The Maze Generation Problem is NP-complete

The Maze Generation problem has been presented as a benchmark for the Second Answer Set Programming Competition. We prove that the problem is NP–complete and identify relevant classes of unsatisfiable instances. The Maze Generation (MG) problem has been presented as a benchmark for the Second Answer Set Programming (ASP) Competition1. The problem has been placed in the NP category, which contains problems in NP for which an algorithm in P is unknown. Here, we prove that the problem is in fact complete for the complexity class NP. In addition, we identify classes of unsatisfiable instances that are efficiently recognizable. We start the discussion by defining a maze. Definition 1 (Maze). A maze is an m × n grid, in which each cell is empty or a wall and two distinct cells on the edges are indicated as entrance and exit, satisfying the following conditions: (1) Each cell in an edge of the grid is a wall, except entrance and exit that are empty; (2) There is no 2 × 2 square of empty cells or walls; (3) If two walls are on a diagonal of a 2× 2 square, then not both of their common neighbors are empty; (4) No wall is completely surrounded by empty cells; (5) There is a path from the entrance to every empty cell. The MG problem is the decision problem concerning the possibility to build a maze by extending a partially fixed grid. Definition 2 (Maze Generation problem). An instance of the MG problem is a structure of the form 〈G, I,O,W,E〉, where G is a set of cells (pairs of integers) representing a grid, I and O are two distinct cells of G, W and E are subsets of G. The MG problem is then defined as follows: Given a structure 〈G, I,O,W,E〉, is there a maze of the same size of G such that I and O are entrance and exit, respectively, each cell in W is a wall and each cell in E is empty?