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Hashiwokakero, or simply Hashi, is a Japanese single-player puzzle played ona rectangular grid with no standard size. Some cells of the grid contain acircle, called island, with a number inside it ranging from one to eight. Theremaining positions of the grid are empty. The player must connect all of theislands by drawing a series of horizontal or vertical bridges between them,respecting a series of rules: the number of bridges incident to an islandequals the number indicated in the circle, at most two bridges are incident toany 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 thispaper, we present some complexity results and relationships between Hashi andwell-known graph theory problems. We give a formulation of the problem by meansof an integer linear mathematical programming model, and apply a branch-and-cutalgorithm to solve the model in which connectivity constraints are dynamicallygenerated. We also develop a puzzle generator. Our experiments on 1440 Hashipuzzles show that the algorithm can consistently solve hard puzzles with up to400 islands.

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