The algorithmic Fried Potato Problem in two dimensions

TL;DR

This paper presents an efficient algorithm to determine optimal cuts for the Fried Potato Problem in two dimensions, improving previous methods by leveraging advanced geometric data structures.

Contribution

It introduces a faster $O(m \, \log^4 m)$ algorithm for finding optimal cutting directions in convex polygons, enhancing prior quadratic-time solutions.

Findings

The algorithm computes optimal cuts in convex polygons efficiently.

Preprocessing the dome structure enables fast linear programming solutions.

The method improves computational complexity over previous approaches.

Abstract

Conway's Fried Potato Problem seeks to determine the best way to cut a convex body in $n$ parts by $n-1$ hyperplane cuts (with the restriction that the $i$-th cut only divides in two one of the parts obtained so far), in a way as to minimize the maxuimum of the inradii of the parts. It was shown by Bezdek and Bezdek that the solution is always attained by $n-1$ parallel cuts. But the algorithmic problem of finding the best direction for these parallel cuts remains. In this note we show that for a convex $m$-gon $P$, this direction (and hence the cuts themselves) can be found in time $O(m \log^4 m)$, which improves on a quadratic algorithm proposed by Ca\~nete-Fern\'andez-M\'arquez (DMD 2022). Our algorithm first preprocesses what we call the dome (closely related to the medial axis) of $P$ using a variant of the Dobkin-Kirkpatrick hierarchy, so that linear programs in the dome and in slices of it can be solved in polylogarithmic time.