## Symmetry breaking in constraint satisfaction with graph-isomorphism comma-free codes

Venue: | Proc. of AI&M 2004. 2004. http://www.informatik.uni-trier.de/~ley/db/conf/amai/amai2004.html |

Citations: | 3 - 0 self |

### BibTeX

@INPROCEEDINGS{Flener_symmetrybreaking,

author = {Pierre Flener and Justin Pearson and Meinolf Sellmann and Pascal Van Hentenryck},

title = {Symmetry breaking in constraint satisfaction with graph-isomorphism comma-free codes},

booktitle = {Proc. of AI&M 2004. 2004. http://www.informatik.uni-trier.de/~ley/db/conf/amai/amai2004.html},

year = {}

}

### OpenURL

### Abstract

Abstract. In recent years, symmetry breaking for constraint satisfaction problems (CSPs) has attracted considerable attention. Various general schemes have been proposed to eliminate symmetries. In general, these schemes may take exponential space or time to eliminate all the symmetries. We identify several classes of CSPs that encompass many practical problems and for which symmetry breaking for various forms of value or variable interchangeability is tractable using dedicated search procedures. We also show the limits of efficient symmetry breaking for such dominance-detection schemes by proving intractability results for some classes of CSPs. 1

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Citation Context ...ickly. Also since the symmetry groups in general would be large and constructing the graphs is quite simple this method avoids generating many no-goods as would be done with SBDS or its optimisations =-=[8]-=-. The technique of using graph isomorphism could also be applied to Balanced Incomplete Block Designs and the Social Golfer and this is work in progress. This work was partially supported by a STINT i... |

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Citation Context ...may drastically improve performance [3, 17, 20, 25]. An important contribution in this area has been the development of various general schemes for symmetry breaking during search in CSPs (e.g., SBDS =-=[2, 15]-=- and SBDD [9, 13, 20], the latter being described briefly in Section 3). Unfortunately, in general, these schemes may require exponential resources to break all the symmetries. Indeed, some schemes ma... |

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Citation Context ... a consequence, practical applications often place limits on how many nogoods can be stored and/or which symmetries to break. Other than eliminating symmetries by re-modelling the problem (see, e.g., =-=[24]-=-), another important approach is to break symmetries by adding constraints before search starts (e.g., [8, 19]). Unfortunately, in general, a super-exponential number of constraints may be needed to b... |

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Citation Context ...Our approach does not aim at discovering this fact; it rather focuses on how to exploit it to break the symmetries it induces. This theoretical paper, which unites and extends 3 our work published in =-=[27, 22]-=-, is structured as follows. First, in Section 2, we define CSPs and assignments 3 Sections 5.3 and 7.2 and the epilogue to Corollary 2 are new, while Section 5.2 was generalized. The originally omitte... |

24 | Structural symmetry breaking
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(Show Context)
Citation Context ...Our approach does not aim at discovering this fact; it rather focuses on how to exploit it to break the symmetries it induces. This theoretical paper, which unites and extends 3 our work published in =-=[27, 22]-=-, is structured as follows. First, in Section 2, we define CSPs and assignments 3 Sections 5.3 and 7.2 and the epilogue to Corollary 2 are new, while Section 5.2 was generalized. The originally omitte... |

21 |
A Generalized Implicit Enumeration Algorithm for Graph Coloring
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Citation Context ...eak value symmetry with minimal overhead. First, in Section 5.1, we describe our new approach in full detail on the class of fully value-interchangeable CSPs, showing how it leads to the known result =-=[16]-=- that all their value symmetries can be broken by a dedicated search procedure with a constant overhead with respect to both time and space at every node explored (Theorem 5). Then, in Section 5.2, we... |

21 | Constraint Techniques for Solving the Protein Structure Prediction Problem
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Citation Context ...gation of local information. Often propagation takes the form of local consistency methods (such are arc-consistency) or specialised propagation algorithms implemented as global-constraints. Recently =-=[9, 6, 2, 3]-=- there has been much interest using symmetries inherent in problems to speed up search. Essentially symmetrically equivalent branches in the search space are pruned. Often this is achieved by either i... |

