Being Fat and Friendly is Not Enough

TL;DR

This paper proves the non-existence of a near-optimal approximation algorithm for covering points with pre-specified fat triangles of similar size, settling the problem's computational complexity and exploring related geometric covering issues.

Contribution

It establishes the hardness of approximating the minimum number of fat triangles needed to cover points, providing a definitive complexity result for this geometric covering problem.

Findings

No $(1+\eps)$-approximation algorithm exists for the problem.

Constant factor approximation algorithm is known, confirming the problem's approximability boundary.

Explores related problems including cover by friendly fat shapes and 3D triangle independent sets.

Abstract

We show that there is no $(1+\eps)$-approximation algorithm for the problem of covering points in the plane by minimum number of fat triangles of similar size (with the minimum angle of the triangles being close to 45 degrees). Here, the available triangles are prespecified in advance. Since a constant factor approximation algorithm is known for this problem \cite{cv-iaags-07}, this settles the approximability of this problem. We also investigate some related problems, including cover by friendly fat shapes, and independent set of triangles in three dimensions.