Maximum independent queen set on polyominoes is NP-complete
2026-07-27 • Computational Complexity
Computational Complexity
AI summaryⓘ
The authors study a puzzle where you pick tiles on a chessboard shaped like connected squares (a polyomino) so that no two tiles can attack each other like queens do in chess. They prove that deciding if such a selection exists is a hard computer problem (NP-complete). They also show that counting how many such selections there are is even harder (#P-complete). Interestingly, they find that a similar counting problem for rooks, which is usually easier, becomes hard on these boards.
Independent setNP-completeQueen graphPolyominoParsimonious reduction#P-completeRook movesComputational complexityCounting problems
Authors
Alexis Langlois-Rémillard, Mia Müßig
Abstract
Finding a set of vertices in a graph with no edges between them, INDSET, is a well-known NP-complete problem. The queen graph of a chessboard is constructed by taking vertices as the tiles of the chessboard and drawing edges between two tiles if a queen can move from one to the other. We call INDQUEENS the independent set problem on a queen graph where the chessboard is a polyomino. We prove that INDQUEENS on polyominoes is NP-complete, proving a conjecture of Langlois-Rémillard--Müßig--Roldán. As our reduction is parsimonious, we can further prove that it is #P-complete. We furthermore prove that INDROOKS on polyominoes is #P-complete, despite being solvable in polynomial time.