How to Catch $k$ Grid Points

Sariel Har-Peled; Elfarouk Harb; Qizheng He

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arXiv · 2607.10824

How to Catch $k$ Grid Points

Sariel Har-Peled·Elfarouk Harb·Qizheng He

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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 $\Theta(k^{1/3})$ boundary grid points, and its longest edge has length $\Theta(k^{1/4})$. 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.

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Sariel Har-Peled, Elfarouk Harb, Qizheng He. 2026-07-12. How to Catch $k$ Grid Points. https://arxiv.org/abs/2607.10824

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