Randomized Parameterized Algorithms for the Kidney Exchange Problem

Qilong Feng,Mugang Lin,Jianxin Wang,Bin Fu

doi:10.3390/a12020050

Abstract

In order to increase the potential kidney transplants between patients and their incompatible donors, kidney exchange programs have been created in many countries. In the programs, designing algorithms for the kidney exchange problem plays a critical role. The graph theory model of the kidney exchange problem is to find a maximum weight packing of vertex-disjoint cycles and chains for a given weighted digraph. In general, the length of cycles is not more than a given constant L (typically 2 ≤ L ≤ 5), and the objective function corresponds to maximizing the number of possible kidney transplants. In this paper, we study the parameterized complexity and randomized algorithms for the kidney exchange problem without chains from theory. We construct two different parameterized models of the kidney exchange problem for two cases L = 3 and L ≥ 3 , and propose two randomized parameterized algorithms based on the random partitioning technique and the randomized algebraic technique, respectively.

Highlights

Kidney transplantation is one of the best treatment methods for those who are suffering from end-stage renal failure
Applying the randomized algebraic technique, we reduce the Kidney Exchange problem (KEP) to the multilinear monomial detection problem, and obtain an O∗ (2(k+ L) (k + L)2 L3 n L ) algorithm based on the results of Koutis and Williams [27,28,29]
We develop a randomized algorithm for the Parameterized KEP (p-KEP) based on the algebraic technique, introduced by Koutis in [27]

Summary

Introduction

Kidney transplantation is one of the best treatment methods for those who are suffering from end-stage renal failure. The KEP becomes more complex, and can be modelled as finding an optimal set of vertex-disjoint cycles and chains from the directed graph of given instance such that the length of each cycle is at most L [5]. Xiao and Wang [8] studied the variant of the KEP which contains chain exchanges and allows some compatible patient-donor pairs to participate in exchanges to find better matched kidneys. They designed two O(2n n3 ) exact algorithms based on dynamic programming and subset convolution for the KEP with and without length constraints, respectively.

Preliminaries and Problem Definitions

Parameterized Complexity

KEP Definitions

A Randomized Algorithm Based on Random Partitioning Technique

A Randomized Algorithm Based on Algebraic Technique

Conclusions