An approximability-related parameter on graphs―-properties and applications

Graph TheoryInternational audienceWe introduce a binary parameter on optimisation problems called separation. The parameter is used to relate the approximation ratios of different optimisation problems; in other words, we can convert approximability (and non-approximability) result for one problem into (non)-approximability results for other problems. Our main application is the problem (weighted) maximum H-colourable subgraph (Max H-Col), which is a restriction of the general maximum constraint satisfaction problem (Max CSP) to a single, binary, and symmetric relation. Using known approximation ratios for Max k-cut, we obtain general asymptotic approximability results for Max H-Col for an arbitrary graph H. For several classes of graphs, we provide near-optimal results under the unique games conjecture. We also investigate separation as a graph parameter. In this vein, we study its properties on circular complete graphs. Furthermore, we establish a close connection to work by Šámal on cubical colourings of graphs. This connection shows that our parameter is closely related to a special type of chromatic number. We believe that this insight may turn out to be crucial for understanding the behaviour of the parameter, and in the longer term, for understanding the approximability of optimisation problems such as Max H-Col

Faster arithmetic for number-theoretic transforms

We show how to improve the efficiency of the computation of fast Fourier transforms over F_p where p is a word-sized prime. Our main technique is optimisation of the basic arithmetic, in effect decreasing the total number of reductions modulo p, by making use of a redundant representation for integers modulo p. We give performance results showing a significant improvement over Shoup's NTL library.Comment: 9 pages, a few minor changes and reorganisation, to appear in JS

Conjunctions of Among Constraints

Many existing global constraints can be encoded as a conjunction of among constraints. An among constraint holds if the number of the variables in its scope whose value belongs to a prespecified set, which we call its range, is within some given bounds. It is known that domain filtering algorithms can benefit from reasoning about the interaction of among constraints so that values can be filtered out taking into consideration several among constraints simultaneously. The present pa- per embarks into a systematic investigation on the circumstances under which it is possible to obtain efficient and complete domain filtering algorithms for conjunctions of among constraints. We start by observing that restrictions on both the scope and the range of the among constraints are necessary to obtain meaningful results. Then, we derive a domain flow-based filtering algorithm and present several applications. In particular, it is shown that the algorithm unifies and generalizes several previous existing results.Comment: 15 pages plus appendi

Representing Fitness Landscapes by Valued Constraints to Understand the Complexity of Local Search

Local search is widely used to solve combinatorial optimisation problems and to model biological evolution, but the performance of local search algorithms on different kinds of fitness landscapes is poorly understood. Here we introduce a natural approach to modelling fitness landscapes using valued constraints. This allows us to investigate minimal representations (normal forms) and to consider the effects of the structure of the constraint graph on the tractability of local search. First, we show that for fitness landscapes representable by binary Boolean valued constraints there is a minimal necessary constraint graph that can be easily computed. Second, we consider landscapes as equivalent if they allow the same (improving) local search moves; we show that a minimal normal form still exists, but is NP-hard to compute. Next we consider the complexity of local search on fitness landscapes modelled by valued constraints with restricted forms of constraint graph. In the binary Boolean case, we prove that a tree-structured constraint graph gives a tight quadratic bound on the number of improving moves made by any local search; hence, any landscape that can be represented by such a model will be tractable for local search. We build two families of examples to show that both the conditions in our tractability result are essential. With domain size three, even just a path of binary constraints can model a landscape with an exponentially long sequence of improving moves. With a treewidth two constraint graph, even with a maximum degree of three, binary Boolean constraints can model a landscape with an exponentially long sequence of improving moves

Evidence for adaptation in pathological tendon in response to chronic load. : Does the pathological tendon adapt in respond to chronic load? A systematic Review

ABSTRACT Background Tendinopathy is a common problem amongst especially athletes but also in nonathletic people. The area of pathology in the tendon and its relationship with pain is questioned. Whether the tendon pathology resolves and if this correlates with decreased pain levels and improved function is also questioned. A greater knowledge of whether the tendon adapts to load could improve knowledge about the pathogenesis and management of tendinopathy.   Objective To systematically review the evidence for tendon adaptation to load.  thereby answering the question: Structural and material adaptation of pathological tendons with loading - Do pathological tendons adapt to mechanical loading?   Data sources A systematic search of the databases PubMed, CINAHL, EBSCO and Google Scholar was undertaken October 2015.   Study eligibility criteria Randomized controlled trials, cohort studies and controlled trials were included. Studies investigating the response of pathological tendons response to chronic load were included.         Keywords: Tendinopathy, Tendinosis, Pathology, Adaptation, Exercise                                   Study appraisal and synthesis methods Included studies were evaluated for risk of bias using the Pedro scale. Guidelines regarding level of evidence were taken from van Tulder et al (114).   Participants and interventions Seven studies met the inclusion criteria. A total of 184 patients underwent either eccentric or heavy slow training programs. Structural and/or mechanical, and/or biochemical outcome measures were collected after intervention.   Results Overall there is limited evidence that structural changes occur within the pathological area of the tendon, especially the core of the tendon as a response to heavy load. However, there is moderate support of HSR and certain biochemical outcomes.   Conclusions and implications of key findings Limited evidence indicates a period of heavy resistance training may result in some changes in the pathological area of the tendon. The presence of tendon pathology may be a risk factor for developing pain. So, longitudinal studies with large cohorts and validated objective outcome measures are needed to further investigate the extent to which a pathological tendon adapt to heavy load and how this relates to pain and function

