Kidney Exchange: Faster Parameterized Algorithms and Tighter Lower Bounds

TL;DR

This work advances the theoretical understanding of the Kidney Exchange Problem by delivering a faster deterministic FPT algorithm parameterized by $t$, achieving $O^*( (4e)^t )$ time through color coding and a subset DP that builds vertex-disjoint cycles and altruistic paths. It also sharpens the hardness landscape, proving no polynomial kernel for parameter $t+\ell_p+\ell_c+|\mathcal{B}|$ under standard assumptions, and establishing W[1]-hardness with respect to pathwidth (including restricted DAG and altruistic-vertex cases), as well as para-NP-hardness for combined parameters. Additionally, NP-hardness on DAGs is shown via a two-step reduction from 3-Partition to Fixed-Size-3-Partition and then to Kidney Exchange on DAGs, clarifying the boundary between tractable and intractable regimes for cycle-and-chain kidney exchanges. Collectively, the results map a near-complete picture of parameterized complexity for Kidney Exchange with respect to pathwidth, cycle/path length constraints, and solution size, informing both algorithm design and theoretical limitations for practical clearing mechanisms.

Abstract

The kidney exchange mechanism allows many patient-donor pairs who are otherwise incompatible with each other to come together and exchange kidneys along a cycle. However, due to infrastructure and legal constraints, kidney exchange can only be performed in small cycles in practice. In reality, there are also some altruistic donors who do not have any paired patients. This allows us to also perform kidney exchange along paths that start from some altruistic donor. Unfortunately, the computational task is NP-complete. To overcome this computational barrier, an important line of research focuses on designing faster algorithms, both exact and using the framework of parameterized complexity. The standard parameter for the kidney exchange problem is the number $t$ of patients that receive a healthy kidney. The current fastest known deterministic FPT algorithm for this problem, parameterized by $t$, is $O^\star\left(14^t\right)$. In this work, we improve this by presenting a deterministic FPT algorithm that runs in time $O^\star\left((4e)^t\right)\approx O^\star\left(10.88^t\right)$. This problem is also known to be W[1]-hard parameterized by the treewidth of the underlying undirected graph. A natural question here is whether the kidney exchange problem admits an FPT algorithm parameterized by the pathwidth of the underlying undirected graph. We answer this negatively in this paper by proving that this problem is W[1]-hard parameterized by the pathwidth of the underlying undirected graph. We also present some parameterized intractability results improving the current understanding of the problem under the framework of parameterized complexity.

Figures (1)

• Figure 1: Reduction from Fixed-Size-3-Partition problem

Theorems & Definitions (17)

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• proof
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