Near-Optimal Dynamic Matching via Coarsening with Application to Heart Transplantation

Online matching has been a mainstay in domains such as Internet advertising and organ allocation, but practical algorithms often lack strong theoretical guarantees. We take an important step toward addressing this by developing new online matching algorithms based on a coarsening approach. Although coarsening typically implies a loss of granularity, we show that, to the contrary, aggregating offline nodes into capacitated clusters can yield near-optimal theoretical guarantees. We apply our methodology to heart transplant allocation to develop theoretically grounded policies based on structural properties of historical data. Furthermore, in simulations based on real data, our policy closely matches the performance of the omniscient benchmark, achieving competitive ratio 0.91, drastically higher than the US status quo policy’s 0.51. Our work bridges the gap between data-driven heuristics and pessimistic theoretical lower bounds.

💡 Research Summary

The paper tackles the online matching problem, a fundamental model for dynamic resource allocation where a set of offline resources (e.g., patients awaiting organ transplants) must be matched irrevocably to a stream of online arrivals (e.g., donor organs). While online matching has been extensively studied in fields such as Internet advertising, practical implementations—especially in high‑stakes domains like organ allocation—often rely on heuristics that lack rigorous performance guarantees. Conversely, the theoretical literature typically assumes worst‑case or overly generic stochastic models, producing pessimistic competitive ratios that are far from what is achievable with real‑world data. This work bridges the gap by introducing a “coarsening” methodology that aggregates offline nodes into capacitated clusters, thereby enabling the use of b‑matching techniques with provable near‑optimal guarantees.

Key Contributions

1. Coarsening Framework – The authors propose to partition the offline node set into clusters of size b, each treated as a single node with capacity b. Within a cluster, individual nodes are heterogeneous but assumed to have edge weights that differ from a representative weight (\bar w_{u,v}) by at most a relative error (\delta(b)). This “bounded cluster variance” assumption captures the empirical observation that many patients share similar expected life‑year gains.
2. Theoretical Guarantees – Building on the stochastic b‑matching algorithm of Brubach et al. (2016), the paper proves that if the representative weights are used in the linear program (LP) and the intra‑cluster error is bounded, the resulting online algorithm achieves a competitive ratio of at least (\alpha(b)(1-2\delta(b))), where (\alpha(b)) is the classic b‑matching bound that approaches 1 as b grows. The authors further extend the analysis to allow cluster‑specific errors and to accommodate estimation error (\eta) between observed and true edge weights.
3. Algorithmic Design (Algorithm 2) – The offline phase enumerates a set of candidate cluster sizes (B), constructs a partition for each size that minimizes (\delta(b)), computes the lower bound (\alpha(b)(1-2\delta(b))), and selects the configuration with the highest bound. The LP is then solved on the representative graph, yielding fractional matching probabilities (f_{u,v}). In the online phase, each arriving donor is matched to a free node inside each cluster with probability (f_{u,v} r_v), where (r_v) is the arrival rate. The algorithm is non‑adaptive but provably near‑optimal under the stated assumptions.
4. Empirical Validation on UNOS Data – Using the United Network for Organ Sharing (UNOS) registry for heart transplants (January–June 2019), the authors identify large patient clusters with low intra‑cluster variance. By optimizing the cluster size (approximately b = 0.8 of the total patient pool), the policy attains a competitive ratio of 0.91 in simulation, dramatically outperforming the status‑quo US allocation policy (0.51) and a non‑capacitated stochastic matching baseline (0.63). Sensitivity analysis shows the method remains robust when the weight estimation error (\eta) is as high as 0.1, with competitive ratios staying above 0.85.
5. Broader Impact – The work demonstrates that data‑driven clustering, traditionally used for statistical robustness, can also be leveraged for algorithmic performance guarantees. By providing a provable bridge between historical medical data and online decision making, the paper offers a template for future policy design in organ allocation, kidney exchange, and other life‑critical matching markets.

Technical Highlights

• The competitive ratio bound (\alpha(b) = 1 - b^{-1/2} + \epsilon - e^{-b^2\epsilon/3}) (from Brubach et al.) is shown to improve with larger b, but larger clusters increase (\delta(b)). The optimal trade‑off is captured by maximizing (\alpha(b)(1-2\delta(b))).
• The analysis accommodates stochastic edge failures (p_{u,v}) and shows that the algorithm’s performance degrades gracefully with failure probability.
• For heterogeneous clusters, the authors derive a weighted average error term, proving that high errors are tolerable in low‑value or rarely selected clusters (Theorem 7, 9).
• The LP formulation remains identical to the classic b‑matching LP, ensuring computational tractability; the only modification is the use of representative weights.

Conclusion
The paper introduces a novel coarsening‑based online matching algorithm that achieves near‑optimal competitive ratios in a realistic organ allocation setting. By rigorously quantifying the trade‑off between cluster size and intra‑cluster heterogeneity, and by validating the approach on extensive real‑world data, the authors provide both theoretical insight and practical guidance for designing allocation policies that are both data‑driven and provably effective. This work is likely to influence future research on dynamic matching in healthcare and other domains where high‑value, time‑critical decisions are made under uncertainty.