The building puzzle is still hard even in the single lined version

Abstract

The Building puzzle (a.k.a., the Skyscraper) is a Latin square completion-type puzzle like Sudoku, KenKen and Futoshiki. Recently, Iwamoto and Matsui showed the NP-completeness of the decision problem version of this puzzle, which asks whether a given instance has a solution or not. We provide a stronger result in the present paper; it is still NP-complete to decide whether we can complete a single line of the grid (i.e., a 1 × n or an n × 1 subgrid) without violating the rule.