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An ETH-Tight FPT Algorithm for Rejection-Proof Set Packing with Applications to Kidney Exchange

Authors Bart M. P. Jansen , Jeroen S. K. Lamme , Ruben F. A. Verhaegh

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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Applied computing → Health informatics
  • Theory of computation → Algorithmic game theory
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • Parameterized complexity
  • Multi-agent kidney exchange
  • Kernelization
  • Set packing

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Abstract

We study the parameterized complexity of a recently introduced multi-agent variant of the Kidney Exchange problem. Given a directed graph G and integers d and k, the standard problem asks whether G contains a packing of vertex-disjoint cycles, each of length ≤ d, covering at least k vertices in total. In the multi-agent setting we consider, the vertex set is partitioned over several agents who reject a cycle packing as solution if it can be modified into an alternative packing that covers more of their own vertices. A cycle packing is called rejection-proof if no agent rejects it and the problem asks whether such a packing exists that covers at least k vertices. 
We exploit the sunflower lemma on a set packing formulation of the problem to give a kernel for this Σ₂^P-complete problem that is polynomial in k for all constant values of d. We also provide a 2^𝒪(k log k) + n^𝒪(1) algorithm based on it and show that this FPT algorithm is asymptotically optimal under the ETH. Further, we generalize the problem by including an additional positive integer c in the input that naturally captures how much agents can modify a given cycle packing to reject it. For every constant c, the resulting problem simplifies from being Σ₂^P-complete to NP-complete. The super-exponential lower bound already holds for c = 2, though. We present an ad-hoc single-exponential algorithm for c = 1. These results reveal an interesting discrepancy between the classical and parameterized complexity of the problem and give a good view of what makes it hard.

Cite As Get BibTex

Bart M. P. Jansen, Jeroen S. K. Lamme, and Ruben F. A. Verhaegh. An ETH-Tight FPT Algorithm for Rejection-Proof Set Packing with Applications to Kidney Exchange. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.IPEC.2025.9

Author Details

Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Jeroen S. K. Lamme
  • Eindhoven University of Technology, The Netherlands
Ruben F. A. Verhaegh
  • Eindhoven University of Technology, The Netherlands

Funding

  • Jansen, Bart M. P.: Supported by the Dutch Research Council (NWO) through Gravitation-grant NETWORKS-024.002.003.
  • Lamme, Jeroen S. K.: Supported by the Dutch Research Council (NWO) through Gravitation-grant NETWORKS-024.002.003.

References

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