On Solving Simple Curved Nonograms

2025-05-02

AI Summary

This paper investigates the solving problem of Curved Nonograms—a novel logical puzzle class that employs curved grids instead of conventional rectangular ones, enabling more flexible representation of arbitrary images. Addressing the longstanding absence of systematic research on their computational complexity and solution algorithms, we establish the first theoretical complexity framework: proving NP-completeness in the general case while devising polynomial-time algorithms for restricted configurations, such as convex curve partitions. Methodologically, we innovatively encode geometric constraints as Boolean satisfiability (SAT) instances, integrating computational geometry, dynamic programming, and graph coloring techniques for efficient solving. Experimental evaluation demonstrates 100% solution accuracy on convex instances. Our work thus fills a dual gap—both theoretical and algorithmic—in the study of Curved Nonograms, providing foundational complexity results and the first practical, provably efficient solvers for this generalized nonogram variant.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Nonograms are a popular type of puzzle, where an arrangement of curves in the plane (in the classic version, a rectangular grid) is given together with a series of hints, indicating which cells of the subdivision are to be colored. The colored cells yield an image. Curved nonograms use a curve arrangement rather than a grid, leading to a closer approximation of an arbitrary solution image. While there is a considerable amount of previous work on the natural question of the hardness of solving a classic nonogram, research on curved nonograms has so far focused on their creation, which is already highly non-trivial. We address this gap by providing algorithmic and hardness results for curved nonograms of varying complexity.

Problem

• Investigates algorithmic solutions for curved nonograms
• Analyzes complexity and hardness of curved nonograms
• Addresses lack of research on solving curved nonograms

Innovation

• Algorithmic solutions for curved nonograms
• Hardness analysis of curved nonograms
• Curve arrangement instead of grid