Online Primal-Dual Algorithms with Configuration Linear Programs

TL;DR

This paper introduces primal-dual algorithms based on configuration linear programs for online problems with non-linear and non-convex objectives, achieving near-optimal competitive ratios across various applications.

Contribution

It extends primal-dual approaches to non-convex objectives in online settings, introducing new concepts like local smoothness for analyzing competitive ratios.

Findings

Achieves optimal competitive ratios for network routing and scheduling.

Develops algorithms for non-convex facility location and submodular minimization.

Generalizes previous convex-focused primal-dual algorithms to non-convex objectives.

Abstract

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. In this paper, we present primal-dual approaches based on configuration linear programs to design competitive online algorithms for problems with arbitrarily-grown objective. This approach is particularly appropriate for non-linear (non-convex) objectives in online setting. 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, energy-efficient 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. 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 $\ell_{k}$-norms and gives competitive solutions for online problems of submodular minimization and some natural non-convex minimization under covering constraints.