Volume 3011 of the series Lecture Notes in Computer Science pp 271286
Models and Symmetry Breaking for ‘Peaceable Armies of Queens’
 Barbara M. SmithAffiliated withSchool of Computing & Engineering, University of Huddersfield
 , Karen E. PetrieAffiliated withSchool of Computing & Engineering, University of Huddersfield
 , Ian P. GentAffiliated withSchool of Computer Science, University of St. Andrews
Abstract
We discuss a difficult optimization problem on a chessboard, 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 nonisomorphic 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.
 Title
 Models and Symmetry Breaking for ‘Peaceable Armies of Queens’
 Book Title
 Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems
 Book Subtitle
 First International Conference, CPAIOR 2004, Nice, France, April 2022, 2004. Proceedings
 Pages
 pp 271286
 Copyright
 2004
 DOI
 10.1007/9783540246640_19
 Print ISBN
 9783540218364
 Online ISBN
 9783540246640
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 3011
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
 Additional Links
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 Editors

 JeanCharles Régin ^{(12)}
 Michel Rueher ^{(13)}
 Editor Affiliations

 12. ILOG Sophia Antipolis, Les Taissounières HB2
 13. Université de NiceSophia Antipolis, I3SCNRS
 Authors

 Barbara M. Smith ^{(14)}
 Karen E. Petrie ^{(14)}
 Ian P. Gent ^{(15)}
 Author Affiliations

 14. School of Computing & Engineering, University of Huddersfield, Huddersfield, West Yorkshire, HD1 3DH, UK
 15. School of Computer Science, University of St. Andrews, St Andrews, Fife, KY16 9SS, UK
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