The algorithmic Fried Potato Problem in two dimensions

2025-01-23

AI Summary

This paper addresses the optimal parallel partitioning problem for a two-dimensional convex $m$-gon: given $n-1$ parallel cuts, partition it into $n$ subregions to minimize the maximum inradius among them—i.e., to equilibrate the “thinnest” part. We present the first subquadratic-time algorithm for this problem. Our method integrates, for the first time, a dome structure—a medial-axis-like geometric construct—with the Dobkin–Kirkpatrick hierarchical decomposition, enabling efficient directional-space search. This integration is supported by dome preprocessing, an enhanced hierarchy, and a multiscale linear programming solver. The algorithm achieves a time complexity of $O(m log^4 m)$, improving upon the previous $O(m^2)$ bound. This represents a significant breakthrough in overcoming the computational bottleneck for optimal parallel partitioning of convex polygons in two dimensions.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Optimal Cutting Direction
• Convex Polygon
• Maximum Inscribed Circle Radius

Innovation

• Convex Geometry
• Optimization Algorithm
• Conway's Potato Problem