[Paper Review] Greedy algorithm for stochastic matching is a 2-approximation

Marek Adamczyk|arXiv (Cornell University)|Jul 18, 2010

Optimization and Search Problems8 references3 citations

TL;DR

This paper proves that the greedy algorithm for stochastic matching—selecting edges with the highest success probability at each step—is a 2-approximation for maximizing expected matching size under patience constraints. The proof uses a novel probabilistic analysis and exchange argument to show that the greedy strategy's expected performance is at least half of the optimal solution’s, confirming a long-standing conjecture in the field of adaptive stochastic optimization.

ABSTRACT

Motivated by applications in online dating and kidney exchange, the stochastic matching problem was introduced by Chen, Immorlica, Karlin, Mahdian and Rudra (2009). They have proven a 4-approximation of a simple greedy strategy, but conjectured that it is in fact a 2-approximation. In this paper we confirm this hypothesis.

Motivation & Objective

• To resolve the conjecture that the greedy algorithm for stochastic matching achieves a 2-approximation ratio.
• To provide a tight analysis of the greedy strategy’s performance in the presence of vertex patience constraints.
• To establish a theoretical foundation for practical applications in online dating and kidney exchange programs.
• To extend prior 4-approximation bounds to a tight 2-approximation using probabilistic and structural arguments.
• To formalize and prove a key inequality involving conditional probabilities of vertex matching under different probing strategies.

Proposed method

• Uses a decision tree representation to model deterministic algorithms and analyze their expected outcomes.
• Applies an exchange argument comparing the greedy algorithm’s behavior to the optimal algorithm’s actions when a high-probability edge is not probed.
• Introduces a key inequality involving the ratio $\frac{1-p_{\alpha\beta}}{p_{\alpha\beta}}$ to bound the expected gain from vertex $\alpha$ when edge $\alpha\beta$ is skipped.
• Leverages the independence of the last probe of a vertex and the decision to probe a specific edge to decouple conditional probabilities.
• Employs a probabilistic comparison between the greedy and optimal algorithms’ success rates, conditioned on not probing a specific edge.
• Uses the fact that $\frac{1-x}{x}$ is decreasing to bound the ratio of failure-to-success probabilities across edges.

Experimental results

Research questions

• RQ1 Is the greedy algorithm for stochastic matching a 2-approximation, as conjectured by Chen et al.?
• RQ2 Can a tighter analysis than the existing 4-approximation be achieved for the greedy strategy in the unweighted stochastic matching problem?
• RQ3 What is the relationship between the expected matching size of the greedy algorithm and the optimal algorithm under patience constraints?
• RQ4 How does the conditional probability of vertex matching change when a high-probability edge is skipped?
• RQ5 Can an exchange argument be constructed to compare the performance of greedy and optimal strategies in the stochastic matching setting?

Key findings

• The greedy algorithm achieves a 2-approximation ratio for the stochastic matching problem under patience constraints.
• The paper confirms the conjecture by Chen et al. that the greedy strategy is a 2-approximation, improving upon the previously known 4-approximation.
• The analysis shows that $\mathbb{E}[GRD(G)] \geq \frac{1}{2} \cdot \mathbb{E}[OPT(G)]$, proving the 2-approximation bound.
• A critical inequality is proven: $\frac{1-p_{\alpha\beta}}{p_{\alpha\beta}} \cdot \mathbb{P}[\text{take } \alpha \text{ in } t_\alpha \mid \neg\text{probe } \alpha\beta] \leq \mathbb{P}[\text{OPT does not take } \alpha \mid \text{not probe } \alpha\beta]$, which enables the approximation proof.
• The proof relies on the independence of the last probe of a vertex and the probing of a specific edge, allowing conditional probability decomposition.
• The result holds for general graphs and does not require bipartite or structural assumptions, making it broadly applicable.