Geometry of Rounding: Near Optimal Bounds and a New Neighborhood Sperner's Lemma

04/10/2023

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by   Jason Vander Woude, et al.

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A partition 𝒫 of ℝ^d is called a (k,ε)-secluded partition if, for every p⃗∈ℝ^d, the ball B_∞(ε, p⃗) intersects at most k members of 𝒫. A goal in designing such secluded partitions is to minimize k while making ε as large as possible. This partition problem has connections to a diverse range of topics, including deterministic rounding schemes, pseudodeterminism, replicability, as well as Sperner/KKM-type results. In this work, we establish near-optimal relationships between k and ε. We show that, for any bounded measure partitions and for any d≥ 1, it must be that k≥(1+2ε)^d. Thus, when k=k(d) is restricted to poly(d), it follows that ε=ε(d)∈ O(ln d/d). This bound is tight up to log factors, as it is known that there exist secluded partitions with k(d)=d+1 and ε(d)=1/2d. We also provide new constructions of secluded partitions that work for a broad spectrum of k(d) and ε(d) parameters. Specifically, we prove that, for any f:ℕ→ℕ, there is a secluded partition with k(d)=(f(d)+1)^⌈d/f(d)⌉ and ε(d)=1/2f(d). These new partitions are optimal up to O(log d) factors for various choices of k(d) and ε(d). Based on the lower bound result, we establish a new neighborhood version of Sperner's lemma over hypercubes, which is of independent interest. In addition, we prove a no-free-lunch theorem about the limitations of rounding schemes in the context of pseudodeterministic/replicable algorithms.