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The power of linear programming for valued CSPs
, 2012
"... A class of valued constraint satisfaction problems (VCSPs) is characterised by a valued constraint language, a fixed set of cost functions on a finite domain. An instance of the problem is specified by a sum of cost functions from the language with the goal to minimise the sum. This framework includ ..."
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Cited by 21 (6 self)
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A class of valued constraint satisfaction problems (VCSPs) is characterised by a valued constraint language, a fixed set of cost functions on a finite domain. An instance of the problem is specified by a sum of cost functions from the language with the goal to minimise the sum. This framework includes and generalises well-studied constraint satisfaction problems (CSPs) and maximum constraint satisfaction problems (Max-CSPs). Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation. Using this result, we obtain tractability of several novel and previously widely-open classes of VCSPs, including problems over valued constraint languages that are: (1) submodular on arbitrary lattices; (2) bisubmodular (also known as k-submodular) on arbitrary finite domains; (3) weakly (and hence strongly) treesubmodular on arbitrary trees.
The complexity of finite-valued CSPs
- Institute of Informatics, University of Warsaw, Poland
, 2013
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Relatively Quantified Constraint Satisfaction
"... The constraint satisfaction problem (CSP) is a convenient framework for modelling search problems; the CSP involves deciding, given a set of constraints on variables, whether or not there is an assignment to the variables satisfying all of the constraints. This paper is concerned with the more gener ..."
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Cited by 5 (2 self)
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The constraint satisfaction problem (CSP) is a convenient framework for modelling search problems; the CSP involves deciding, given a set of constraints on variables, whether or not there is an assignment to the variables satisfying all of the constraints. This paper is concerned with the more general framework of quantified constraint satisfaction, in which variables can be quantified both universally and existentially. We study the relatively quantified constraint satisfaction problem (RQCSP), in which the values for each individual variable can be arbitrarily restricted. We give a complete complexity classification of the cases of the RQCSP where the types of constraints that may appear are specified by a constraint language.
On the definition of suitable orderings to generate adjunctions over an unstructured codomain
, 2014
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The Complexity of Finite-Valued CSPs (Extended Abstract)
, 2013
"... Let Γ be a set of rational-valued functions on a fixed finite do-main; such a set is called a finite-valued constraint language. The valued constraint satisfaction problem, VCSP(Γ), is the problem of minimising a function given as a sum of functions from Γ. We establish a dichotomy theorem with resp ..."
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Let Γ be a set of rational-valued functions on a fixed finite do-main; such a set is called a finite-valued constraint language. The valued constraint satisfaction problem, VCSP(Γ), is the problem of minimising a function given as a sum of functions from Γ. We establish a dichotomy theorem with respect to exact solvability for all finite-valued languages defined on domains of arbitrary finite size. We show that every core language Γ either admits a bi-nary idempotent and symmetric fractional polymorphism in which case the basic linear programming relaxation solves any instance of VCSP(Γ) exactly, or Γ satisfies a simple hardness condition that allows for a polynomial-time reduc-tion from Max-Cut to VCSP(Γ). In other words, there is a single algorithm for all tractable cases and a single reason
On Galois Connections and Soft Computing
, 2013
"... After recalling the different interpretations usually assigned to the term Galois connection, both in the crisp and in the fuzzy case, we survey on several of their applications in Computer Science and, specifically, in Soft Computing. ..."
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After recalling the different interpretations usually assigned to the term Galois connection, both in the crisp and in the fuzzy case, we survey on several of their applications in Computer Science and, specifically, in Soft Computing.
Min CSP on Four Elements: Moving Beyond
"... We report new results on the complexity of the valued constraint satisfaction problem (VCSP). Under the unique games conjecture, the approximability of finite-valued VCSP is fairly well-understood. However, there is yet no characterisation of VCSPs that can be solved exactly in polynomial time. Th ..."
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We report new results on the complexity of the valued constraint satisfaction problem (VCSP). Under the unique games conjecture, the approximability of finite-valued VCSP is fairly well-understood. However, there is yet no characterisation of VCSPs that can be solved exactly in polynomial time. This is unsatisfactory, since such results are interesting from a combinatorial optimisation perspective; there are deep connections with, for instance, submodular and bisubmodular minimisation. We consider the Min and Max CSP problems (i.e. where the cost functions only attain values in {0, 1}) over four-element domains and identify all tractable fragments. Similar classi cations were previously known for two- and three-element domains. In the process, we introduce a new class of tractable VCSPs based on a generalisation of submodularity. We also extend and modify a graph-based technique by Kolmogorov and šivný (originally introduced by Takhanov) for e ciently obtaining hardness results in our setting. This allow us to prove the result without relying on computer-assisted case analyses (which is fairly common when studying VCSPs). The hardness results are further simpli ed by the introduction of powerful reduction techniques.
On fuzzy preordered sets and monotone Galois connections
"... Abstract—In this work, we focus on the study of necessary and sufficient conditions in order to ensure the existence (under some constraints) of monotone Galois connections between fuzzy preordered sets. ..."
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Abstract—In this work, we focus on the study of necessary and sufficient conditions in order to ensure the existence (under some constraints) of monotone Galois connections between fuzzy preordered sets.
