On Approximating the Weighted Region Problem in Square Tessellations

TL;DR

This paper improves the approximation ratio for finding shortest weighted paths in square-tessellated planes from 2√2 to √2+1, using grid graph methods.

Contribution

It provides a tighter approximation bound for the weighted region problem specifically in square tessellations, enhancing previous results.

Findings

Achieves a $(\sqrt{2}+1)$-approximation ratio.

Improves upon the previous $2\sqrt{2}$ approximation.

Offers a more efficient approach for shortest path approximation in square grids.

Abstract

The weighted region problem is the problem of finding the weighted shortest path on a plane consisting of polygonal regions with different weights. For the case when the plane is tessellated by squares, we can solve the problem approximately by finding the shortest path on a grid graph defined by placing a vertex at the center of each grid. In this note, we show that the obtained path admits $(\sqrt{2}+1)$-approximation. This improves the previous result of $2\sqrt{2}$.