Nonogram: Complexity of Inference and Phase Transition Behavior

Aaron Foote; Danny Krizanc

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arXiv · 2507.07283

Nonogram: Complexity of Inference and Phase Transition Behavior

Aaron Foote·Danny Krizanc

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Abstract

Nonogram is a popular combinatorial puzzle (similar in nature to Sudoku or Minesweeper) in which a puzzle solver must determine if there exists a setting of the puzzle parameters that satisfy a given set of constraints. It has long been known that the problem of deciding if a solution exists is a computationally difficult problem. Despite this fact, humans still seem to enjoy playing it. This work aims to reconcile these seemingly contradictory facts by (1) analyzing the complexity of the inference problem for Nonogram (the problem of determining if there exists a puzzle parameter that can be inferred from the constraints without guessing) and (2) experimentally establishing the existence of a phase transition behavior for this inference problem. Our results show that the difficulty of the inference problem is largely determined by the density of filled cells (positive parameters) in a given puzzle. Along the way we implement an efficient encoding of a Nonogram board as a Boolean formula in Conjunctive Normal Form (CNF) through the use of regular expressions in order to make our experiments feasible.

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Aaron Foote, Danny Krizanc. 2025-07-09. Nonogram: Complexity of Inference and Phase Transition Behavior. https://arxiv.org/abs/2507.07283

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