Egalitarian pairwise kidney exchange: fast algorithms vialinear programming and parametric flow

We revisit the pairwise kidney exchange problem established by Roth Sonmez and Unver [23]. Our goal, explained in terms of graph theory, is to find a maximum fractional matching on an undirected graph, that Lorenz-dominates any other fractional matching. The Lorenz-dominant fractional matching, which can be implemented as a lottery of integral matchings, is in some sense the fairest allocation and also enjoys the property of being incentive compatible. The original algorithm by Roth et al. runs in time exponential in the size of input. In this paper, we target at designing practically efficient polynomial time algorithms for finding the Lorenz-dominant fractional matching. We start with a conceptually very simple algorithm, coined the water-filling algorithm. The water-filling algorithm is natural and allows us to present a simpler constructive proof that there exists a unique Lorenz-dominant fractional matching. The algorithm can be readily realized by a series of linear programs, each having an exponential number of constraints, and thus can be solved by the ellipsoid algorithm in polynomial time. However, it is known that the ellipsoid algorithm is not efficient in practice. To make the algorithm practical, we propose the second implementation based on a parametric flow computation on a carefully constructed flow network. Notably, the evolution of the parametric flow simulates exactly the water-filling algorithm. The second implementation only consists of a maximum matching computation and a parametric flow computation, both of which admit very efficient algorithms in practice.