Tight Bounds for Online Balanced Partitioning in the Generalized Learning Model

TL;DR

This paper establishes tight bounds for online balanced partitioning in a generalized learning model, providing two deterministic algorithms with competitive ratios close to optimal and demonstrating their theoretical limits.

Contribution

It introduces two new deterministic online algorithms with tight competitive bounds for the generalized learning model in balanced partitioning.

Findings

First algorithm achieves competitive ratio $O(\max(\sqrt{k\ell \log k}, \ell \log k))$ with $1+\epsilon$ augmentation.

Second algorithm achieves competitive ratio $O(\sqrt{k})$ with $2+\epsilon$ augmentation.

Lower bounds prove the optimality of these competitive ratios.

Abstract

Resource allocation in distributed and networked systems such as the Cloud is becoming increasingly flexible, allowing these systems to dynamically adjust toward the workloads they serve, in a demand-aware manner. Online balanced partitioning is a fundamental optimization problem underlying such self-adjusting systems. We are given a set of $\ell$ servers. On each server we can schedule up to $k$ processes simultaneously. The demand is described as a sequence of requests $\sigma_t=\{p_i, p_{j}\}$, which means that the two processes $p_i,p_{j}$ communicate. A process can be migrated from one server to another which costs 1 unit per process move. If the communicating processes are on different servers, it further incurs a communication cost of 1 unit for this request. The objective is to minimize the competitive ratio: the cost of serving such a request sequence compared to the cost incurred by an optimal offline algorithm. Henzinger et al. (at SIGMETRICS 2019) introduced a learning variant of this problem where the cost of an online algorithm is compared to the cost of a static offline algorithm that does not perform any communication, but which simply learns the communication graph and keeps the discovered connected components together. This problem variant was recently also studied at SODA 2021. In this paper, we consider a more general learning model (i.e., stronger adversary), where the offline algorithm is not restricted to keep connected components together. Our main contribution are tight bounds for this problem. In particular, we present two deterministic online algorithms: (1) an online algorithm with competitive ratio $O(\max(\sqrt{k\ell \log k}, \ell \log k))$ and augmentation $1+\epsilon$; (2) an online algorithm with competitive ratio $O(\sqrt{k})$ and augmentation $2+\epsilon$. We further present lower bounds showing optimality of these bounds.