Small Representations of Big Kidney Exchange Graphs
This addresses scalability issues in kidney exchange matching for patients and donors, offering a potential solution to improve efficiency in large-scale exchanges, though it is incremental as it builds on existing graph encoding methods.
The paper tackles the NP-complete problem of optimally matching patients to donors in large kidney exchanges by showing that if compatibility graphs can be encoded with a constant number of attributes, the clearing problem becomes solvable in polynomial time, and experiments with real UNOS data confirm that small numbers of attributes suffice.
Kidney exchanges are organized markets where patients swap willing but incompatible donors. In the last decade, kidney exchanges grew from small and regional to large and national---and soon, international. This growth results in more lives saved, but exacerbates the empirical hardness of the $\mathcal{NP}$-complete problem of optimally matching patients to donors. State-of-the-art matching engines use integer programming techniques to clear fielded kidney exchanges, but these methods must be tailored to specific models and objective functions, and may fail to scale to larger exchanges. In this paper, we observe that if the kidney exchange compatibility graph can be encoded by a constant number of patient and donor attributes, the clearing problem is solvable in polynomial time. We give necessary and sufficient conditions for losslessly shrinking the representation of an arbitrary compatibility graph. Then, using real compatibility graphs from the UNOS nationwide kidney exchange, we show how many attributes are needed to encode real compatibility graphs. The experiments show that, indeed, small numbers of attributes suffice.