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arXiv · 2509.11965

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

Bart M. P. Jansen·Jeroen S. K. Lamme·Ruben F. A. Verhaegh
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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 $\leq d$, 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 $Σ_2^P$-complete problem that is polynomial in $k$ for all constant values of $d$. We also provide a $2^{\mathcal{O}(k \log k)} + n^{\mathcal{O}(1)}$ 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 $Σ_2^P$-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.

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Bart M. P. Jansen, Jeroen S. K. Lamme, Ruben F. A. Verhaegh. 2025-10-10. An ETH-Tight FPT Algorithm for Rejection-Proof Set Packing with Applications to Kidney Exchange. https://arxiv.org/abs/2509.11965

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