# Edge-matching Problems with Rotations

## TL;DR

This paper explores the computational complexity of edge-matching puzzles with rotations, revealing that allowing rotations can change the problem from polynomial-time solvable to NP-hard, depending on puzzle size and parameters.

## Contribution

It provides new complexity results for edge-matching puzzles with rotations, including NP-hardness proofs and fixed-parameter tractability based on the number of colors.

## Key findings

- Allowing rotations makes certain puzzles NP-hard.
- Some edge-matching problems remain polynomial-time solvable.
- The problem is fixed-parameter tractable with respect to the number of colors.

## Abstract

Edge-matching problems, also called edge matching puzzles, are abstractions of placement problems with neighborhood conditions. Pieces with colored edges have to be placed on a board such that adjacent edges have the same color. The problem has gained interest recently with the (now terminated) Eternity~II puzzle, and new complexity results. In this paper we consider a number of settings which differ in size of the puzzles and the manipulations allowed on the pieces. We investigate the effect of allowing rotations of the pieces on the complexity of the problem, an aspect that is only marginally treated so far. We show that some problems have polynomial time algorithms while others are NP-complete. Especially we show that allowing rotations in one-row puzzles makes the problem NP-hard. The proofs of the hardness result uses a large number of colors. This is essential because we also show that this problem (and another related one) is fixed-parameter tractable, where the relevant parameter is the number of colors.