We consider the (offline) vertex-weighted Online Matching problem under Known Identical and Independent Distributions (KIID) with integral arrival rates. We propose a meta-algorithm, denoted as $\mathsf{RTB}$, featuring Real-Time Boosting, where the core idea is as follows. Consider a bipartite graph $G=(I,J,E)$, where $I$ and $J$ represent the sets of offline and online nodes, respectively. Let $\mathbf{x}=(x_{ij}) \in [0,1]^{|E|}$, where $x_{ij}$ for $(i,j) \in E$ represents the probability that edge $(i,j)$ is matched in an offline optimal policy (a.k.a. a clairvoyant optimal policy), typically obtained by solving a benchmark linear program (LP). Upon the arrival of an online node $j$ at some time $t \in [0,1]$, $\mathsf{RTB}$ samples a safe (available) neighbor $i \in I_{j,t}$ with probability $x_{ij}/\sum_{i' \in I_{j,t}} x_{i'j}$ and matches it to $j$, where $I_{j,t}$ denotes the set of safe offline neighbors of $j$. In this paper, we showcase the power of Real-Time Boosting by demonstrating that $\mathsf{RTB}$, when fed with $\mathbf{X}^*$, achieves a competitive ratio of $(2e^4 - 8e^2 + 21e - 27) / (2e^4) \approx 0.7341$, where $\mathbf{X}^* \in \{0,1/3,2/3\}^{|E|}$ is a random vector obtained by applying a customized dependent rounding technique due to Brubach et al. (Algorithmica, 2020). Our result improves upon the state-of-the-art ratios of 0.7299 by Brubach et al. (Algorithmica, 2020) and 0.725 by Jaillet and Lu (Mathematics of Operations Research, 2013). Notably, this improvement does not stem from the algorithm itself but from a new competitive analysis methodology: We introduce an Ordinary Differential Equation (ODE) system-based approach that enables a {holistic} analysis of $\mathsf{RTB}$. We anticipate that utilizing other well-structured vectors from more advanced rounding techniques could potentially yield further improvements in the competitiveness.

翻译：我们研究具有整数到达率的已知独立同分布(KIID)模型下的（离线）顶点加权在线匹配问题。我们提出一种元算法，记作$\mathsf{RTB}$，其核心特征是实时增强机制，基本思想如下。考虑二分图$G=(I,J,E)$，其中$I$和$J$分别表示离线节点与在线节点集合。令$\mathbf{x}=(x_{ij}) \in [0,1]^{|E|}$，其中对于边$(i,j) \in E$，$x_{ij}$表示该边在离线最优策略（即全知最优策略）中被匹配的概率，该概率通常通过求解基准线性规划(LP)获得。当在线节点$j$在时刻$t \in [0,1]$到达时，$\mathsf{RTB}$以概率$x_{ij}/\sum_{i' \in I_{j,t}} x_{i'j}$从安全（可用）邻居集合$I_{j,t}$中采样一个离线邻居$i$并将其与$j$匹配，此处$I_{j,t}$表示$j$在时刻$t$的安全离线邻居集合。本文通过证明$\mathsf{RTB}$算法在输入$\mathbf{X}^*$时能达到$(2e^4 - 8e^2 + 21e - 27) / (2e^4) \approx 0.7341$的竞争比，展示了实时增强机制的有效性。其中$\mathbf{X}^* \in \{0,1/3,2/3\}^{|E|}$是通过应用Brubach等人（Algorithmica, 2020）提出的定制化依赖舍入技术获得的随机向量。我们的结果改进了当前最佳竞争比：Brubach等人（Algorithmica, 2020）的0.7299以及Jaillet与Lu（Mathematics of Operations Research, 2013）的0.725。值得注意的是，这一改进并非源于算法本身，而是源于一种新的竞争分析方法：我们提出了基于常微分方程(ODE)系统的分析方法，实现了对$\mathsf{RTB}$算法的整体性分析。我们预期，若采用更先进舍入技术产生的其他结构良好向量，可能进一步提升算法的竞争性能。