The Stackelberg Kidney Exchange Problem is $\Sigma_2^p$-complete

TL;DR

This paper introduces the Stackelberg kidney exchange problem, analyzing its computational complexity and showing it is $\, extstyle ext{Sigma}_2^p$-complete for exchanges involving three or more pairs, but polynomial for pairwise exchanges.

Contribution

It formally defines the Stackelberg kidney exchange problem and establishes its computational complexity, revealing a complexity jump from polynomial to $\, ext{Sigma}_2^p$-complete when increasing exchange size.

Findings

$\, ext{Sigma}_2^p$-completeness for $K \, ext{or more}$ pairs exchanges

Polynomial-time solvability for pairwise exchanges

Complexity classification of the Stackelberg kidney exchange problem

Abstract

We introduce the Stackelberg kidney exchange problem. In this problem, an agent (e.g. a hospital or a national organization) has control over a number of incompatible patient-donor pairs whose patients are in need of a transplant. The agent has the opportunity to join a collaborative effort which aims to increase the maximum total number of transplants that can be realized. However, the individual agent is only interested in maximizing the number of transplants within the set of patients under its control. Then, the question becomes which patients to submit to the collaborative effort. We show that, whenever we allow exchanges involving at most a fixed number $K \ge 3$ pairs, answering this question is $\Sigma_2^p$-complete. However, when we restrict ourselves to pairwise exchanges only, the problem becomes solvable in polynomial time