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Rainbow cycle number and efx allocations: (almost) closing the gap

Chashm Jahan, S ; Sharif University of Technology | 2023

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  1. Type of Document: Article
  2. DOI: 10.24963/ijcai.2023/286
  3. Publisher: International Joint Conferences on Artificial Intelligence , 2023
  4. Abstract:
  5. Recently, some studies on the fair allocation of indivisible goods notice a connection between a purely combinatorial problem called the Rainbow Cycle problem and a fairness notion known as EFX: assuming that the rainbow cycle number for parameter d(i.e. R(d)) is O(dβ logγ d), we can find a (1 − ϵ)-EFX allocation with Oϵ(nβ+1 β logβ+1 γ n) number of discarded goods. The best upper bound on R(d) is improved in a series of works to O(d4), O(d2+o(1)), and finally to O(d2). Also, via a simple observation, we have R(d) ∈ Ω(d). In this paper, we introduce another problem in extremal combinatorics. For a parameter ℓ, we define the rainbow path degree and denote it by H(ℓ). We show that any lower bound on H(ℓ) yields an upper bound on R(d). Next, we prove that H(ℓ) ∈ Ω(ℓ2/log ℓ) which yields an almost tight upper bound of R(d) ∈ Ω(dlog d). This in turn proves the existence of (1−ϵ)-EFX allocation with Oϵ(√nlog n) number of discarded goods. In addition, for the special case of the Rainbow Cycle problem that the edges in each part form a permutation, we improve the upper bound to R(d) ≤ 2d−4. We leverage H(ℓ) to achieve this bound. Our conjecture is that the exact value of H(ℓ) is ⌊ℓ22 ⌋ − 1. We provide some experiments that support this conjecture. Assuming this conjecture is correct, we have R(d) ∈ Θ(d). © 2023 International Joint Conferences on Artificial Intelligence. All rights reserved
  6. Keywords:
  7. Combinatorial problem ; Cycle number ; Extremal combinatorics ; Fair allocation ; Indivisible good ; Low bound ; Rainbow cycles
  8. Source: IJCAI International Joint Conference on Artificial Intelligence ; Volume 2023-August , 2023 , Pages 2572-2580 ; 10450823 (ISSN); 978-195679203-4 (ISBN)
  9. URL: https://www.ijcai.org/proceedings/2023/286