Complexity of Planar Graph Orientation Consistency, Promise-Inference, and Uniqueness, with Applications to Minesweeper Variants

TL;DR

This paper investigates the computational complexity of three Minesweeper problems—consistency, inference, and solvability—proving their NP-completeness or coNP-completeness, and correcting previous proofs regarding inference.

Contribution

It establishes the complexity classifications for Minesweeper problems and corrects a flaw in prior coNP-completeness proofs for inference.

Findings

Consistency is NP-complete.

Inference is coNP-complete with corrected proof.

Solvability is coNP-complete.

Abstract

We study three problems related to the computational complexity of the popular game Minesweeper. The first is consistency: given a set of clues, is there any arrangement of mines that satisfies it? This problem has been known to be NP-complete since 2000, but our framework proves it as a side effect. The second is inference: given a set of clues, is there any cell that the player can prove is safe? The coNP-completeness of this problem has been in the literature since 2011, but we discovered a flaw that we believe is present in all published results, and we provide a fixed proof. Finally, the third is solvability: given the full state of a Minesweeper game, can the player win the game by safely clicking all non-mine cells? This problem has not yet been studied, and we prove that it is coNP-complete.