9]. Our framework allows us to study these problems using a unified approach. Department of Computer Science, the University of Hong Kong. {hubert,zhiyi,nkang}@cs.hku.hk