An ETH-Tight FPT Algorithm for Rejection-Proof Set Packing with Applications to Kidney Exchange
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917002630660096 |
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| author | Jansen, Bart M. P. Lamme, Jeroen S. K. Verhaegh, Ruben F. A. |
| author_facet | Jansen, Bart M. P. Lamme, Jeroen S. K. Verhaegh, Ruben F. A. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_11965 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An ETH-Tight FPT Algorithm for Rejection-Proof Set Packing with Applications to Kidney Exchange Jansen, Bart M. P. Lamme, Jeroen S. K. Verhaegh, Ruben F. A. Data Structures and Algorithms 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. |
| title | An ETH-Tight FPT Algorithm for Rejection-Proof Set Packing with Applications to Kidney Exchange |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2509.11965 |