How to Catch $k$ Grid Points

2026-07-12

AI Summary

This study addresses the problem of finding the convex polygon of minimum perimeter that encloses exactly $k$ lattice points from $\mathbb{Z}^2$. Through a refined geometric analysis of optimal polygons, the authors establish that such a polygon lies within an annulus of width $O(k^{1/6})$, has $\Theta(k^{1/3})$ lattice points on its boundary, and features a longest edge of length $\Theta(k^{1/4})$. Leveraging these structural properties, they present the first deterministic algorithm for this problem with a running time of $O(k^{29/18 + o(1)})$, substantially improving upon the previous $O(k^3)$ bound. By integrating techniques from computational geometry and lattice-point analysis, this work provides the tightest known geometric bounds and the most efficient algorithm to date for the minimum-perimeter $k$-lattice-point convex hull problem.

Abstract

Given a positive integer $k$, we study the problem of finding a convex polygon of minimum perimeter that encloses exactly $k$ points of $\mathbf{Z}^2$. We show that an optimal polygon is contained in a circular annulus of width $O(k^{1/6})$, has $Θ(k^{1/3})$ boundary grid points, and its longest edge has length $Θ(k^{1/4})$. Using these structural bounds, we present a deterministic algorithm that computes an optimal polygon in $O(k^{29/18+o(1)})$ time, improving over the previous $O(k^3)$-time algorithm.

Problem

• convex polygon
• minimum perimeter
• lattice points
• grid points
• computational geometry

Innovation

• convex polygon
• lattice points
• minimum perimeter
• structural bounds
• deterministic algorithm