Slant/Gokigen Naname is NP-complete, and Some Variations are in P

TL;DR

This paper proves that a generalized version of the Slant puzzle is NP-complete, but identifies certain variants solvable in polynomial time by analyzing grid graph connectivity and degree constraints.

Contribution

It establishes NP-completeness for a generalized Slant puzzle and provides polynomial algorithms for specific constrained variants.

Findings

Generalized Slant is NP-complete

Polynomial algorithms exist for certain constrained variants

Connectivity and degree constraints are key to solvability

Abstract

In this paper we show that a generalized version of the Nikoli puzzle Slant is NP-complete. We also give polynomial time algorithms for versions of the puzzle where some constraints are omitted. These problems correspond to simultaneously satisfying connectivity and vertex degree constraints in a grid graph and its dual.