Packing identical simple polygons is NP-hard

TL;DR

This paper proves that determining whether multiple copies of a simple polygon can be packed into a larger simple polygon without overlap is NP-hard, extending previous results to the case of simple polygons.

Contribution

It introduces a novel NP-hardness proof for packing identical simple polygons, using a reduction from Planar-Circuit-SAT that encodes circuit information within the polygon.

Findings

NP-hardness of packing identical simple polygons established

Reduction from Planar-Circuit-SAT demonstrated

Extends previous NP-hardness results to simple polygons

Abstract

Given a small polygon S, a big simple polygon B and a positive integer k, it is shown to be NP-hard to determine whether k copies of the small polygon (allowing translation and rotation) can be placed in the big polygon without overlap. Previous NP-hardness results were only known in the case where the big polygon is allowed to be non-simple. A novel reduction from Planar-Circuit-SAT is presented where a small polygon is constructed to encode the entire circuit.