Online and Random-order Load Balancing Simultaneously

TL;DR

This paper introduces algorithms for online load balancing under lp-norms that provide optimal guarantees in both worst-case and random-order models, advancing beyond traditional analyses and addressing the challenge of simultaneous guarantees.

Contribution

The paper presents a new algorithm, SIMULTANEOUSLB, achieving near-optimal guarantees in both models, and introduces a novel OLO algorithm with improved regret and a smoothing technique for lp-norms.

Findings

The greedy algorithm with restart has improved random-order guarantees over worst-case.

The proposed SIMULTANEOUSLB algorithm achieves simultaneous optimal guarantees in both models.

A new smoothing technique for lp-norms enhances algorithm analysis and performance.

Abstract

We consider the problem of online load balancing under lp-norms: sequential jobs need to be assigned to one of the machines and the goal is to minimize the lp-norm of the machine loads. This generalizes the classical problem of scheduling for makespan minimization (case l_infty) and has been thoroughly studied. However, despite the recent push for beyond worst-case analyses, no such results are known for this problem. In this paper we provide algorithms with simultaneous guarantees for the worst-case model as well as for the random-order (i.e. secretary) model, where an arbitrary set of jobs comes in random order. First, we show that the greedy algorithm (with restart), known to have optimal O(p) worst-case guarantee, also has a (typically) improved random-order guarantee. However, the behavior of this algorithm in the random-order model degrades with p. We then propose algorithm SIMULTANEOUSLB that has simultaneously optimal guarantees (within constants) in both worst-case and random-order models. In particular, the random-order guarantee of SIMULTANEOUSLB improves as p increases. One of the main components is a new algorithm with improved regret for Online Linear Optimization (OLO) over the non-negative vectors in the lq ball. Interestingly, this OLO algorithm is also used to prove a purely probabilistic inequality that controls the correlations arising in the random-order model, a common source of difficulty for the analysis. Another important component used in both SIMULTANEOUSLB and our OLO algorithm is a smoothing of the lp-norm that may be of independent interest. This smoothness property allows us to see algorithm SIMULTANEOUSLB as essentially a greedy one in the worst-case model and as a primal-dual one in the random-order model, which is instrumental for its simultaneous guarantees.