Multi-Level Aggregation via Dual Fitting: An $O(D)$-Competitive Algorithm

2026-08-04

AI Summary

This work addresses online optimization for the Multi-Level Aggregation Problem (MLAP) with arbitrary delay functions, aiming to narrow the gap in competitive ratios between this general setting and the special case with only deadlines. The authors propose a novel online algorithm based on dual fitting, featuring an innovative retrospective dual construction and a time-dependent dual packing mechanism that effectively overcomes infeasibility issues inherent in traditional primal-dual approaches. The algorithm achieves a competitive ratio of 2D on trees of depth D, significantly improving upon the previous O(D²) bound. Moreover, in the deadline-only setting, it attains a competitive ratio of D, which is asymptotically optimal in both cases.

Abstract

We present a new online algorithm for the well-known Multi-Level Aggregation Problem (MLAP) with arbitrary delay functions, achieving a $2D$-competitive ratio, where $D$ is the depth of the underlying tree. This result improves the current best-known competitive ratio of $O(D^2)$ and asymptotically matches the $D$-competitive bound previously known only for the deadline variant, thereby closing the asymptotic gap between the two settings. Our key technical contribution is a novel dual fitting framework that provides a unified analysis for both settings; in particular, it also establishes a $D$-competitive ratio for MLAP with deadlines. Our analysis is built upon two new ideas: a hindsight dual construction, which resolves the infeasibility issues in traditional online primal-dual methods, and a time-dependent dual packing that maintains feasibility over dynamic request sets.

Problem

• Multi-Level Aggregation Problem
• online algorithm
• competitive ratio
• delay functions
• deadline variant

Innovation

• Multi-Level Aggregation Problem
• online algorithm
• dual fitting
• competitive ratio
• primal-dual method