A Deterministic Polylogarithmic Competitive Algorithm for Matching with Delays

2025-10-23

AI Summary

This paper studies the online Minimum-Cost Delayed Perfect Matching (MPMD) problem under the challenging setting where the underlying metric space is unknown or infinite. To address this, we propose the first deterministic algorithm that requires no prior knowledge of the metric structure or the number of requests. Our approach integrates recursive metric partitioning, construction of a virtual matching tree, and hierarchical analysis of delay costs to enable online dynamic clustering and matching decisions. The algorithm achieves a competitive ratio of $O(log^5 m)$, substantially improving upon the previous best exponential bound of $O(m^{0.59})$. It is the first solution for arbitrary unknown metric spaces to attain a polynomial-logarithmic performance guarantee, and it significantly narrows the gap to the known lower bound of $Omega(log m / log log m)$. This work constitutes a key theoretical advance in tightening the fundamental limits of MPMD.

Technology Category

• Search and Optimization: Distributed Search
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In the online Min-cost Perfect Matching with Delays (MPMD) problem, $m$ requests in a metric space are submitted at different times by an adversary. The goal is to match all requests while (i) minimizing the sum of the distances between matched pairs as well as (ii) how long each request remained unmatched after it appeared. While there exist almost optimal algorithms when the metric space is finite and known a priori, this is not the case when the metric space is infinite or unknown. In this latter case, the best known algorithm, due to Azar and Jacob-Fanani, has competitiveness $mathcal{O}(m^{0.59})$ which is exponentially worse than the best known lower bound of $Ω(log m / log log m)$ by Ashlagi et al. We present a $mathcal{O}(log^5 m)$-competitive algorithm for the MPMD problem. This algorithm is deterministic and does not need to know the metric space or $m$ in advance. This is an exponential improvement over previous results and only a polylogarithmic factor away from the lower bound.

Problem

• Minimizing matching costs and delays in online metric space
• Addressing exponential competitiveness gap for unknown infinite metrics
• Providing deterministic polylogarithmic competitive algorithm without prior knowledge

Innovation

• Deterministic algorithm for online matching with delays
• Polylogarithmic competitive ratio without metric knowledge
• Exponential improvement over prior matching delay solutions