Benchmark Instances and Branch-and-Cut Algorithm for the Hashiwokakero Puzzle

TL;DR

This paper introduces a new benchmark and a branch-and-cut algorithm for solving Hashiwokakero puzzles, demonstrating its effectiveness on large, complex instances with up to 400 islands.

Contribution

It provides a formal mathematical model and a dynamic constraint generation algorithm for Hashi puzzles, along with a new puzzle generator.

Findings

Algorithm solves puzzles with up to 400 islands.

Effective for hard puzzles, demonstrating scalability.

Provides complexity analysis and graph theory relationships.

Abstract

Hashiwokakero, or simply Hashi, is a Japanese single-player puzzle played on a rectangular grid with no standard size. Some cells of the grid contain a circle, called island, with a number inside it ranging from one to eight. The remaining positions of the grid are empty. The player must connect all of the islands by drawing a series of horizontal or vertical bridges between them, respecting a series of rules: the number of bridges incident to an island equals the number indicated in the circle, at most two bridges are incident to any side of an island, bridges cannot cross each other or pass through islands, and each island must eventually be reachable from any other island. In this paper, we present some complexity results and relationships between Hashi and well-known graph theory problems. We give a formulation of the problem by means of an integer linear mathematical programming model, and apply a branch-and-cut algorithm to solve the model in which connectivity constraints are dynamically generated. We also develop a puzzle generator. Our experiments on 1440 Hashi puzzles show that the algorithm can consistently solve hard puzzles with up to 400 islands.