Zig-Zag Numberlink is NP-Complete

TL;DR

This paper proves that the Zig-Zag Numberlink puzzle, involving connecting terminal pairs with disjoint paths covering all grid vertices, is NP-complete, establishing its computational difficulty.

Contribution

It demonstrates NP-completeness of Zig-Zag Numberlink, a variant of Numberlink, expanding understanding of its computational complexity.

Findings

Proves NP-completeness of Zig-Zag Numberlink.

Connects the problem to previous NP-hardness results.

Highlights the problem's relation to popular puzzles and applications.

Abstract

When can $t$ terminal pairs in an $m \times n$ grid be connected by $t$ vertex-disjoint paths that cover all vertices of the grid? We prove that this problem is NP-complete. Our hardness result can be compared to two previous NP-hardness proofs: Lynch's 1975 proof without the ``cover all vertices'' constraint, and Kotsuma and Takenaga's 2010 proof when the paths are restricted to have the fewest possible corners within their homotopy class. The latter restriction is a common form of the famous Nikoli puzzle \emph{Numberlink}; our problem is another common form of Numberlink, sometimes called \emph{Zig-Zag Numberlink} and popularized by the smartphone app \emph{Flow Free}.