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

TL;DR

This work develops a unifying gadget-based framework grounded in planar graph orientation (PGO) to analyze Minesweeper decision problems. It formalizes consistency, inference, and solvability, corrects a flaw in prior inference proofs, and proves $coNP$-hardness of inference and solvability, with solvability remaining hard even after a single click. The framework relies on silent, parsimonious gadget implementations and network simulations to carry reductions from planar circuit problems to Minesweeper variants, enabling systematic hardness proofs across many variants, including those with empty initial boards. By providing a modular reduction toolkit, the paper guides both theoretical analysis and the design of Minesweeper-like puzzles, offering transferable insights for a broad class of grid-based reasoning games.

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.

Figures (11)

• Figure 1: Left: an example of a consistent board. Right: one possible way to satisfy all clues.
• Figure 2: An example of an inconsistent board.
• Figure 3: Left: an example of a board with an inference. Right: green squares mark cells that can be inferred.
• Figure 4: An example of a board that doesn't have an inference.
• Figure 5: The OR gate from inf Figure 16. There is a minor typo: the lower cell with $b$ should have $\overline b$.

Theorems & Definitions (51)

• Definition 2.1
• Definition 2.2: Minesweeper consistency problem
• Proposition 2.3
• proof
• Definition 2.4: Minesweeper inference problem
• Proposition 2.5
• proof
• Definition 2.6: Minesweeper solvability problem
• Proposition 2.7
• proof