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89
Breaking row and column symmetries in matrix models
 Proceedings of Eighth International Conference on Principles and Practice of Constraint Programming (CPO2
, 2002
"... Abstract. We identify an important class of symmetries in constraint programming, arising from matrices of decision variables where rows and columns can be swapped. Whilst lexicographically ordering the rows (columns) breaks all the row (column) symmetries, lexicographically ordering both the rows ..."
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Cited by 115 (37 self)
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Abstract. We identify an important class of symmetries in constraint programming, arising from matrices of decision variables where rows and columns can be swapped. Whilst lexicographically ordering the rows (columns) breaks all the row (column) symmetries, lexicographically ordering both the rows and the columns fails to break all the compositions of the row and column symmetries. Nevertheless, our experimental results show that this is effective at dealing with these compositions of symmetries. We extend these results to cope with symmetries in any number of dimensions, with partial symmetries, and with symmetric values. Finally, we identify special cases where all compositions of the row and column symmetries can be eliminated by the addition of only a linear number of symmetrybreaking constraints. 1
Groups and Constraints: Symmetry Breaking during Search
 In Proceedings of CP02, LNCS 2470
, 2002
"... We present an interface between the ECL constraint logic programming system and the GAPcompu tational abstract algebra system. The interface provides a method for e#ciently dealing with large nu mbers of symmetries of constraint satisfaction problems for minimal programming e#ort. We als ..."
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Cited by 61 (14 self)
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We present an interface between the ECL constraint logic programming system and the GAPcompu tational abstract algebra system. The interface provides a method for e#ciently dealing with large nu mbers of symmetries of constraint satisfaction problems for minimal programming e#ort. We also report an implementation of SBDSu sing the GAPECL interface which is capable of handling many more symmetries than previou s implementations and provides improved search performance for symmetric constraint satisfaction problems.
Satisfiability Solvers
, 2008
"... The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worstcase exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and h ..."
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Cited by 50 (0 self)
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The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worstcase exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and hardware verification [29–31, 228], automatic test pattern generation [138, 221], planning [129, 197], scheduling [103], and even challenging problems from algebra [238]. Annual SAT competitions have led to the development of dozens of clever implementations of such solvers [e.g. 13,
Exploiting symmetries within constraint satisfaction search
, 2001
"... Symmetry often appears in realworld constraint satisfaction problems, but strategies for exploiting it are only beginning to be developed. Here, a framework for exploiting symmetry within depthfirst search is proposed, leading to two heuristics for variable selection and a domain pruning procedure ..."
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Cited by 44 (1 self)
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Symmetry often appears in realworld constraint satisfaction problems, but strategies for exploiting it are only beginning to be developed. Here, a framework for exploiting symmetry within depthfirst search is proposed, leading to two heuristics for variable selection and a domain pruning procedure. These strategies are then applied to two highly symmetric combinatorial problems, namely the Ramsey problem and the generation of balanced incomplete block designs. Experimental results show that these generalpurpose strategies can compete with, and in some cases outperform, previous more ad hoc procedures.
Propositional Satisfiability and Constraint Programming: a Comparative Survey
 ACM Computing Surveys
, 2006
"... Propositional Satisfiability (SAT) and Constraint Programming (CP) have developed as two relatively independent threads of research, crossfertilising occasionally. These two approaches to problem solving have a lot in common, as evidenced by similar ideas underlying the branch and prune algorithms ..."
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Cited by 38 (4 self)
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Propositional Satisfiability (SAT) and Constraint Programming (CP) have developed as two relatively independent threads of research, crossfertilising occasionally. These two approaches to problem solving have a lot in common, as evidenced by similar ideas underlying the branch and prune algorithms that are most successful at solving both kinds of problems. They also exhibit differences in the way they are used to state and solve problems, since SAT’s approach is in general a blackbox approach, while CP aims at being tunable and programmable. This survey overviews the two areas in a comparative way, emphasising the similarities and differences between the two and the points where we feel that one technology can benefit from ideas or experience acquired
Generic SBDD using computational group theory
 In Proceedings of CP’03
, 2003
"... Abstract. We introduce a novel approach for symmetry breaking by dominance detection (SBDD). The essence of SBDD is to perform ‘dominance checks ’ at each node in a search tree to ensure that no symmetrically equivalent node has been visited before. While a highly effective technique for dealing wit ..."
