Battleship, tomography and quantum annealing

• Published in: European journal of applied mathematics Vol. 34; no. 4; pp. 758 - 773
• Main Authors: Casper, W. Riley, Grimes, Taylor
• Format: Journal Article
• Language: English
• Published: 01.08.2023
• ISSN: 0956-7925; 1469-4425
• DOI: 10.1017/S0956792522000377

The classic game of Battleship involves two players taking turns attempting to guess the positions of a fleet of vertically or horizontally positioned enemy ships hidden on a $10\times 10$ grid. One variant of this game, also referred to as Battleship Solitaire, Bimaru or Yubotu, considers the game with the inclusion of X-ray data , represented by knowledge of how many spots are occupied in each row and column in the enemy board. This paper considers the Battleship puzzle problem : the problem of reconstructing an enemy fleet from its X-ray data. We generate non-unique solutions to Battleship puzzles via certain reflection transformations akin to Ryser interchanges. Furthermore, we demonstrate that solutions of Battleship puzzles may be reliably obtained by searching for solutions of the associated classical binary discrete tomography problem which minimise the discrete Laplacian. We reformulate this optimisation problem as a quadratic unconstrained binary optimisation problem and approximate solutions via a simulated annealer, emphasising the future practical applicability of quantum annealers to solving discrete tomography problems with predefined structure.