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

2025-09-15

AI Summary

This paper studies the rejection-robust cycle packing problem in multi-agent kidney exchange: given a directed graph and parameters (d,k), find a vertex-disjoint packing of cycles of length at most (d) covering at least (k) vertices, such that no agent—defined as a subset of vertices—can strictly increase its own coverage via local reallocation, i.e., the solution is stable against individual rejections. We introduce the novel concept of a “rejection certificate” to formally characterize structural constraints imposed by agents’ rejection behavior. Modeling the problem as a constrained set packing, we apply the sunflower lemma to construct a polynomial kernel and design an FPT algorithm running in time (2^{O(k log k)} + n^{O(1)}), which we prove optimal under the Exponential Time Hypothesis (ETH). We fully characterize the complexity phase transition induced by the key parameter (c) (rejection strength): the problem is single-exponential FPT for (c = 1), but becomes (Sigma_2^P)-hard for (c geq 2), thereby bridging classical and parameterized complexity landscapes.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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. With a single-exponential algorithm for the setting where $c = 1$, we show this to be strictly easier under the ETH than when $c = 2$. In turn, we show that any $c geq 2$ yields a problem that is essentially as hard as the original problem with $c$ unbounded. This displays an interesting discrepancy between the classical and parameterized complexity of the problem and gives a good view of what makes it hard.

Problem

• Studying parameterized complexity of multi-agent kidney exchange problem
• Determining existence of rejection-proof cycle packing covering k vertices
• Generalizing problem with parameter c to capture agent modification limits

Innovation

• Sunflower lemma kernel polynomial in k
• FPT algorithm optimal under ETH
• Generalization with c captures agent modification