The Pareto cover problem

TL;DR

This paper introduces the Pareto cover problem, aiming to select a set of points minimizing expected domination cost for random points, with complexity results and an approximation scheme for fixed parameters.

Contribution

It formalizes the Pareto cover problem, proves NP-hardness for small cases, and develops an FPTAS for fixed k under certain distribution assumptions.

Findings

NP-hardness for k=2 with binary distributions

Existence of an FPTAS for fixed k

Applicability to feature-based customer request satisfaction

Abstract

We introduce the problem of finding a set $B$ of $k$ points in $[0,1]^n$ such that the expected cost of the cheapest point in $B$ that dominates a random point from $[0,1]^n$ is minimized. We study the case where the coordinates of the random points are independently distributed and the cost function is linear. This problem arises naturally in various application areas where customers' requests are satisfied based on predefined products, each corresponding to a subset of features. We show that the problem is NP-hard already for $k=2$ when each coordinate is drawn from $\{0,1\}$, and obtain an FPTAS for general fixed $k$ under mild assumptions on the distributions.