Simple Wriggling is Hard unless You Are a Fat Hippo

TL;DR

This paper proves the NP-hardness of connecting two points with a wire in a polygonal domain with holes, but shows that for 'fat' snakes, the shortest path can be efficiently computed.

Contribution

It establishes NP-hardness for the general snake connection problem and identifies a polynomial-time solution for 'fat' snakes with small length-to-width ratio.

Findings

NP-hardness of wire connection in polygonal domains with holes

Polynomial-time algorithm for 'fat' snakes with small length-to-width ratio

Implication for approximation difficulty of shortest path for long snakes

Abstract

We prove that it is NP-hard to decide whether two points in a polygonal domain with holes can be connected by a wire. This implies that finding any approximation to the shortest path for a long snake amidst polygonal obstacles is NP-hard. On the positive side, we show that snake's problem is "length-tractable": if the snake is "fat", i.e., its length/width ratio is small, the shortest path can be computed in polynomial time.