Geometry of Rounding

Abstract

Rounding has proven to be a fundamental tool in theoretical computer science. By observing that rounding and partitioning of $\mathbb{R}^d$ are equivalent, we introduce the following natural partition problem which we call the {\em secluded hypercube partition problem}: Given $k\in \mathbb{N}$ (ideally small) and $\epsilon>0$ (ideally large), is there a partition of $\mathbb{R}^d$ with unit hypercubes such that for every point $p \in \mathbb{R}^d$, its closed $\epsilon$-neighborhood (in the $\ell_{\infty}$ norm) intersects at most $k$ hypercubes? We undertake a comprehensive study of this partition problem. We prove that for every $d\in \mathbb{N}$, there is an explicit (and efficiently computable) hypercube partition of $\mathbb{R}^d$ with $k = d+1$ and $\epsilon = \frac{1}{2d}$. We complement this construction by proving that the value of $k=d+1$ is the best possible (for any $\epsilon$) for a broad class of ``reasonable'' partitions including hypercube partitions. We also investigate the optimality of the parameter $\epsilon$ and prove that any partition in this broad class that has $k=d+1$, must have $\epsilon\leq\frac{1}{2\sqrt{d}}$. These bounds imply limitations of certain deterministic rounding schemes existing in the literature. Furthermore, this general bound is based on the currently known lower bounds for the dissection number of the cube, and improvements to this bound will yield improvements to our bounds. While our work is motivated by the desire to understand rounding algorithms, one of our main conceptual contributions is the introduction of the {\em secluded hypercube partition problem}, which fits well with a long history of investigations by mathematicians on various hypercube partitions/tilings of Euclidean space.