The complexity of any optimisation problem depends critically on the form of the objective functi... more The complexity of any optimisation problem depends critically on the form of the objective function. Valued constraint satisfaction problems are discrete optimisation problems where the function to be minimised is given as a sum of cost functions defined on specified subsets of variables. These cost functions are chosen from some fixed set of available cost functions, known as a valued constraint language. We show in this paper that when the costs are non-negative rational numbers or infinite, then the complexity of a valued constraint problem is determined by certain algebraic properties of this valued constraint language, which we call weighted polymorphisms. We define a Galois connection between valued constraint languages and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterised. These results provide a new approach in the search for tractable valued constraint languages.
The connection between the complexity of constraint languages and clone theory, discovered by Coh... more The connection between the complexity of constraint languages and clone theory, discovered by Cohen and Jeavons in a series of papers, has been a fruitful line of research on the complexity of CSPs. In a recent result, Cohen et al. have established a Galois connection between the complexity of valued constraint languages and so-called weighted clones. In this paper, we initiate the study of weighted clones. Firstly, we prove an analogue of Rosenberg's classification of minimal clones for weighted clones. Secondly, we show minimality of several weighted clones whose support clone is generated by a single minimal operation. Finally, we classify all Boolean weighted clones. This classification implies a complexity classification of Boolean valued constraint languages obtained by Cohen et al.
Discrete optimisation problems arise in many different areas and are studied under many different... more Discrete optimisation problems arise in many different areas and are studied under many different names. In many such problems the quantity to be optimised can be expressed as a sum of functions of a restricted form. Here we present a unifying theory of complexity for problems of this kind. We show that the complexity of a finite-domain discrete optimisation problem is determined by certain algebraic properties of the objective function, which we call weighted polymorphisms. We define a Galois connection between sets of rational-valued functions and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterised.
We consider the problem of sampling from the uniform distribution on the set of Eulerian orientat... more We consider the problem of sampling from the uniform distribution on the set of Eulerian orientations of subgraphs of the triangular lattice. Although it is known that this can be achieved in polynomial time for any graph, the algorithm studied here is more natural in the context of planar Eulerian graphs. We analyse the mixing time of a Markov chain on the Eulerian orientations of a planar graph which moves between orientations by reversing the edges of directed faces. Using path coupling and the comparison method we obtain a polynomial upper bound on the mixing time of this chain for any solid subgraph of the triangular lattice. By considering the conductance of the chain we show that there exist subgraphs with holes for which the chain will always take an exponential amount of time to converge. Finally, as an additional justification for studying a Markov chain on the set of Eulerian orientations of planar graphs, we show that the problem of counting Eulerian orientations remains #P-complete when restricted to planar graphs.
Principles and Practice of Constraint Programming–CP 2011, 2011
The connection between the complexity of constraint languages and clone theory, discovered by Coh... more The connection between the complexity of constraint languages and clone theory, discovered by Cohen and Jeavons in a series of papers, has been a fruitful line of research on the complexity of CSPs. In a recent result, Cohen et al. have established a Galois connection between the complexity of valued constraint languages and so-called weighted clones. In this paper, we initiate the study of weighted clones. Firstly, we prove an analogue of Rosenberg's classification of minimal clones for weighted clones. Secondly, we show minimality of several weighted clones whose support clone is generated by a single minimal operation. Finally, we classify all Boolean weighted clones. This classification implies a complexity classification of Boolean valued constraint languages obtained by Cohen et al.
Mathematical Foundations of Computer Science 2011, 2011
The complexity of any optimisation problem depends critically on the form of the objective functi... more The complexity of any optimisation problem depends critically on the form of the objective function. Valued constraint satisfaction problems are discrete optimisation problems where the function to be minimised is given as a sum of cost functions defined on specified subsets of variables. These cost functions are chosen from some fixed set of available cost functions, known as a valued constraint language. We show in this paper that when the costs are non-negative rational numbers or infinite, then the complexity of a valued constraint problem is determined by certain algebraic properties of this valued constraint language, which we call weighted polymorphisms. We define a Galois connection between valued constraint languages and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterised. These results provide a new approach in the search for tractable valued constraint languages.
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Papers by Páidí Creed