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Twentyfive comparators is optimal when sorting nine inputs (and twentynine for ten
, 2014
"... This paper describes a computerassisted nonexistence proof of 9input sorting networks consisting of 24 comparators, hence showing that the 25comparator sorting network found by Floyd in 1964 is optimal. As a corollary, we obtain that the 29comparator network found by Waksman in 1969 is optimal ..."
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This paper describes a computerassisted nonexistence proof of 9input sorting networks consisting of 24 comparators, hence showing that the 25comparator sorting network found by Floyd in 1964 is optimal. As a corollary, we obtain that the 29comparator network found by Waksman in 1969 is optimal when sorting 10 inputs. This closes the two smallest open instances of the optimalsize sorting network problem, which have been open since the results of Floyd and Knuth from 1966 proving optimality for sorting networks of up to 8 inputs. The proof involves a combination of two methodologies: one based on exploiting the abundance of symmetries in sorting networks, and the other based on an encoding of the problem to that of satisfiability of propositional logic. We illustrate that, while each of these can singlehandedly solve smaller instances of the problem, it is their combination that leads to the more efficient solution that scales to handle 9 inputs. 1
Backbones for Equality
"... Abstract. This paper generalizes the notion of the backbone of a CNF formula to capture also equations between literals. Each such equation applies to remove a variable from the original formula thus simplifying the formula without changing its satisfiability, or the number of its satisfying assignm ..."
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Abstract. This paper generalizes the notion of the backbone of a CNF formula to capture also equations between literals. Each such equation applies to remove a variable from the original formula thus simplifying the formula without changing its satisfiability, or the number of its satisfying assignments. We prove that for a formula with n variables, the generalized backbone is computed with at most n+1 satisfiable calls and exactly one unsatisfiable call to the SAT solver. We illustrate the integration of generalized backbone computation to facilitate the encoding of finite domain constraints to SAT. In this context generalized backbones are computed for small groups of constraints and then propagated to simplify the entire constraint model. A preliminary experimental evaluation is provided. 1
A Novel SATBased Approach to Model Based Diagnosis
"... This paper introduces a novel encoding of Model Based Diagnosis (MBD) to Boolean Satisfaction (SAT) focusing on minimal cardinality diagnosis. The encoding is based on a combination of sophisticated MBD preprocessing algorithms and the application of a SAT compiler which optimizes the encoding to ..."
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This paper introduces a novel encoding of Model Based Diagnosis (MBD) to Boolean Satisfaction (SAT) focusing on minimal cardinality diagnosis. The encoding is based on a combination of sophisticated MBD preprocessing algorithms and the application of a SAT compiler which optimizes the encoding to provide more succinct CNF representations than obtained with previous works. Experimental evidence indicates that our approach is superior to all published algorithms for minimal cardinality MBD. In particular, we can determine, for the first time, minimal cardinality diagnoses for the entire standard ISCAS85 and 74XXX benchmarks. Our results open the way to improve the stateoftheart on a range of similar MBD problems. 1.
Blocked Clause Decomposition
"... Abstract. We demonstrate that it is fairly easy to decompose any propositional formula into two subsets such that both can be solved by blocked clause elimination. Such a blocked clause decomposition is useful to cheaply detect backbone variables and equivalent literals. Blocked clause decompositio ..."
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Abstract. We demonstrate that it is fairly easy to decompose any propositional formula into two subsets such that both can be solved by blocked clause elimination. Such a blocked clause decomposition is useful to cheaply detect backbone variables and equivalent literals. Blocked clause decompositions are especially useful when they are unbalanced, i.e., one subset is much larger in size than the other one. We present algorithms and heuristics to obtain unbalanced decompositions efficiently. Our techniques have been implemented in the stateoftheart solver Lingeling. Experiments show that the performance of Lingeling is clearly improved due to these techniques on application benchmarks of the SAT Competition 2013. 1
unknown title
"... The MiniZinc Challenge compares different solvers on aset of MiniZinc models and problems instances. MiniZinc1 (Nethercote et al. 2007) was our response to the call for a standard constraintprogramming modeling language. MiniZinc is high level enough to express most combinatorial optimization pr ..."
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The MiniZinc Challenge compares different solvers on aset of MiniZinc models and problems instances. MiniZinc1 (Nethercote et al. 2007) was our response to the call for a standard constraintprogramming modeling language. MiniZinc is high level enough to express most combinatorial optimization problems easily and in a largely solverindependent way; for example, it supports sets, arrays, and userdefined predicates, some overloading, and some automatic coercions. However, MiniZinc is low level enough that it can be mapped easily onto many solvers. For example, it is first order, and it only supports decision variable types that are supported by most existing constraintprogramming solvers: integers, floats, Booleans, and sets of integers. MiniZinc also allows separation of a model from its data; provides a library containing declarative definitions of many global constraints; and has a system of annotations that allows nondeclarative information (such as search strategies) and solverspecific information (such as variable representations) to be layered on top of declarative models.
Aspartame: Solving Constraint Satisfaction Problems with Answer Set Programming
"... Abstract. Encoding finite linear CSPs as Boolean formulas and solving them by using modern SAT solvers has proven to be highly effective, as exemplified by the awardwinning sugar system. We here develop an alternative approach based on ASP. This allows us to use firstorder encodings providing us w ..."
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Abstract. Encoding finite linear CSPs as Boolean formulas and solving them by using modern SAT solvers has proven to be highly effective, as exemplified by the awardwinning sugar system. We here develop an alternative approach based on ASP. This allows us to use firstorder encodings providing us with a high degree of flexibility for easy experimentation with different implementations. The resulting system aspartame reuses parts of sugar for parsing and normalizing CSPs. The obtained set of facts is then combined with an ASP encoding that can be grounded and solved by offtheshelf ASP systems. We establish the competitiveness of our approach by empirically contrasting aspartame and sugar. 1
Breaking Symmetries in Graph Coloring Problems with Degree Matrices: the Ramsey Number R(4,3,3)=30?
"... Abstract. This paper introduces a general methodology that applies to solve graph edgecoloring problems and demonstrates its use to compute the Ramsey number R(4, 3, 3). The number R(4, 3, 3) is often presented as the unknown Ramsey number with the best chances of being found “soon”. Yet, its preci ..."
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Abstract. This paper introduces a general methodology that applies to solve graph edgecoloring problems and demonstrates its use to compute the Ramsey number R(4, 3, 3). The number R(4, 3, 3) is often presented as the unknown Ramsey number with the best chances of being found “soon”. Yet, its precise value has remained unknown for more than 50 years. The proposed technique is based on two wellstudied concepts, abstraction and symmetry. First, we introduce an abstraction on graph colorings, degree matrices, that specify the degree of each vertex in each color. We compute, using a SAT solver, an overapproximation of the set of degree matrices of all solutions of the graph coloring problem. Then, for each degree matrix in the overapproximation, we compute, again using a SAT solver, the set of all solutions with matching degrees. Breaking symmetries, on degree matrices in the first step and with respect to graph isomorphism in the second, is cardinal to the success of the approach. We illustrate the approach via two applications: proving that R(4, 3, 3) = 30 and computing the previously unknown number of (3, 3, 3; 13) Ramsey colorings (78,892). 1