Complexity of minesweeper with restricted number

Abstract

We consider the Minesweeper consistency problem (is a given partially completed board consistent with some mine placement?) when the set of numbers that may appear on a Minesweeper board is restricted. First, we analyze the possible sets of numbers that could exist in legal rectangular Minesweeper boards, proving either possibility or impossibility for 509 of the 512 subsets of {0, 1, 2, 3, 4, 5, 6, 7, 8} (leaving 3 subsets as open problems), and thus make conclusions on the relations among some restricted-set Minesweeper consistency problems. We prove either inclusion in P or NP-completeness for the restricted-set Minesweeper consistency problem for 134 of the 512 subsets of the set of numbers above. In particular, we show that {0,1}-Minesweeper consistency is NP-complete, while {0}-Minesweeper consistency and {1}-Minesweeper consistency are in P. We also suggest a few more dimensions in which the Minesweeper consistency problem could be analyzed.