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29
The Complexity of Soft Constraint Satisfaction
, 2006
"... Over the past few years there has been considerable progress in methods to systematically analyse the complexity of constraint satisfaction problems with specified constraint types. One very powerful theoretical development in this area links the complexity of a set of constraints to a corresponding ..."
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Cited by 19 (8 self)
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Over the past few years there has been considerable progress in methods to systematically analyse the complexity of constraint satisfaction problems with specified constraint types. One very powerful theoretical development in this area links the complexity of a set of constraints to a corresponding set of algebraic operations, known as polymorphisms. In this paper we extend the analysis of complexity to the more general framework of combinatorial optimisation problems expressed using various forms of soft constraints. We launch a systematic investigation of the complexity of these problems by extending the notion of a polymorphism to a more general algebraic operation, which we call a multimorphism. We show that many tractable sets of soft constraints, both established and novel, can be characterised by the presence of particular multimorphisms. We also show that a simple set of NPhard constraints has very restricted multimorphisms. Finally, we use the notion of multimorphism to give a complete classification of complexity for the Boolean case which extends several earlier classification results for particular special cases.
Minimum Cost and List Homomorphisms to Semicomplete Digraphs
 Discrete Appl. Math
"... For digraphs D and H, a mapping f: V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). Let H be a fixed directed or undirected graph. The homomorphism problem for H asks whether a directed or undirected graph input digraph D admits a homomorphism to H. The list homomorphis ..."
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Cited by 18 (9 self)
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For digraphs D and H, a mapping f: V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). Let H be a fixed directed or undirected graph. The homomorphism problem for H asks whether a directed or undirected graph input digraph D admits a homomorphism to H. The list homomorphism problem for H is a generalization of the homomorphism problem for H, where every vertex x ∈ V (D) is assigned a set Lx of possible colors (vertices of H). The following optimization version of these decision problems was introduced in [16], where it was motivated by a realworld problem in defence logistics. Suppose we are given a pair of digraphs D, H and a positive cost ci(u) for each u ∈ V (D) and i ∈ V (H). The cost of a homomorphism f of D to H is � u∈V (D) cf(u)(u). For a fixed digraph H, the minimum cost homomorphism problem for H, MinHOMP(H), is stated as follows: For an input digraph D and costs ci(u) for each u ∈ V (D) and i ∈ V (H), verify whether there is a homomorphism of D to H and, if it exists, find
A dichotomy for minimum cost graph homomorphisms
 European J. Combin
, 2007
"... For graphs G and H, a mapping f: V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f(u)f(v) ∈ E(H). If, moreover, each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is � u∈V (G) cf(u)(u). For each fixed graph H, we have the minimum cost h ..."
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Cited by 13 (6 self)
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For graphs G and H, a mapping f: V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f(u)f(v) ∈ E(H). If, moreover, each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is � u∈V (G) cf(u)(u). For each fixed graph H, we have the minimum cost homomorphism problem, written as MinHOM(H). The problem is to decide, for an input graph G with costs ci(u), u ∈ V (G), i ∈ V (H), whether there exists a homomorphism of G to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems. We prove a dichotomy of the minimum cost homomorphism problems for graphs H, with loops allowed. When each connected component of H is either a reflexive proper interval graph or an irreflexive proper interval bigraph, the problem MinHOM(H) is polynomial time solvable. In all other cases the problem MinHOM(H) is NPhard. This solves an open problem from an earlier paper. 1
An algebraic characterisation of complexity for valued constraints
 In: Proceedings CP’06. Volume 4204 of Lecture Notes in Computer Science., SpringerVerlag
, 2006
"... Classical constraint satisfaction is concerned with the feasibility of satisfying a collection of constraints. The extension of this framework to include optimisation is now also being investigated and a theory of socalled soft constraints is being developed. In this extended framework, tuples of ..."
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Cited by 13 (5 self)
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Classical constraint satisfaction is concerned with the feasibility of satisfying a collection of constraints. The extension of this framework to include optimisation is now also being investigated and a theory of socalled soft constraints is being developed. In this extended framework, tuples of values allowed by constraints are given desirability weightings, or costs, and the goal is to find the most desirable (or least cost) assignment. The complexity of any optimisation problem depends critically on the type of function which has to be minimized. For soft constraint problems this function is a sum of cost functions 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 rational numbers or infinite the complexity of a soft constraint problem is determined by certain algebraic properties of the valued constraint language, which we call feasibility polymorphisms and fractional polymorphisms. As an immediate application of these results, we show that the existence of a nontrivial fractional polymorphism is a necessary condition for the tractability of a valued constraint language with rational or infinite costs over any finite domain (assuming P ≠ NP).
Minimum Cost Homomorphisms to Semicomplete Multipartite Digraphs
"... Abstract. For digraphs D and H, a mapping f: V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). If, moreover, each vertex u ∈ V (D) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is � u∈V (D) c f(u)(u). For each fixed digraph H, we have the ..."
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Cited by 12 (8 self)
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Abstract. For digraphs D and H, a mapping f: V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). If, moreover, each vertex u ∈ V (D) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is � u∈V (D) c f(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem for H. The problem is to decide, for an input graph D with costs ci(u), u ∈ V (D), i ∈ V (H), whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems. We describe a dichotomy of the minimum cost homomorphism problem for semicomplete bipartite digraphs H. This solves an open problem from an earlier paper. To obtain the dichotomy of this paper, we introduce and study a new notion, a kMinMax ordering of digraphs. Key words. homomorphisms, minimum cost homomorphisms, semicomplete bipartite digraphs
Virtual Arc Consistency for Weighted CSP
"... Optimizing a combination of local cost functions on discrete variables is a central problem in many formalisms such as in probabilistic networks, maximum satisfiability, weighted CSP or factor graphs. Recent results have shown that maintaining a form of local consistency in a Branch and Bound search ..."
