Online Matching under KIID: Enhanced Competitive Analysis through Ordinary Differential Equation Systems

TL;DR

This work tackles vertex-weighted online matching under Known Identical and Independent Distributions with integral arrival rates. It introduces a Real-Time Boosting (RTB) meta-algorithm that, when paired with a structured randomized rounding vector, provably attains a competitive ratio of $\frac{2 e^4 - 8 e^2 + 21 e - 27}{2 e^4} \approx 0.7341$, surpassing prior results. A novel Ordinary Differential Equation (ODE) system-based analysis provides a holistic view of how real-time boosting interacts across offline neighbors, enabling identification of Worst-Scenario structures and enabling precise performance guarantees. The results underscore that gains stem from a principled analysis framework rather than algorithmic tweaks and point to potential improvements with more advanced rounding. The approach has potential implications for broader online matching variants and practical deployment in dynamic markets with integral arrival rates.

Abstract

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.

Figures (15)

• Figure 1: Two plots highlighting the positive correlation between offline nodes ($i_1$ and $i_2$) staying safe (or getting matched) at any time due to the real-time boosting in $\mathsf{RTB}\xspace$. In Figure \ref{['fig:20/a/l']}, we have a complete bipartite graph where every edge has a value of $\epsilon := (1 - \mathsf{e}^{-K})/K$, forming a feasible (and also optimal) solution to the natural LP in huang2021online. In Figure \ref{['fig:20/a/r']}, we see a cycle where small edges of mass $1/3$ and big edges of mass $2/3$ are marked in black and red, respectively.
• Figure 2: An example highlighting (i) the need for modifications to sampling distributions of online nodes and (ii) the difference between our approach and those in bib:Jailletbrubach2020online. Throughout this paper, we assume the following unless specified otherwise: (i) Big edges of mass $2/3$ and small edges of mass $1/3$ are marked in red and black, respectively; (ii) The value next to each offline node represents its total mass in the well-structured vector $\mathbf{x} \in \{0,1/3,2/3\}^{|E|}$. Figure \ref{['fig:29/a']} shows that even under the current holistic competitive analysis, the Matching Probability per Mass (MPM) achieved by node $\mathbf{i}$ in $\mathsf{RTB}\xspace$ equals $1 - 22/(9 \mathsf{e}^2) \approx 0.6692 < \kappa^*_B$, the target MPM for an offline node of mass one in the form of 1B1S (see Appendix \ref{['app:mot/one']}). Figure \ref{['fig:29/b']} proposes modified sampling distributions for nodes $j_b$ and $j_s$, where the input vector for $j_b$ is updated from $(1/3, 2/3)$ to $(z_1, 1 - z_1)$ with $z_1 \in [0,1]$. Similarly, the input vector for $j_s$ is updated from $(1/3, 1/3, 1/3)$ to $(1 - 2z_2, z_2, z_2)$, where $z_2 \in [0,1/2]$. In Section \ref{['sec:a+c']}, we establish that an aggressive setting of $z_1 = z_2 = 0$ in Figure \ref{['fig:29/b']} suffices to ensure every offline node achieves an MPM greater than the target specified in Proposition \ref{['pro:main-1']}. Specifically, we show that under the aggressive setting, node $\mathbf{i}$ achieves an MPM equal to $1 - \mathsf{e}^{-2} > \kappa^*_B$, and nodes ${\bar{i}}$ and ${\tilde{i}}$ each achieve an MPM of $3(1 - 2/\mathsf{e}) \approx 0.7927 > \kappa^*_s$. This contrasts with the fact that in the same aggressive setting, nodes ${\bar{i}}$ and ${\tilde{i}}$ each achieve an MPM equal to $3\left(1 - 9/(4\mathsf{e})\right) \approx 0.5168 < \kappa^*_s$ following the approach in bib:Jailletbrubach2020online.
• Figure 3: The target offline node $i_0$ is in the form of 1B1S (one big and one small edge, marked in red and black, respectively), where the small online neighbor $j_s$ has another big edge $(i_2, j_s)$. The Folding Procedure merges the two offline neighbors of $i_0$, i.e.,$i_1$ and $i_2$, into one ($i_3$), resulting in the structure shown on the right.
• Figure 4: Three possible WS structures for an offline node of mass one after the rounding, where big edges of mass $2/3$ and small edges of mass $1/3$ are marked in red and black, respectively.
• Figure 5: An example showing that for any target offline node, its WS structure must be instantiated when any of its offline neighbors shares a single online neighbor with it; otherwise, we can decompose it into another strictly worse structure for it. In the example above, the target offline node is $\mathbf{i}_1$, and it shares two online neighbors, $j_1$ and $j_2$, with its offline neighbor $i_2$. Note that all edges have a mass of $1/3$. The decomposition consists of (1) pruning the edge $(i_2, j_1)$ (marked in blue) and (2) adding another offline neighbor ${\tilde{i}}_2$ of $\mathbf{i}_1$ and edge $({\tilde{i}}_2, j_1)$ to compensate for the role played by the pruned edge $(i_2,j_1)$. We can verify that at any time $t \in [0,1]$: (1) Given $i_1$ is safe at $t$, the probability of ${\tilde{i}}_2$ staying safe at $t$ and that of $\hat{i}_2$ are both larger than that of $i_2$; (2) The real-time boosting effect for matching $\mathbf{i}_1$ from the unavailability of $i_2$ is equivalent to that from the unavailability of both ${\tilde{i}}_2$ and $\hat{i}_2$. Thus, we conclude that $\mathbf{i}_1$ has a strictly worse structure in terms of a larger probability of staying safe after decomposition.

Theorems & Definitions (16)

• Lemma 1: Appendix \ref{['app:well-bt']}
• Theorem 1
• proposition 1: brubach2020online
• proposition 2
• Example 1
• Lemma 2: Appendix \ref{['app:vir']}
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6