Kidney Exchange: Faster Parameterized Algorithms and Tighter Lower Bounds

Palash Dey, Aritra Banik, Abhishek Sahu, Sujoy Bhore

Keywords: kidney exchange, FPT algorithm, parameterized complexity, W-hardness

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
get around 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 $O^⋆ (14^t)$. In this work, we improve this by presenting a deterministic FPT algorithm that runs in
time $O^⋆ ((4e)^t) \approx O^⋆ (10.88^t)$. 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.

Area: Search, Optimization, Planning, and Scheduling (SOPS)

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Submission Number: 1534