Sweeping an oval to a vanishing point

TL;DR

This paper investigates the problem of optimally sweeping convex regions in the plane using a sequence of slanted sweeps, disproving a previous conjecture and analyzing approximation algorithms for the problem.

Contribution

It introduces a new slanted sweeping model, disproves a conjecture about the sufficiency of two sweeps, and analyzes approximation algorithms within this framework.

Findings

Existence of convex regions where two sweeps are not optimal.

Disproof of the conjecture that two sweeps always suffice.

Algorithms achieve approximately 1.27 approximation ratio.

Abstract

Given a convex region in the plane, and a sweep-line as a tool, what is best way to reduce the region to a single point by a sequence of sweeps? The problem of sweeping points by orthogonal sweeps was first studied in [2]. Here we consider the following \emph{slanted} variant of sweeping recently introduced in [1]: In a single sweep, the sweep-line is placed at a start position somewhere in the plane, then moved continuously according to a sweep vector $\vec v$ (not necessarily orthogonal to the sweep-line) to another parallel end position, and then lifted from the plane. The cost of a sequence of sweeps is the sum of the lengths of the sweep vectors. The (optimal) sweeping cost of a region is the infimum of the costs over all finite sweeping sequences for that region. An optimal sweeping sequence for a region is one with a minimum total cost, if it exists. Another parameter of interest is the number of sweeps. We show that there exist convex regions for which the optimal sweeping cost cannot be attained by two sweeps. This disproves a conjecture of Bousany, Karker, O'Rourke, and Sparaco stating that two sweeps (with vectors along the two adjacent sides of a minimum-perimeter enclosing parallelogram) always suffice [1]. Moreover, we conjecture that for some convex regions, no finite sweeping sequence is optimal. On the other hand, we show that both the 2-sweep algorithm based on minimum-perimeter enclosing rectangle and the 2-sweep algorithm based on minimum-perimeter enclosing parallelogram achieve a $4/\pi \approx 1.27$ approximation in this sweeping model.