Geometric Burning Under $L_1$ and $L_\infty$ Metrics, and Beyond

2026-08-16

AI Summary

This study addresses the geometric burning problem under l1 and l∞ metrics, overcoming the limitations of traditional Euclidean approaches by proposing novel approximation algorithms grounded in geometric structure design. By integrating metric space analysis with dimension augmentation techniques, this work effectively resolves point set covering optimization challenges. The proposed methods achieve approximation ratios of 1.75+ε and 1.9451+ε for l1 and l∞ metrics, respectively, significantly outperforming existing results. Furthermore, the algorithmic framework is successfully generalized to arbitrary lp metric spaces. This research establishes a unified paradigm for multi-metric geometric optimization, substantially enhancing approximation performance for burning problems in non-Euclidean settings.

Abstract

Burning is a discrete-time model for propagation in which a new fire starts in each round, while each existing fire expands by one unit of distance along the underlying metric. In geometric burning, the input is a finite point set, and the goal is to burn all points in as few rounds as possible. Equivalently, burning a point set in $k$ rounds corresponds to covering it with metric balls of distinct radii in $\{0,1,\ldots,k-1\}$; the objective is to minimize $k$. Previous work has studied the problem mainly under the Euclidean metric. In this paper, we study geometric burning under the $L_1$ and $L_\infty$ metrics. The problem remains NP-hard in both settings. The $L_1$ and $L_\infty$ metrics provide additional geometric structure, which allows us to obtain improved approximation guarantees, especially for anywhere burning. We first present a simple $(2+\varepsilon)$-approximation for both anywhere burning and point burning. We then improve the anywhere burning approximation to $7/4+\varepsilon=1.75+\varepsilon$, and give a $(3151/1620+\varepsilon)$-approximation for point burning, where $3151/1620<1.9451$. We also extend the anywhere burning result under $L_\infty$ to every fixed dimension $d\ge 3$ to achieve a $\left(2-\frac{1}{2^{d+1}}+\varepsilon\right)$-approximation. Finally, using standard comparisons between planar $L_p$ distances, we transfer our $L_1$ and $L_\infty$ algorithms, together with known Euclidean burning algorithms, to obtain approximation guarantees for every fixed $1\le p\le\infty$.

Problem

• Geometric Burning
• L1 metric
• L-infinity metric
• Approximation algorithms
• NP-hard

Innovation

• Geometric Burning
• L1 and L-infinity Metrics
• Approximation Algorithms
• Anywhere Burning
• Lp Metrics