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

TL;DR

This work analyzes the parameterized complexity of rejection-proof kidney exchange problems in multi-agent settings. It introduces RPKE-$d$ and $c$-RPKE-$d$, and develops a sunflower-based kernel yielding a polynomial-sized kernel for constant $d$, paired with a $2^{\mathcal{O}(k \log k)}$ FPT algorithm. Complementary ETH-based lower bounds establish tightness of the algorithmic results, and the authors further explore the boundary case $c=1$ with a single-exponential $3^n$ algorithm, while showing NP-hardness for $c\ge2$. The results illuminate a striking discrepancy between classical and parameterized complexity for rejection-proof packings and provide a foundation for further exploration of multi-agent combinatorial problems in kidney exchange and beyond.

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.

Figures (1)

• Figure 1: A socially optimal packing of cycles is shown on the left and covers six vertices. However, the blue agent rejects this solution as they could modify it to cover more blue vertices as shown on the right. They can do this by rejecting the left-most $3$-cycle and including an internal $2$-cycle.

Theorems & Definitions (22)

• Definition 1
• Lemma 1
• Lemma 2: Sunflower lemma
• Lemma 2
• Definition 3
• Lemma 4: CyganFGKMPS16
• Lemma 5
• Lemma 7
• Lemma 7
• Lemma 7