Fast and Efficient Matching Algorithm with Deadline Instances

TL;DR

This paper studies the online weighted bipartite matching problem with deadlines and uses a sketching matrix to approximate edge weights, reducing edge-weight computation from $O(nd)$ to $ ilde{O}(rac{1}{ ext{eps}^2}(n+d))$. It introduces FastGreedy and FastPostponedGreedy, achieving competitive ratios $(1- ext{eps})/2$ and $(1- ext{eps})/4$, with space $O(nd + ext{eps}^{-2}(n+d) ext{log}(n/ ext{delta}))$ and per-operation time $O( ext{eps}^{-2}(n+d) ext{log}(n/ ext{delta}))$, by applying a Johnson-Lindenstrauss sketch to distance-based edge weights. Empirical results on four real-world datasets show 10–20x speedups while maintaining total matching values close to the original algorithms, validating practical viability for large-scale, high-dimensional data. The framework enables efficient deadline-aware matching in large-scale systems and suggests extending sketching techniques to other variants of online matching and related optimization problems.

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 $ε\in (0,0.1)$ 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 $\frac{1 - ε}{2}$ and $\frac{1 - ε}{4}$ 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 $\widetilde{O}(ε^{-2} \cdot (n + d))$, where for any function $f$, we use $\widetilde{O}(f)$ to denote $f \cdot \mathrm{poly}(\log f)$.

Figures (23)

• Figure 1: The relationship between running time and parameter $s$ and $dl$ on GECRS data set. The parameters are defined as follows: $n$ is the node count, $d$ is the original node dimension, $s$ is the dimension after transformation, and $\mathrm{dl}$ is the maximum matching time per node (referred to as the deadline). Here GECRS denotes gene expression cancer RNA-Seq Data Set. PGreedy denotes Postponed Greedy.
• Figure 2: The relationship between running time and parameter $s$ and $dl$ on Arcene data set. The parameters are defined as follows: $n$ is the node count, $d$ is the original node dimension, $s$ is the dimension after transformation, and $\mathrm{dl}$ is the maximum matching time per node (referred to as the deadline).
• Figure 3: The relationship between running time and parameter $s$ and $dl$ on ARBT data set. The parameters are defined as follows: $n$ is the node count, $d$ is the original node dimension, $s$ is the dimension after transformation, and $\mathrm{dl}$ is the maximum matching time per node (referred to as the deadline). Let ARBT denote a study of Asian Religious and Biblical Texts Data Set.
• Figure 4: The relationship between running time and parameter $s$ and $dl$ on REJAFADA data set. The parameters are defined as follows: $n$ is the node count, $d$ is the original node dimension, $s$ is the dimension after transformation, and $\mathrm{dl}$ is the maximum matching time per node (referred to as the deadline).
• Figure 5: Comparison between the running time of each two algorithms.

Theorems & Definitions (47)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5: JL Lemma, jl84
• Lemma 3.1: Restatement of Lemma \ref{['lem:standard_greedy_correctness_formal']}
• Lemma 3.2: Restatement of Lemma \ref{['lem:calculate_the_competive_ratio_formal']}
• Lemma 3.3: Restatement of Lemma \ref{['lem:standard_postponed_greedy_correctness_formal']}
• Theorem 3.4
• proof