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Cited by 34 (9 self)
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Abstract. We introduce a novel approach for symmetry breaking by dominance detection (SBDD). The essence of SBDD is to perform ‘dominance checks ’ at each node in a search tree to ensure that no symmetrically equivalent node has been visited before. While a highly effective technique for dealing with symmetry in constraint programs, SBDD forces a major overhead on the programmer, of writing a dominance checker for each new problem to be solved. Our novelty here is an entirely generic dominance checker. This in itself is new, as are the algorithms to implement it. It can be used for any symmetry group arising in a constraint program. A constraint programmer using our system merely has to define a small number (typically 2–6) of generating symmetries, and our system detects and breaks all resulting symmetries. Our dominance checker also performs some propagation, again generically, so that values are removed from variables if setting them would lead to a successful dominance check. We have implemented this generic SBDD and report results on its use. Our implementation easily handles problems involving 10 36 symmetries, with only four permutations needed to direct the dominance checks during search. 1
Algorithms for quantified constraint satisfaction problems
 In Proceedings of CP2004
, 2004
"... Abstract. Many propagation and search algorithms have been developed for constraint satisfaction problems (CSPs). In a standard CSP all variables are existentially quantified. The CSP formalism can be extended to allow universally quantified variables, in which case the complexity of the basic reaso ..."
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Cited by 21 (2 self)
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Abstract. Many propagation and search algorithms have been developed for constraint satisfaction problems (CSPs). In a standard CSP all variables are existentially quantified. The CSP formalism can be extended to allow universally quantified variables, in which case the complexity of the basic reasoning tasks rises from NPcomplete to PSPACEcomplete. Such problems have, so far, been studied mainly in the context of quantified Boolean formulae. Little work has been done on problems with discrete nonBoolean domains. We attempt to fill this gap by extending propagation and search algorithms from standard CSPs to the quantified case. We also show how the notion of value interchangeability can be exploited to break symmetries and speed up search by orders of magnitude. Finally, we test experimentally the algorithms and methods proposed. 1
Partial Symmetry Breaking
 In Proceedings of CP02, LNCS 2470
, 2002
"... In this paper we de ne partial symmetry breaking, a concept that has been used in many previous papers without being the main topic of any research. This paper is the rst systematic study of partial symmetry breaking in constraint programming. We show experimentally that performing symmetry br ..."
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Cited by 21 (2 self)
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In this paper we de ne partial symmetry breaking, a concept that has been used in many previous papers without being the main topic of any research. This paper is the rst systematic study of partial symmetry breaking in constraint programming. We show experimentally that performing symmetry breaking with only a subset of all symmetries can result in greatly reduced runtimes. We also look at the consequences of using partial symmetry breaking in terms of variable and value ordering heuristics. Finally, dierent methods of selecting symmetries are considered before presenting a general algorithm for selecting subsets of symmetries.
Modelling and Solving English Peg Solitaire
, 2003
"... Peg Solitaire is a well known puzzle, which can prove difficult despite its simple rules. Pegs are arranged on a board such that at least one ‘hole’ remains. By making draughts/checkerslike moves, pegs are gradually removed until no further moves are possible or some goal configuration is achieved. ..."
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Cited by 20 (9 self)
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Peg Solitaire is a well known puzzle, which can prove difficult despite its simple rules. Pegs are arranged on a board such that at least one ‘hole’ remains. By making draughts/checkerslike moves, pegs are gradually removed until no further moves are possible or some goal configuration is achieved. This paper considers the English variant, consisting of a board in a cross shape with 33 holes. Modelling Peg Solitaire via constraint or integer programming techniques presents a considerable challenge and is examined in detail. The merits of the resulting models are discussed and they are compared empirically. The sequential nature of the puzzle naturally conforms to a planning problem, hence we also present an experimental comparison with several leading AI planning systems. Other variants of the puzzle, such as ‘Fool’s Solitaire’ and ‘Longhop’ Solitaire are also considered.
Breaking Symmetry of Interchangeable Variables and Values
"... A common type of symmetry is when both variables and values partition into interchangeable sets. Polynomial methods have been introduced to eliminate all symmetric solutions introduced by such interchangeability. Unfortunately, whilst eliminating all symmetric solutions is tractable in this case, p ..."
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Cited by 18 (13 self)
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A common type of symmetry is when both variables and values partition into interchangeable sets. Polynomial methods have been introduced to eliminate all symmetric solutions introduced by such interchangeability. Unfortunately, whilst eliminating all symmetric solutions is tractable in this case, pruning all symmetric values is NPhard. We introduce a new global constraint called SIGLEX and its GAC propagator for pruning some (but not necessarily all) symmetric values. We also investigate how different postings of the SIGLEX constraints affect the pruning performance during constraint solving. Finally, we test these static symmetry breaking constraints experimentally for the first time.