Hexasort -- The Complexity of Stacking Colors on Graphs

TL;DR

Hexasort, a stacking game involving merging colored stacks on graphs, is proven to be NP-hard even in restricted cases, but certain settings allow for polynomial-time solutions using dynamic programming.

Contribution

The paper establishes the NP-hardness of Hexasort on various graph classes and develops fixed-parameter tractable algorithms for specific cases.

Findings

Hexasort is NP-hard on trees of bounded height or degree.

Polynomial-time algorithms are possible in certain constrained settings.

The problem's complexity varies with graph structure and constraints.

Abstract

Many popular puzzle and matching games have been analyzed through the lens of computational complexity. Prominent examples include Sudoku, Candy Crush, and Flood-It. A common theme among these widely played games is that their generalized decision versions are NP-hard, which is often thought of as a source of their inherent difficulty and addictive appeal to human players. In this paper, we study a popular single-player stacking game commonly known as Hexasort. The game can be modelled as placing colored stacks onto the vertices of a graph, where adjacent stacks of the same color merge and vanish according to deterministic rules. We prove that Hexasort is NP-hard, even when restricted to single-color stacks and progressively more constrained classes of graphs, culminating in strong NP-hardness on trees of either bounded height or degree. Towards fixed-parameter tractable algorithms, we identify settings in which the problem becomes polynomial-time solvable and present dynamic programming algorithms.