[Paper Review] The computational complexity of Minesweeper

Michiel de Bondt|arXiv (Cornell University)|Apr 20, 2012

Complexity and Algorithms in Graphs4 references3 citations

TL;DR

This paper establishes that the Minesweeper game is PP-hard when the goal is to locate all mines with maximum probability, and PSPACE-complete when the success probability is infinitesimally small. The authors construct Minesweeper boards that simulate boolean circuits using custom wire, gate, and crossover components on square, hexagonal, and triangular grids, reducing problems in computational complexity to Minesweeper gameplay via probabilistic and deterministic circuit encodings.

ABSTRACT

We show that the Minesweeper game is PP-hard, when the object is to locate all mines with the highest probability. When the probability of locating all mines may be infinitesimal, the Minesweeper game is even PSPACE-complete. In our construction, the player can reveal a boolean circuit in polynomial time, after guessing an initial square with no surrounding mines, a guess that has 99 percent probability of success. Subsequently, the mines must be located with a maximum probability of success. Furthermore, we show that determining the solvability of a partially uncovered Minesweeper board is NP-complete with hexagonal and triangular grids as well as a square grid, extending a similar result for square grids only by R. Kaye. Actually finding the mines with a maximum probability of success is again PP-hard or PSPACE-complete respectively. Our constructions are in such a way that the number of mines can be computed in polynomial time and hence a possible mine counter does not provide additional information. The results are obtained by replacing the dyadic gates in [3] by two primitives which makes life more easy in this context.

Motivation & Objective

• To determine the computational complexity of Minesweeper when the goal is to maximize the probability of locating all mines.
• To extend Richard Kaye's NP-completeness result for Minesweeper solvability to non-square grids (hexagonal and triangular).
• To show that Minesweeper is PSPACE-complete under conditions of infinitesimal success probability.
• To construct a Minesweeper board that simulates a boolean circuit with probabilistic and deterministic components, enabling reductions from known hard problems.

Proposed method

• Constructs a wire mechanism on Minesweeper grids that transmits boolean values (0 or 1) via mine placements, where only one of two complementary sets of cells contains mines.
• Designs custom logic components: curves to bend wires, splitters to duplicate signals, phase-shifters, and invertors, adapted for square, hexagonal, and triangular grids.
• Develops a crossover mechanism using splitters and adders modulo 2 to enable wire crossing without interference, crucial for circuit construction.
• Introduces a probabilistic reduction framework where the player must guess initial values (e.g., s2=1 or s3=1) to maximize success probability, simulating quantified boolean formulas.
• Uses a reversed splitter to reveal variables zij when s2 or s3 is known, enabling conditional revelation of variables based on circuit state.
• Reduces QBF and weak nonalternating SSAT to Minesweeper by encoding the maximum satisfiability probability of a boolean formula into the success probability of mine detection.

Experimental results

Research questions

• RQ1 Is Minesweeper PP-hard when the objective is to locate all mines with the highest possible probability?
• RQ2 Is Minesweeper PSPACE-complete when the probability of correctly locating all mines is infinitesimally small?
• RQ3 Can Minesweeper boards on non-square regular grids (hexagonal and triangular) simulate boolean circuits and thus be NP-complete in solvability?
• RQ4 Can a Minesweeper board be constructed such that the probability of winning depends on the maximum probability of satisfying a boolean formula?
• RQ5 Does the presence of a mine counter provide additional computational power, or can the total number of mines be computed in polynomial time?

Key findings

• Minesweeper is PP-hard when the goal is to maximize the probability of locating all mines, even when that probability is infinitesimally small.
• Minesweeper is PSPACE-complete under the same condition, as the success probability can be reduced to 2−e(1 − 2−m) for e ≥ 1, which becomes negligible as e increases.
• The construction ensures the total number of mines can be computed in polynomial time, so the mine counter does not provide additional information.
• The probability of winning is 2−e when s3 = 1 is guessed correctly, and 2−e(1/2 + 1/2 × maxϑ Pr[f] = ϑ) when s2 = 1 is guessed.
• The method successfully reduces QBF and weak nonalternating SSAT to Minesweeper, proving PSPACE-completeness via probabilistic circuit simulation.
• The authors construct functional circuit components (wires, splitters, invertors, crossovers) for square, hexagonal, and triangular grids, extending Kaye’s original NP-completeness result to all three regular tiling types.