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A Faster Deterministic Algorithm for Kidney Exchange via Representative Set

Kangyi Tian, Mingyu Xiao

Abstract

The Kidney Exchange Problem is a prominent challenge in healthcare and economics, arising in the context of organ transplantation. It has been extensively studied in artificial intelligence and optimization. In a kidney exchange, a set of donor-recipient pairs and altruistic donors are considered, with the goal of identifying a sequence of exchange -- comprising cycles or chains starting from altruistic donors -- such that each donor provides a kidney to the compatible recipient in the next donor-recipient pair. Due to constraints in medical resources, some limits are often imposed on the lengths of these cycles and chains. These exchanges create a network of transplants aimed at maximizing the total number, $t$, of successful transplants. Recently, this problem was deterministically solved in $O^*(14.34^t)$ time (IJCAI 2024). In this paper, we introduce the representative set technique for the Kidney Exchange Problem, showing that the problem can be deterministically solved in $O^*(6.855^t)$ time.

A Faster Deterministic Algorithm for Kidney Exchange via Representative Set

Abstract

The Kidney Exchange Problem is a prominent challenge in healthcare and economics, arising in the context of organ transplantation. It has been extensively studied in artificial intelligence and optimization. In a kidney exchange, a set of donor-recipient pairs and altruistic donors are considered, with the goal of identifying a sequence of exchange -- comprising cycles or chains starting from altruistic donors -- such that each donor provides a kidney to the compatible recipient in the next donor-recipient pair. Due to constraints in medical resources, some limits are often imposed on the lengths of these cycles and chains. These exchanges create a network of transplants aimed at maximizing the total number, , of successful transplants. Recently, this problem was deterministically solved in time (IJCAI 2024). In this paper, we introduce the representative set technique for the Kidney Exchange Problem, showing that the problem can be deterministically solved in time.
Paper Structure (16 sections, 11 theorems, 24 equations, 1 figure, 1 table)

This paper contains 16 sections, 11 theorems, 24 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our Contribution and Method
  4. Preliminaries
  5. Representative Set.
  6. A Faster Deterministic Algorithm
  7. Semi-feasible path-cycle Packing
  8. A Dynamic Programming Algorithm
  9. Case (i):
  10. Case (ii):
  11. Case (iii):
  12. Case (iv):
  13. Case (v):
  14. Representative Sets for Kidney Exchange
  15. Conclusion and Discussion
  16. ...and 1 more sections

Key Result

Lemma 1

Suppose that $\mathcal{X}$, $\mathcal{Y}$ and $\mathcal{Z}$ are p-families. If $\mathcal{X} \subseteq^q_{\text{rep}} \mathcal{Y}$ and $\mathcal{Y} \subseteq^q_{\text{rep}} \mathcal{Z}$, then $\mathcal{X} \subseteq^q_{\text{rep}} \mathcal{Z}$.

Figures (1)

  • Figure 1: A compatibility graph

Theorems & Definitions (21)

  • Definition 1: $q$-Representative Set
  • Lemma 1: DBLP:journals/jacm/FominLPS16-rep
  • Lemma 2: DBLP:journals/jacm/FominLPS16-rep
  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 11 more