# Symmetric Assembly Puzzles are Hard, Beyond a Few Pieces

## TL;DR

This paper investigates the computational complexity of symmetric assembly puzzles, proving NP-completeness in general but identifying cases where the problem is solvable efficiently.

## Contribution

It establishes the NP-completeness of symmetric assembly puzzles with polyomino pieces and provides a polynomial-time solution for a fixed number of pieces.

## Key findings

- NP-complete for polyomino pieces
- Polynomial-time solvable with a fixed number of pieces
- Complexity results for symmetric assembly puzzles

## Abstract

We study the complexity of symmetric assembly puzzles: given a collection of simple polygons, can we translate, rotate, and possibly flip them so that their interior-disjoint union is line symmetric? On the negative side, we show that the problem is strongly NP-complete even if the pieces are all polyominos. On the positive side, we show that the problem can be solved in polynomial time if the number of pieces is a fixed constant.