Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The NTU Partitioned Matching Game for International Kidney Exchange Programs

Gergely Csáji, Tamás Király, Zsuzsa Mészáros-Karkus

TL;DR

The core of this game suitably captures the notion of stability of an IKEP, as it precludes incentives to deviate from the proposed solution for any possible coalition of the players, and it is proved computational complexity results about the weak and strong cores under various assumptions on the players are proved.

Abstract

Motivated by the real-world problem of international kidney exchange (IKEP), [Biró et al., Generalized Matching Games for International Kidney Exchange, 2019] introduced a generalized transferable utility matching game featuring a partition of the vertex set of a graph into players, and analyzed its complexity. We explore the non-transferable utility (NTU) variant of the game, where the utility of players is given by the number of their matched vertices. The NTU version is arguably a more natural model of the international kidney exchange program, as the utility of a participating country mostly depends on how many of its patients receive a kidney, which is non-transferable by nature. We study the core of this game, which suitably captures the notion of stability of an IKEP, as it precludes incentives to deviate from the proposed solution for any possible coalition of the players. We prove computational complexity results about the weak and strong cores under various assumptions on the players. In particular, we show that if every player has two vertices, which can be considered as an NTU matching game with couples, then the weak core is always non-empty, and the existence of a strong core solution can be decided in polynomial time. In contrast, it is NP-hard to decide whether the strong core is empty when each player has three vertices. We also show that if the number of players is constant, then the non-emptiness of the weak and strong cores is polynomial-time decidable.

The NTU Partitioned Matching Game for International Kidney Exchange Programs

TL;DR

The core of this game suitably captures the notion of stability of an IKEP, as it precludes incentives to deviate from the proposed solution for any possible coalition of the players, and it is proved computational complexity results about the weak and strong cores under various assumptions on the players are proved.

Abstract

Motivated by the real-world problem of international kidney exchange (IKEP), [Biró et al., Generalized Matching Games for International Kidney Exchange, 2019] introduced a generalized transferable utility matching game featuring a partition of the vertex set of a graph into players, and analyzed its complexity. We explore the non-transferable utility (NTU) variant of the game, where the utility of players is given by the number of their matched vertices. The NTU version is arguably a more natural model of the international kidney exchange program, as the utility of a participating country mostly depends on how many of its patients receive a kidney, which is non-transferable by nature. We study the core of this game, which suitably captures the notion of stability of an IKEP, as it precludes incentives to deviate from the proposed solution for any possible coalition of the players. We prove computational complexity results about the weak and strong cores under various assumptions on the players. In particular, we show that if every player has two vertices, which can be considered as an NTU matching game with couples, then the weak core is always non-empty, and the existence of a strong core solution can be decided in polynomial time. In contrast, it is NP-hard to decide whether the strong core is empty when each player has three vertices. We also show that if the number of players is constant, then the non-emptiness of the weak and strong cores is polynomial-time decidable.
Paper Structure (18 sections, 20 theorems, 7 figures)

This paper contains 18 sections, 20 theorems, 7 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. The Gallai-Edmonds decomposition
  4. Matroids
  5. The NTU matching game with couples
  6. Weak core
  7. Strong core
  8. Existence of a strong core matching
  9. Case 1: $a \in Q_j$ for some $j\neq i$, $c' \in Q_i$.
  10. Case 2: $c' \in Q_j$ for some $j\neq i$, $a \in Q_i$.
  11. Case 3: $a \in Q_j$ for some $j\neq i$, $c' \in Q_{\ell}$ for some $\ell \neq i$.
  12. Case 1: $i\in \{k+1,k+2\}$.
  13. Case 2: $i\in [k]$.
  14. Examples
  15. Constant number of players
  16. ...and 3 more sections

Key Result

Lemma 1

Given a graph $G = (V;E)$, a partition $V = V_1 \cup V_2 \cup \dots \cup V_m$ of $V$, and a vector $q \in \mathbb{Z}_+^m$, it can be decided in polynomial time if there is a matching $M$ such that $|V(M) \cap V_i| \geq q_i$ for every $i \in [m]$.

Figures (7)

  • Figure 1: An $(A,B;C)$-$\delta$-path. Solid edges denote the player edges, i.e. edges of $M_0$, while dashed edges are in $E$.
  • Figure 2: An $M_0$-alternating path in $G'$ from $Q_i$ to $Q_j$, indicated by thick edges. Edges in $M_0$ are solid, while edges in $E$ are dashed.
  • Figure 3: Solid edges are in $M_0$, dashed egdes are in $E$. The example on the right is obtained from the example on the left by adding $ab'$. The strong core is empty on the left, while bold edges on the right form a matching in the strong core.
  • Figure 4: A special edge between $u$ and $v$.
  • Figure 5: The variable gadget. The dotted lines represent special edges.
  • ...and 2 more figures

Theorems & Definitions (51)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Lemma 5
  • proof
  • ...and 41 more