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The algorithmic Fried Potato Problem in two dimensions

Francisco Criado, Francisco Santos

TL;DR

This work addresses Conway's Fried Potato Problem in two dimensions, seeking a division of a convex polygon into $n$ parts via $n-1$ cuts that minimizes the largest inradius. Building on Bezdek–Bezdek's result that the optimum is realized by $n-1$ parallel cuts, the authors reduce the problem to finding a distance parameter $\rho$ and a direction $v$ such that $\operatorname{width}(P^{\rho})=2n\rho$, and develop a quasi-linear algorithm to compute these in $O(m \log^4 m)$ time for polygons with $m$ edges. A key innovation is the 3D dome $\widetilde{P}$ construction and its efficient preprocessing via a Dobkin–Kirkpatrick hierarchy, enabling fast linear-programming queries on the dome and its slices. This yields a substantial improvement over prior quadratic-time approaches and provides a practical framework for efficiently solving the Baker's Potato problem in the plane.

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 Cañete-Fernández-Márquez (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.

The algorithmic Fried Potato Problem in two dimensions

TL;DR

This work addresses Conway's Fried Potato Problem in two dimensions, seeking a division of a convex polygon into parts via cuts that minimizes the largest inradius. Building on Bezdek–Bezdek's result that the optimum is realized by parallel cuts, the authors reduce the problem to finding a distance parameter and a direction such that , and develop a quasi-linear algorithm to compute these in time for polygons with edges. A key innovation is the 3D dome construction and its efficient preprocessing via a Dobkin–Kirkpatrick hierarchy, enabling fast linear-programming queries on the dome and its slices. This yields a substantial improvement over prior quadratic-time approaches and provides a practical framework for efficiently solving the Baker's Potato problem in the plane.

Abstract

Conway's Fried Potato Problem seeks to determine the best way to cut a convex body in parts by hyperplane cuts (with the restriction that the -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 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 -gon , this direction (and hence the cuts themselves) can be found in time , which improves on a quadratic algorithm proposed by Cañete-Fernández-Márquez (DMD 2022). Our algorithm first preprocesses what we call the dome (closely related to the medial axis) of 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.
Paper Structure (3 sections, 7 theorems, 19 equations)

This paper contains 3 sections, 7 theorems, 19 equations.

Table of Contents

  1. From fried potatoes to baker's potatoes
  2. The Dobkin-Kirkpatrick hyerarchy
  3. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 2

Let $C$ be a convex body in $\mathbb{R}^d$ and $n\in \mathbb{N}$. Let $P$ be a division of $C$ into $n$ subsets $C_1,\dots, C_n$ given by $n-1$ successive hyperplane cuts. These cuts of $P$ do not extend beyond previously made cuts, therefore $(n-1)$ cuts produce $n$ pieces. Then, where $\rho>0$ is the unique number satisfying Furthermore, equality holds for the division of $C$ given by $n-1$ pa

Theorems & Definitions (12)

  • Definition 1
  • Theorem 2: Bezdek-Bezdek Bezdek^2
  • Theorem 3
  • Theorem 4: Dobkin-Kirkpatrick Hierarchy DK1DK2, see also orourke
  • Definition 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Theorem 8
  • ...and 2 more