Art Gallery Problem with Rook and Queen Vision

How many chess rooks or queens does it take to guard all the squares of a given polyomino, the union of square tiles from a square grid? This question is a version of the art gallery problem in which the guards can "see" whichever squares the rook or queen attacks. We show that floor(n/2) rooks or floor(n/3) queens are sufficient and sometimes necessary to guard a polyomino with n tiles. We also prove that finding the minimum number of rooks or the minimum number of queens needed to guard a polyomino is NP-hard. These results also apply to d-dimensional rooks and queens on d-dimensional polycubes. We also use bipartite matching theorems to describe sets of non-attacking rooks on polyominoes.

[1]  Joseph S. B. Mitchell,et al.  The Art Gallery Theorem for Polyominoes , 2012, Discret. Comput. Geom..

[2]  J. Mark Keil,et al.  Polygon Decomposition and the Orthogonal Art Gallery Problem , 2007, Int. J. Comput. Geom. Appl..

[3]  E. J. COCKAYNE,et al.  Chessboard domination problems , 1991, Discret. Math..

[4]  J. O'Rourke.  Art gallery theorems and algorithms , 1987 .

[5]  Ian P. Gent,et al.  Complexity of n-Queens Completion , 2017, J. Artif. Intell. Res..

[6]  Martin Nöllenburg,et al.  Using ILP/SAT to determine pathwidth, visibility representations, and other grid-based graph drawings , 2013, Graph Drawing.

[7]  Yahya Ould Hamidoune,et al.  Representing a planar graph by vertical lines joining different levels , 1983, Discret. Math..

[8]  Val Pinciu,et al.  Guarding Polyominoes, Polycubes and Polyhypercubes , 2015, Electron. Notes Discret. Math..

[9]  Valentin Gledel,et al.  A generalization of Arc-Kayles , 2019, Int. J. Game Theory.

[10]  Bruce A. Reed,et al.  Partition into cliques for cubic graphs: Planar case, complexity and approximation , 2008, Discret. Appl. Math..

[11]  László Lovász,et al.  Chessboard Complexes and Matching Complexes , 1994 .

[12]  Chuzo Iwamoto,et al.  Computational Complexity of the r -visibility Guard Set Problem for Polyominoes , 2013, JCDCGG.

[13]  Roberto Tamassia,et al.  A unified approach to visibility representations of planar graphs , 1986, Discret. Comput. Geom..

[14]  Simeon C. Ntafos,et al.  On Gallery Watchmen in Grids , 1986, Inf. Process. Lett..

[15]  D. T. Lee,et al.  Computational complexity of art gallery problems , 1986, IEEE Trans. Inf. Theory.

[16]  Brett Stevens,et al.  A survey of known results and research areas for n-queens , 2009, Discret. Math..

[17]  Charles M. Grinstead,et al.  On the queen domination problem , 1990, Discret. Math..

[18]  V. Chvátal.  A combinatorial theorem in plane geometry , 1975 .

[19]  Frédéric Maire,et al.  Polyominos and Perfect Graphs , 1994, Inf. Process. Lett..