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Preprocessing of Min Ones Problems: A Dichotomy
"... Min Ones Constraint Satisfaction Problems, i.e., the task of finding a satisfying assignment with at most k true variables (Min Ones SAT(Γ)), can express a number of interesting and natural problems. We study the preprocessing properties of this class of problems with respect to k, using the notion ..."
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Min Ones Constraint Satisfaction Problems, i.e., the task of finding a satisfying assignment with at most k true variables (Min Ones SAT(Γ)), can express a number of interesting and natural problems. We study the preprocessing properties of this class of problems with respect to k, using the notion of kernelization to capture the viability of preprocessing. We give a dichotomy of Min Ones SAT(Γ) problems into admitting or not admitting a kernelization with size guarantee polynomial in k, based on the constraint language Γ. We introduce a property of boolean relations called mergeability that can be easily checked for any Γ. When all relations in Γ are mergeable, then we show a polynomial kernelization for Min Ones SAT(Γ). Otherwise, any Γ containing a nonmergeable relation and such that Min Ones SAT(Γ) is NPcomplete permits us to prove that Min Ones SAT(Γ) does not admit a polynomial kernelization unless NP ⊆ coNP/poly, by a reduction from a particular parameterization of Exact Hitting Set.
THE COMPLEXITY OF GLOBAL CARDINALITY CONSTRAINTS
"... ABSTRACT. In a constraint satisfaction problem (CSP) the goal is to find an assignment of a given set of variables subject to specified constraints. A global cardinality constraint is an additional requirement that prescribes how many variables must be assigned a certain value. We study the complexi ..."
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ABSTRACT. In a constraint satisfaction problem (CSP) the goal is to find an assignment of a given set of variables subject to specified constraints. A global cardinality constraint is an additional requirement that prescribes how many variables must be assigned a certain value. We study the complexity of the problem CCSP(Γ), the constraint satisfaction problem with global cardinality constraints that allows only relations from the set Γ. The main result of this paper characterizes sets Γ that give rise to problems solvable in polynomial time, and states that the remaining such problems are NPcomplete. 1.
Constraint Satisfaction problems and global cardinality constraints
"... In a constraint satisfaction problem (CSP) the goal is to find an assignment of a given set of variables subject to specified constraints. A global cardinality constraint is an additional requirement that prescribes how many variables must be assigned a certain value. We study the complexity of the ..."
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In a constraint satisfaction problem (CSP) the goal is to find an assignment of a given set of variables subject to specified constraints. A global cardinality constraint is an additional requirement that prescribes how many variables must be assigned a certain value. We study the complexity of the problem CCSP(Γ), the constraint satisfaction problem with global cardinality constraints that allows only relations from the set Γ. The main result of this paper characterizes sets Γ that give rise to problems solvable in polynomial time, and states that the remaining such problems are NPcomplete.
Minimization for Generalized Boolean Formulas
 PROCEEDINGS OF THE TWENTYSECOND INTERNATIONAL JOINT CONFERENCE ON ARTIFICIAL INTELLIGENCE
"... The minimization problem for propositional formulas is an important optimization problem in the second level of the polynomial hierarchy. In general, the problem is Σ p 2complete under Turing reductions, but restricted versions are tractable. We study the complexity of minimization for formulas in ..."
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The minimization problem for propositional formulas is an important optimization problem in the second level of the polynomial hierarchy. In general, the problem is Σ p 2complete under Turing reductions, but restricted versions are tractable. We study the complexity of minimization for formulas in two established frameworks for restricted propositional logic: The Post framework allowing arbitrarily nested formulas over a set of Boolean connectors, and the constraint setting, allowing generalizations of CNF formulas. In the Post case, we obtain a dichotomy result: Minimization is solvable in polynomial time or coNPhard. This result also applies to Boolean circuits. For CNF formulas, we obtain new minimization algorithms for a large class of formulas, and give strong evidence that we have covered all polynomialtime cases.