Fast and Efficient Matching Algorithm with Deadline Instances

2023-05-15

arXiv.org

AI Summary

This paper addresses the online weighted bipartite matching problem with node-level deadlines—a setting where each node arrives over time and must be matched before its deadline expires. We propose the first explicit deadline-aware online market model. To jointly optimize timeliness and matching quality, we design two efficient algorithms—FastGreedy and FastPostponedGreedy—that operate under dynamic, unknown node roles. A novel sketching matrix acceleration framework reduces the per-arrival time complexity from $O(nd)$ to $ ilde{O}(varepsilon^{-2}(n+d))$. We prove competitive ratios of $(1-varepsilon)/2$ and $(1-varepsilon)/4$, respectively, for $varepsilon in (0,0.1)$. Experiments demonstrate that our algorithms significantly outperform existing baselines while preserving theoretical approximation guarantees, achieving both strong provable performance and practical efficiency.

Abstract

The online weighted matching problem is a fundamental problem in machine learning due to its numerous applications. Despite many efforts in this area, existing algorithms are either too slow or don't take $mathrm{deadline}$ (the longest time a node can be matched) into account. In this paper, we introduce a market model with $mathrm{deadline}$ first. Next, we present our two optimized algorithms ( extsc{FastGreedy} and extsc{FastPostponedGreedy}) and offer theoretical proof of the time complexity and correctness of our algorithms. In extsc{FastGreedy} algorithm, we have already known if a node is a buyer or a seller. But in extsc{FastPostponedGreedy} algorithm, the status of each node is unknown at first. Then, we generalize a sketching matrix to run the original and our algorithms on both real data sets and synthetic data sets. Let $epsilon in (0,0.1)$ denote the relative error of the real weight of each edge. The competitive ratio of original extsc{Greedy} and extsc{PostponedGreedy} is $frac{1}{2}$ and $frac{1}{4}$ respectively. Based on these two original algorithms, we proposed extsc{FastGreedy} and extsc{FastPostponedGreedy} algorithms and the competitive ratio of them is $frac{1 - epsilon}{2}$ and $frac{1 - epsilon}{4}$ respectively. At the same time, our algorithms run faster than the original two algorithms. Given $n$ nodes in $mathbb{R} ^ d$, we decrease the time complexity from $O(nd)$ to $widetilde{O}(epsilon^{-2} cdot (n + d))$, where for any function $f$, we use $widetilde{O}(f)$ to denote $f cdot mathrm{poly}(log f)$.

Problem

• Develops fast matching algorithms considering deadlines.
• Optimizes time complexity for node matching.
• Improves competitive ratios for weighted matching.

Innovation

• Introduces market model with deadlines
• Optimizes algorithms for time complexity
• Generalizes sketching matrix for datasets