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Cited by 10 (4 self)
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Optimizing a combination of local cost functions on discrete variables is a central problem in many formalisms such as in probabilistic networks, maximum satisfiability, weighted CSP or factor graphs. Recent results have shown that maintaining a form of local consistency in a Branch and Bound search provides bounds that are strong enough to solve many practical instances. In this paper, we introduce Virtual Arc Consistency (VAC) which iteratively identifies and applies sequences of cost propagation over rational costs that are guaranteed to transform a WCSP in another WCSP with an improved constant cost. Although not as strong as Optimal Soft Arc Consistency, VAC is faster and powerful enough to solve submodular problems. Maintaining VAC inside branch and bound leads to important improvements in efficiency on large difficult problems and allowed us to close two famous frequency assignment problem instances.
Minimum Cost Homomorphisms to reflexive digraphs
 8th Latin American Theoretical Informatics (LATIN), Rio de Janeiro, Brazil
"... For digraphs G and H, a homomorphism of G to H is a mapping f: V (G)→V (H) such that uv ∈ A(G) implies f(u)f(v) ∈ A(H). If moreover each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of a homomorphism f is u∈V (G) c f(u)(u). For each fixed digraph H, the minimum cost hom ..."
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Cited by 10 (8 self)
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For digraphs G and H, a homomorphism of G to H is a mapping f: V (G)→V (H) such that uv ∈ A(G) implies f(u)f(v) ∈ A(H). If moreover each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of a homomorphism f is u∈V (G) c f(u)(u). For each fixed digraph H, the minimum cost homomorphism problem for H, denoted MinHOM(H), is the following problem. Given an input digraph G, together with costs ci(u), u ∈ V (G), i ∈ V (H), and an integer k, decide if G admits a homomorphism to H of cost not exceeding k. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems such as chromatic partition optimization and applied problems in repair analysis. For undirected graphs the complexity of the problem, as a function of the parameter H, is well understood; for digraphs, the situation appears to be more complex, and only partial results are known. We focus on the minimum cost homomorphism problem for reflexive digraphs H (every vertex of H has a loop). It is known that the problem MinHOM(H) is polynomial time solvable if the digraph H has a MinMax ordering, i.e., if its vertices can be linearly ordered by < so that i < j, s < r and ir, js ∈ A(H) imply that is ∈ A(H) and jr ∈ A(H). We give a forbidden induced subgraph characterization of reflexive digraphs with a MinMax ordering; our characterization implies a polynomial time test for the existence of a MinMax ordering. Using this characterization, we show that for a reflexive digraph H which does not admit a MinMax ordering, the minimum cost homomorphism problem is NPcomplete. Thus we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for reflexive digraphs. 1
Soft arc consistency revisited
 Artificial Intelligence
"... The Valued Constraint Satisfaction Problem (VCSP) is a generic optimization problem defined by a network of local cost functions defined over discrete variables. It has applications in Artificial Intelligence, Operations Research, Bioinformatics and has been used to tackle optimization problems in o ..."
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Cited by 9 (3 self)
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The Valued Constraint Satisfaction Problem (VCSP) is a generic optimization problem defined by a network of local cost functions defined over discrete variables. It has applications in Artificial Intelligence, Operations Research, Bioinformatics and has been used to tackle optimization problems in other graphical models (including discrete Markov Random Fields and Bayesian Networks). The incremental lower bounds produced by local consistency filtering are used for pruning inside Branch and Bound search. In this paper, we extend the notion of arc consistency by allowing fractional weights and by allowing several arc consistency operations to be applied simultaneously. Over the rationals and allowing simultaneous operations, we show that an optimal arc consistency closure can theoretically be determined in polynomial time by reduction to linear programming. This defines Optimal Soft Arc Consistency (OSAC). To reach a more practical algorithm, we show that the existence of a sequence of arc consistency operations which increases the lower bound can be detected by establishing arc consistency in a classical Constraint Satisfaction Problem (CSP) derived from the original cost function network. This leads to a new soft arc consistency method, called,Virtual Arc Consistency which produces improved lower bounds compared with previous techniques and which can solve submodular cost functions. These algorithms have been implemented and evaluated on a variety of problems, including two difficult frequency assignment problems which are solved to optimality for the first time. Our implementation is available in the open source toulbar2 platform.
On solving soft temporal constraints using SAT techniques
 In CP’05, LNCS 3709
, 2005
"... Abstract. In this paper, we present an algorithm for finding utilitarian optimal solutions to Simple and Disjunctive Temporal Problems with Preferences (STPPs and DTPPs) based on Benders ’ decomposition and adopting SAT techniques. In our approach, each temporal constraint is replaced by a Boolean i ..."
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Cited by 8 (3 self)
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Abstract. In this paper, we present an algorithm for finding utilitarian optimal solutions to Simple and Disjunctive Temporal Problems with Preferences (STPPs and DTPPs) based on Benders ’ decomposition and adopting SAT techniques. In our approach, each temporal constraint is replaced by a Boolean indicator variable and the decomposed problem is solved by a tightly integrated STP solver and SAT solver. Several hybridization techniques that take advantage of each solver’s strengths are introduced. Finally, empirical evidence is presented to demonstrate the effectiveness of our method compared to other algorithms. 1
Introduction to the minimum cost homomorphism problem for directed and undirected graphs
 Lecture Notes of the Ramanujan Math
"... For digraphs D and H, a mapping f: V (D)→V (H) is a homomorphism of D ..."
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Cited by 7 (3 self)
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For digraphs D and H, a mapping f: V (D)→V (H) is a homomorphism of D