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We prove the endpoint case of a conjecture of Khot and Moshkovitz related to the unique games conjecture, less a small error. Letn ≥ 2. Suppose a subset Ω ofn‐dimensional Euclidean spacesatisfies −Ω = Ωcand Ω + v = Ωc(up to measure zero sets) for every standard basis vector. For anyand for anyq ≥ 1, letand let. For anyx ∈ ∂Ω, letN(x) denote the exterior normal vector atxsuch that ‖N(x)‖2 = 1. Let. Our main result shows thatBhas the smallest Gaussian surface area among all such subsets Ω, less a small error:In particular,Standard arguments extend these results to a corresponding weak inequality for noise stability. Removing the factor 6 × 10−9would prove the endpoint case of the Khot‐Moshkovitz conjecture. Lastly, we prove a Euclidean analogue of the Khot and Moshkovitz conjecture. The full conjecture of Khot and Moshkovitz provides strong evidence for the truth of the unique games conjecture, a central conjecture in theoretical computer science that is closely related to the P versus NP problem. So, our results also provide evidence for the truth of the unique games conjecture. Nevertheless, this paper does not prove any case of the unique games conjecture.
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Disproof of a packing conjecture of Alon and Spencer
A 1992 conjecture of Alon and Spencer says, roughly, that the ordinary random graphGn,1/2typically admits a covering of a constant fraction of its edges by edge‐disjoint, nearly maximum cliques. We show that this is not the case. The disproof is based on some (partial) understanding of a more basic question: forandA1,…,Atchosen uniformly and independently from thek‐subsets of {1,…,n}, what can one say aboutOur main concern is trying to understand how closely the answers to this and a related question about matchings follow heuristics gotten by pretending that certain (dependent) choices are made independently.
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- Award ID(s):
- 1502650
- PAR ID:
- 10114264
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Random Structures & Algorithms
- Volume:
- 55
- Issue:
- 3
- ISSN:
- 1042-9832
- Page Range / eLocation ID:
- p. 531-544
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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