From Proximity to Utility: A Voronoi Partition of Pareto Optima

TL;DR

This paper extends Voronoi diagrams to incorporate additional site attributes like prices, creating a more expressive partition that considers client preferences and attributes, with complexity analysis showing near-linear expected complexity under certain conditions.

Contribution

The paper introduces a new Voronoi-based diagram considering site attributes, providing complexity bounds and technical results on Pareto optima in high-dimensional spaces.

Findings

Expected complexity of the candidate diagram is near linear for sites with attributes from the same distribution.

Derived high-probability bounds on the number of Pareto optima in uniformly sampled point sets.

Revisited and improved classical backward analysis techniques for high-dimensional Pareto analysis.

Abstract

We present an extension of Voronoi diagrams where when considering which site a client is going to use, in addition to the site distances, other site attributes are also considered (for example, prices or weights). A cell in this diagram is then the locus of all clients that consider the same set of sites to be relevant. In particular, the precise site a client might use from this candidate set depends on parameters that might change between usages, and the candidate set lists all of the relevant sites. The resulting diagram is significantly more expressive than Voronoi diagrams, but naturally has the drawback that its complexity, even in the plane, might be quite high. Nevertheless, we show that if the attributes of the sites are drawn from the same distribution (note that the locations are fixed), then the expected complexity of the candidate diagram is near linear. To this end, we derive several new technical results, which are of independent interest. In particular, we provide a high-probability, asymptotically optimal bound on the number of Pareto optima points in a point set uniformly sampled from the $d$-dimensional hypercube. To do so we revisit the classical backward analysis technique, both simplifying and improving relevant results in order to achieve the high-probability bounds.