Fast and Efficient Matching Algorithm with Deadline Instances

• URL: http://arxiv.org/abs/2305.08353v2
• Date: Mon, 12 Feb 2024 21:17:03 GMT
• Title: Fast and Efficient Matching Algorithm with Deadline Instances
• Authors: Zhao Song, Weixin Wang, Chenbo Yin, Junze Yin
• Abstract summary: We introduce a market model with $mathrmdeadline$ first. We present our two optimized algorithms (textscFastGreedy and textscFastPostponedGreedy) At the same time, our algorithms run faster than the original two algorithms.
• Score: 7.613259578185218
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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 (\textsc{FastGreedy} and \textsc{FastPostponedGreedy}) and offer theoretical proof of the time complexity and correctness of our algorithms. In \textsc{FastGreedy} algorithm, we have already known if a node is a buyer or a seller. But in \textsc{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 \textsc{Greedy} and \textsc{PostponedGreedy} is $\frac{1}{2}$ and $\frac{1}{4}$ respectively. Based on these two original algorithms, we proposed \textsc{FastGreedy} and \textsc{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 $\widetilde{O}(\epsilon^{-2} \cdot (n + d))$.

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