Improved and Parameterized Algorithms for Online Multi-level Aggregation: A Memory-based Approach

2025-11-28

AI Summary

This paper studies the online Multi-Level Aggregation problem with Deadlines (MLAP-D): requests arrive dynamically at nodes of a weighted tree and must be served by transmission subtrees rooted at the root before their deadlines; service cost equals the sum of edge weights along the transmission paths. To overcome the limitations of conventional competitive analysis parameterized by tree depth $D$, we introduce “caterpillar dimension” $H$—a novel structural parameter capturing tree complexity—and propose a memory-based online algorithmic framework that requires no topological preprocessing. Our algorithm achieves competitive ratios of $e(D+1)$ and $e(4H+2)$, improving upon the prior best bounds of $6(D+1)$ and $O(log|V|)$. Notably, when $H ll D, log|V|$—as in path graphs, caterpillar graphs, and lobster graphs—the performance gain is substantial.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Heuristic Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the online multi-level aggregation problem with deadlines (MLAP-D) introduced by Bienkowski et al. (ESA 2016, OR 2020). In this problem, requests arrive over time at the vertices of a given vertex-weighted tree, and each request has a deadline that it must be served by. The cost of serving a request equals the cost of a path from the root to the vertex where the request resides. Instead of serving each request individually, requests can be aggregated and served by transmitting a subtree from the root that spans the vertices on which the requests reside, to potentially be more cost-effective. The aggregated cost is the weight of the transmission subtree. The goal of MLAP-D is to find an aggregation solution that minimizes the total cost while serving all requests. We present improved and parameterized algorithms for MLAP-D. Our result is twofold. First, we present an $e(D+1)$-competitive algorithm where $D$ is the depth of the tree. Second, we present an $e(4H+2)$-competitive algorithm where $H$ is the caterpillar dimension of the tree. Here, $H le D$ and $H le log_2 |V|$ where $|V|$ is the number of vertices in the given tree. The caterpillar dimension remains constant for rich but simple classes of trees, such as line graphs ($H=1$), caterpillar graphs ($H=2$), and lobster graphs ($H=3$). To the best of our knowledge, this is the first online algorithm parameterized on a measure better than depth. The state-of-the-art online algorithms are $6(D+1)$-competitive by Buchbinder, Feldman, Naor, and Talmon (SODA 2017) and $O(log |V|)$-competitive by Azar and Touitou (FOCS 2020). Our framework outperforms the state-of-the-art ratios when $H = o(min{D,log_2 |V|})$. Our simple framework directly applies to trees with any structure and differs from the previous frameworks that reduce the problem to trees with specific structures.

Problem

• Minimizing total cost for serving requests with deadlines on weighted trees
• Developing competitive algorithms parameterized by tree depth and caterpillar dimension
• Improving state-of-the-art competitive ratios for online multi-level aggregation

Innovation

• Memory-based approach for online multi-level aggregation
• Parameterized algorithm using caterpillar dimension
• Framework applicable to trees with any structure