Models and Symmetry Breaking for ‘Peaceable Armies of Queens’

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Abstract

We discuss a difficult optimization problem on a chess-board, requiring equal numbers of black and white queens to be placed on the board so that the white queens cannot attack the black queens. We show how the symmetry of the problem can be straightforwardly eliminated using SBDS, allowing a set of non-isomorphic optimal solutions to be found. We present three different ways of modelling the problem in constraint programming, starting from a basic model. An improvement on this model reduces the number of constraints in the problem by introducing ancillary variables representing the lines on the board. The third model is based on the insight that only the white queens need be placed, so long as there are sufficient unattacked squares to accommodate the black queens. We also discuss variable ordering heuristics: we present a heuristic which finds optimal solutions very quickly but is poor at proving optimality, and the opposite heuristic for which the reverse is true. We suggest that in designing heuristics for optimization problems, the different requirements of the two tasks (finding an optimal solution and proving optimality) should be taken into account.