Tetravex is NP-complete

TL;DR

This paper proves that deciding whether a Tetravex puzzle has a valid tiling is NP-complete, demonstrating the computational difficulty of the game and relating it to other tiling problems.

Contribution

It establishes the NP-completeness of Tetravex, linking it to other known NP-hard tiling and puzzle problems, and discusses open questions for future research.

Findings

Deciding Tetravex tilings is NP-complete.

Tetravex shares complexity properties with other tiling puzzles.

Raises open questions about infinite versions and puzzle generation.

Abstract

Tetravex is a widely played one person computer game in which you are given $n^2$ unit tiles, each edge of which is labelled with a number. The objective is to place each tile within a $n$ by $n$ square such that all neighbouring edges are labelled with an identical number. Unfortunately, playing Tetravex is computationally hard. More precisely, we prove that deciding if there is a tiling of the Tetravex board is NP-complete. Deciding where to place the tiles is therefore NP-hard. This may help to explain why Tetravex is a good puzzle. This result compliments a number of similar results for one person games involving tiling. For example, NP-completeness results have been shown for: the offline version of Tetris, KPlumber (which involves rotating tiles containing drawings of pipes to make a connected network), and shortest sliding puzzle problems. It raises a number of open questions. For example, is the infinite version Turing-complete? How do we generate Tetravex problems which are truly puzzling as random NP-complete problems are often surprising easy to solve? Can we observe phase transition behaviour? What about the complexity of the problem when it is guaranteed to have an unique solution? How do we generate puzzles with unique solutions?