Abstract-We present online algorithms for covering and packing problems with (non-linear) convex objectives. The convex covering problem is defined as: min x∈R n + f (x) s.t. Ax ≥ 1, where f : R n + → R+ is a monotone convex function, and A is an m × n matrix with non-negative entries. In the online version, a new row of the constraint matrix, representing a new covering constraint, is revealed in each step and the algorithm is required to maintain a feasible and monotonically non-decreasing assignment x over time. We also consider a convex packing problem defined as: + → R+ is a monotone convex function. In the online version, each variable yj arrives online and the algorithm must decide the value of yj on its arrival. This represents the Fenchel dual of the convex covering program, when g is the convex conjugate of f . We use a primal-dual approach to give online algorithms for these generic problems, and use them to simplify, unify, and improve upon previous results for several applications.
Recently, REF considered packing/covering problems with general (non-separable) convex objectives, but (as discussed previously) this result requires a monotone gradient assumption on the convex function.
Online Algorithms for Covering and Packing Problems with Convex Objectives
2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
Mathematics Computer Science