彩虹循环数与近似EFX的线性界
A Linear Bound on the Rainbow Cycle Number and Approximate EFX
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中文总结 AI 辅助
该研究证明彩虹循环数R(d)满足线性界R(d)<ed，据此得到未分配物品数量最优渐近保证的部分(1−ε)-EFX分配，并给出对应随机算法。
中文摘要 AI 辅助
关于具有加性估值的公平分配实例是否都存在完全无嫉妒至任意物品（EFX）分配，目前尚无定论。一个被广泛研究的松弛方案允许部分物品未分配，要求其满足(1−ε)-EFX。彩虹循环数R(d)被引入用于研究该问题：R(d)的上界可得到未分配物品数量较少的近似EFX分配。此前最优界为R(d)=O(d log d)，对应O_ε(√(n log n))个未分配物品。我们通过证明R(d)<ed，解决了R(d)为线性的猜想。由此可得，每个含n个智能体的实例都存在一个部分(1−ε)-EFX分配，其未分配物品数量为O(√(n/ε))，这是彩虹循环归约能得到的未分配物品数量的最优渐近保证。我们还给出了一个随机算法，可在输入规模和1/ε的期望多项式时间内找到该分配。
英文摘要
It is open whether every fair-division instance with additive valuations admits a complete envy-free-up-to-any-good (EFX) allocation. A well-studied relaxation allows some goods to remain unallocated and asks for $(1-\varepsilon)$-EFX. The rainbow cycle number $R(d)$ was introduced to study this problem: upper bounds on $R(d)$ yield approximate EFX allocations with few unallocated goods. The best previous bound, $R(d)=O(d\log d)$, gives $O_\varepsilon(\sqrt{n\log n})$ unallocated goods. We resolve the conjecture that $R(d)$ is linear by proving $R(d)<ed$. It follows that every instance with $n$ agents admits a partial $(1-\varepsilon)$-EFX allocation with $O(\sqrt{n/\varepsilon})$ unallocated goods. This is the best possible asymptotic guarantee on the number of unallocated goods obtainable from the rainbow-cycle reduction. We also give a randomized algorithm that finds such an allocation in expected time polynomial in the input size and $1/\varepsilon$.