20 | Matrix modelling - Flener, Frisch, et al. |

20 | Constraint and Integer Programming in OPL
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Citation Context ...g search to break symmetries efficiently. In [28], we address the companion issue of how to automatically detect symmetries in CSP models. Example 1. Consider the scene allocation problem featured in =-=[26]-=-. It aims at producing a movie (or a series) at minimal cost by deciding when to shoot which scenes. Each scene involves a number of actors and at most five scenes a day can be shot. All the actors of... |

14 | Static and dynamic structural symmetry breaking
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(Show Context)
Citation Context ...ominates β, which makes the dominance detection problem at least as hard as bipartite matching. In other words, we can show that dominance detection takes time T, where T ∈ Ω(M) ∩ O(M + m 2 + mn). In =-=[12]-=- we have provided a static counterpart of the here considered dynamic structural symmetry breaking for piecewise interchangeable CSPs, that is we have exploited the concept of signature to devise a se... |

12 |
L.R.: Comma-free codes
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Citation Context ...troduced, then Symmetry Breaking via Dominance Detection (SBDD) is outlined the required graph is then constructed and finally experimental results are presented. 2 Comma-Free Codes A comma-free code =-=[10, 11, 12] over -=-an alphabet A is a set, C ⊆ A ∗ ,ofwords over A such that given any two words, w, v ∈ C, any sub-word, u, ofthe concatenation, wv, is not in the code. Here we will be only interested in codes wh... |

8 |
Recent results in comma-free codes
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Citation Context ...troduced, then Symmetry Breaking via Dominance Detection (SBDD) is outlined the required graph is then constructed and finally experimental results are presented. 2 Comma-Free Codes A comma-free code =-=[10, 11, 12] over -=-an alphabet A is a set, C ⊆ A ∗ ,ofwords over A such that given any two words, w, v ∈ C, any sub-word, u, ofthe concatenation, wv, is not in the code. Here we will be only interested in codes wh... |

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(Show Context)
Citation Context ...any interesting classes of CSPs. This paper, in contrast, assumes that the symmetries in a CSP are known. It focuses on how to exploit this knowledge during search to break symmetries efficiently. In =-=[28]-=-, we address the companion issue of how to automatically detect symmetries in CSP models. Example 1. Consider the scene allocation problem featured in [26]. It aims at producing a movie (or a series) ... |

4 |
Permutation groups. Number 45
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(Show Context)
Citation Context ...xed value in D1, all values in D2 are interchangeable as well. These problems are here called wreath valueinterchangeable CSPs, because the symmetry group corresponds to a wreath 30sproduct of groups =-=[4]-=-. Such problems arise naturally in a variety of applications, e.g., in resource allocation and scheduling. Example 11. Consider the problem of scheduling a meeting where different groups must meet som... |

2 |
Comma-free codes
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(Show Context)
Citation Context ...s our labelling procedure for the set variables. Unfortunately, that global constraint is very efficient because it is able also to filter the domains while our labelling cannot do the same filtering =-=[18]-=-. Let us now return to the full interchangeability of the v varieties. Breaking these extra v! symmetries at the same time is hard, as they compose with the b! block symmetries into v! · b! symmetries... |

1 | Completing comma-free codes
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(Show Context)
Citation Context ...troduced, then Symmetry Breaking via Dominance Detection (SBDD) is outlined the required graph is then constructed and finally experimental results are presented. 2 Comma-Free Codes A comma-free code =-=[10, 11, 12] over -=-an alphabet A is a set, C ⊆ A ∗ ,ofwords over A such that given any two words, w, v ∈ C, any sub-word, u, ofthe concatenation, wv, is not in the code. Here we will be only interested in codes wh... |

1 |
nauty user's guide (version 2.2). Available via http://cs.anu.edu.au/people/bdm/nauty
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(Show Context)
Citation Context ...d for both the partial assignment and the no-good such that the graphs are isomorphic if and only if the assignments are symmetrically equivalent. Isomorphism of the graphs is checked using the nauty =-=[13]-=- system, which is able to return a canonical graph such that two graphs are 4sisomorphic if and only if they have the same canonical graph. Hence if the no-goods are stored as nauty canonical graphs t... |