Online Primal-Dual Algorithms with Configuration Linear Programs

In this paper, we present primal-dual approaches based on configuration linear programs to design competitive online algorithms for problems with arbitrarily-grown objective. Non-linear, especially convex, objective functions have been extensively studied in recent years in which approaches relies crucially on the convexity property of cost functions. Besides, configuration linear programs have been considered typically in offline setting and the main approaches are rounding schemes. In our framework, we consider configuration linear programs coupled with a primal-dual approach. This approach is particularly appropriate for non-linear (non-convex) objectives in online setting. By the approach, we first present a simple greedy algorithm for a general cost-minimization problem. The competitive ratio of the algorithm is characterized by the mean of a notion, called smoothness, which is inspired by a similar concept in the context of algorithmic game theory. The algorithm gives optimal (up to a constant factor) competitive ratios while applying to different contexts such as network routing, vector scheduling, energyefficient scheduling and non-convex facility location. Next, we consider the online 0 − 1 covering problems with non-convex objective. Building upon the resilient ideas from the primal-dual framework with configuration LPs, we derive a competitive algorithm for these problems. Our result generalizes the online primal-dual algorithm developed recently by Azar et al. [8] for convex objectives with monotone gradients to non-convex objectives. The competitive ratio is now characterized by a new concept, called local smoothness — a notion inspired by the smoothness. Our algorithm yields tight competitive ratio for the objectives such as the sum of lk-norms and gives competitive solutions for online problems of submodular minimization and some natural non-convex minimization under covering constraints. ∗Research supported by the ANR project OATA n ANR-15-CE40-0015-01