Tractable constraints on ordered domains

AbstractFinding solutions to a constraint satisfaction problem is known to be an NP-complete problem in general, but may be tractable in cases where either the set of allowed constraints or the graph structure is restricted. In this paper we identify a restricted set of contraints which gives rise to a class of tractable problems. This class generalizes the notion of a Horn formula in propositional logic to larger domain sizes. We give a polynomial time algorithm for solving such problems, and prove that the class of problems generated by any larger set of constraints is NP-complete

The expressive power of valued constraints: Hierarchies and collapses

In this paper we investigate the ways in which a fixed collection of valued constraints can be combined to express other valued constraints. We show that in some cases a large class of valued constraints, of all possible arities, can be expressed by using valued constraints of a fixed finite arity. We also show that some simple classes of valued constraints, including the set of all monotonie valued constraints with finite cost values, cannot be expressed by a subset of any fixed finite arity, and hence form an infinite hierarchy. © Springer-Verlag Berlin Heidelberg 2007

An algebraic theory of complexity for valued constraints: Establishing a Galois connection

Abstract. 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 power of propagation:when GAC is enough

Considerable effort in constraint programming has focused on the development of efficient propagators for individual constraints. In this paper, we consider the combined power of such propagators when applied to collections of more than one constraint. In particular we identify classes of constraint problems where such propagators can decide the existence of a solution on their own, without the need for any additional search. Sporadic examples of such classes have previously been identified, including classes based on restricting the structure of the problem, restricting the constraint types, and some hybrid examples. However, there has previously been no unifying approach which characterises all of these classes: structural, language-based and hybrid. In this paper we develop such a unifying approach and embed all the known classes into a common framework. We then use this framework to identify a further class of problems that can be solved by propagation alone

Epithelial-myoepithelial carcinoma of the tongue base: a case for the case-report and review of the literature

A 60 year old lady was referred to the Princess Alexandra Hospital (Brisbane, Queensland, Australia) tertiary Otolaryngology, Head and Neck Unit from a peripheral hospital for investigation and management of a tumour at the base of the tongue. Biopsy of the tumour revealed it to be an epithelial-myoepithelial carcinoma of the base of the tongue. This is an extremely rare tumour in this location with only 2 other case reports in the world literature: the patients were treated with chemo-radiotherapy and surgery respectively. Our patient was made aware of the world literature and was able to make a fully informed decision on her choice of treatment modality and was treated with radiotherapy. Increasingly journals are limiting publication of case reports to "world firsts" only. We present a case where such a policy would have denied patient choice and possibly led to detrimental treatment

Steering Evolution with Sequential Therapy to Prevent the Emergence of Bacterial Antibiotic Resistance

The increasing rate of antibiotic resistance and slowing discovery of novel antibiotic treatments presents a growing threat to public health. Here, we consider a simple model of evolution in asexually reproducing populations which considers adaptation as a biased random walk on a fitness landscape. This model associates the global properties of the fitness landscape with the algebraic properties of a Markov chain transition matrix and allows us to derive general results on the non-commutativity and irreversibility of natural selection as well as antibiotic cycling strategies. Using this formalism, we analyze 15 empirical fitness landscapes of E. coli under selection by different β-lactam antibiotics and demonstrate that the emergence of resistance to a given antibiotic can be either hindered or promoted by different sequences of drug application. Specifically, we demonstrate that the majority, approximately 70%, of sequential drug treatments with 2–4 drugs promote resistance to the final antibiotic. Further, we derive optimal drug application sequences with which we can probabilistically ‘steer’ the population through genotype space to avoid the emergence of resistance. This suggests a new strategy in the war against antibiotic–resistant organisms: drug sequencing to shepherd evolution through genotype space to states from which resistance cannot emerge and by which to maximize the chance of successful therapy

Counting Representable Sets On Simple Graphs

The graph-colouring problem may be generalised by allowing arbitrary constraints to be specified on the colour combinations permitted at each pair of adjacent nodes. A set of colourings which is the solution to some network of specified constraints is said to be a representable set. This paper derives exact expressions for the number of representable sets when the corresponding graph is cycle-free or series-parallel. 1 Introduction The problem of assigning colours to the vertices of a graph in such a way that adjacent vertices have different colours is one of the oldest topics in graph theory [1]. One way to generalise this problem is to allow arbitrary restrictions to be specified on the colour combinations permitted at each pair of adjacent vertices. These restrictions will be referred to here as `constraints' and a graph which has a constraint specified for each edge will be referred to as a `constraint network' Any colouring which simultaneously satisifies all of the constraints i..

Perfect Constraints Are Tractable

By using recent results from graph theory, including the Strong Perfect Graph Theorem, we obtain a unifying framework for a number of tractable classes of constraint problems. These include problems with chordal microstructure; problems with chordal microstructure complement; problems with tree structure; and the "all-different" constraint. In each of these cases we show that the associated microstructure of the problem is a perfect graph, and hence they are all part of the same larger family of tractable problems. © 2008 Springer-Verlag Berlin Heidelberg