KidneyExchange.jl: a Julia package for solving the kidney exchange problem with branch-and-price

Abstract

The kidney exchange problem (KEP) is an increasingly important healthcare management problem in most European and North American countries which consists of matching incompatible patient-donor pairs in a centralized system. Despite the significant progress in the exact solution of KEP instances in recent years, larger instances still pose a challenge especially when non-directed donors are taken into account. In this article, we present a branch-and-price algorithm for the exact solution of KEP in the presence of non-directed donors. This algorithm is based on a disaggregated cycle and chains formulation where subproblems are managed through graph copies. We additionally present a branch-and-price algorithm based on the position-indexed chain-edge formulation as well as two compact formulations. We formalize and analyze the complexity of the resulting pricing problems and identify the conditions under which they can be solved using polynomial-time algorithms. We propose several algorithmic improvements for the branch-and-price algorithms as well as for the solution of pricing problems. We extensively test all of our implementations using a benchmark made up of different types of instances. Our numerical results show that the proposed algorithm can be significantly faster compared to the state-of-the-art. All models and algorithms presented in the paper are gathered in an open-access Julia package, KidneyExchange.jl.

Appendix

1.1 6.1 Position-indexed chain-edge formulation (PICEF)

$$\begin{aligned} \max \;\;&\sum _{s\in S}\sum _{c\in \mathcal {C}_K^s} w_{c} z_{c}+ \sum _{(u,v) \in A}\sum _{l=1}^L w_{u,v} y_{u,v,l} \end{aligned}$$

(32)

$$\begin{aligned} \text{ s.t. }\;\;&\sum _{s\in S}\sum _{c\in \mathcal {C}_K^s(v)} z_c+\sum _{u\mid (u,v)\in A}\sum _{l=1}^{L} y_{u,v,l}\le 1&\forall v\in V \end{aligned}$$

(33)

$$\begin{aligned}&\sum _{v\mid (v,u)\in A} y_{v,u,l}\ge \sum _{v\mid (u,v)\in A} y_{u,v,(l+1)}&\forall u \in P, l=1,\ldots ,L-1 \end{aligned}$$

(34)

$$\begin{aligned}&\sum _{v\mid (u,v)\in A}y_{u,v,1}=0&\forall u\in P\end{aligned}$$

(35)

$$\begin{aligned}&\sum _{v\mid (u,v)\in A} y_{u,v,l}= 0&\forall u\in D, l=2,\ldots ,L \end{aligned}$$

(36)

$$\begin{aligned}&z_c\in \{0,1\}&\forall s\in S, c\in \mathcal {C}_K^s \end{aligned}$$

(37)

$$\begin{aligned}&y_{u,v,l}\in \{0,1\}&\forall (u,v)\in A, l=1,\ldots ,L. \end{aligned}$$

(38)

Here, constraints (33) impose that each vertex is involved in at most one chain or cycle, (34) are flow balance constraints and ensure that a chain is formed by the chain-flow (y) variables, (35)–(36) impose that only non-directed donors can initiate a chain and therefore appear in position $l=1$ and only in position $l=1$.

1.2 6.2 Detailed numerical results

See Table 4 here.

Table 4 Average solution time (for instances solved to optimality) using each compact MIP formulation