Children’s theatre- not only “painted”… : A study of the state of children’s theatre today and how a good children’s theatre performance should be, according to children, teachers and performers.

Syftet med det här arbetet är att undersöka hur barnteatern kan se ut idag och om den uppfyller barnens krav. Observationer gjordes på tre slumpmässigt utvalda barnföreställningar för lågstadiet. Därefter intervjuades medverkande och publik som i detta fall bestod av barnen och dess lärare. Intervjufrågorna behandlade dels hur medverkande tänker inför en föreställning för barn och hur publiken uppfattade den, dels hur de anser att en bra barnföreställning bör vara. Resultatet i undersökningen visar några tydliga tendenser. De tre barnföreställningarna höll en god kvalitet i avseende på framförande, musik och dekor. Men barn är ofta ovana teaterbesökare och har svårt att ta in information genom bara iakttagande. Därför behöver de en annan typ av föreställning än de tre jag observerade. En föreställning med mera delaktighet.The purpose of this study is to investigate the current state of children’s theatre, and if it meets the needs of the children. Observations were carried out in three randomly chosen children’s theatre performances. After the observations, performers and the audience - in this case, children and their teachers - were interviewed. The questions dealt with how performers think prior to a performance for children, and how the audience reacted to it, in addition to how they think a good children’s theatre performance should be. The result of the study indicates some clear tendencies. The three children’s theatre performances received good reviews with regard to performance, music and décor. But children are seldom familiar with the theatre and have trouble attaining information through watching alone. Thus, they need a different kind of theatrical performance than the three I observed. A performance in which they are more involved in the action

Exploiting Structure in CSP-related Problems

In this thesis we investigate the computational complexity and approximability of computational problems from the constraint satisfaction framework. An instance of a constraint satisfaction problem (CSP) has three components; a set V of variables, a set D of domain values, and a set of constraints C. The constraints specify a set of variables and associated local conditions on the domain values allowed for each variable, and the objective of a CSP is to assign domain values to the variables, subject to these constraints. The first main part of the thesis is concerned with studying restrictions on the structure induced by the constraints on the variables for different computational problems related to the CSP. In particular, we examine how to exploit various graph, and hypergraph, acyclicity measures from the literature to find classes of relational structures for which our computational problems become efficiently solvable. Among the problems studied are, such where, in addition to the constraints of a CSP, lists of allowed domain values for each variable are specified (LHom). We also study variants of the CSP where the objective is changed to: counting the number of possible assignments of domain values to the variables given the constraints of a CSP (#CSP), minimising or maximising the cost of an assignment satisfying all constraints given various different ways of assigning costs to assignments (MinHom, Max Sol, and CSP), or maximising the number of satisfied constraints (Max CSP). In several cases, our investigations uncover the largest known (or possible) classes of relational structures for which our problems are efficiently solvable. Moreover, we take a different view on our optimisation problems MinHom and VCSP; instead of considering fixed arbitrary values for some (hyper)graph acyclicity measure associated with the underlying CSP, we consider the problems parameterised by such measures in combination with other basic parameters such as domain size and maximum arity of constraints. In this way, we identify numerous combinations of the considered parameters which make these optimisation problems admit fixed-parameter algorithms. In the second part of the thesis, we explore the approximability properties of the (weighted) Max CSP problem for graphs. This is a problem which is known to be approximable within some constant ratio, but not believed to be approximable within an arbitrarily small constant ratio. Thus it is of interest to determine the best ratio within which the problem can be approximated, or at least give some bound on this constant. We introduce a novel method for studying approximation ratios which, in the context of Max CSP for graphs, takes the form of a new binary parameter on the space of all graphs. This parameter may, informally, be thought of as a sort of distance between two graphs; knowing the distance between two graphs, we can bound the approximation ratio of one of them, given a bound for the other

“Closest in time, it’s the wind instruments that have had a down.” : A study of how music teachers and principals look at children’s instrument preferences and municipal music- and culture schools methods of recruitment.

The purpose of this study is, on one hand, to obtain an understanding of how children in municipal music- and culture schools choose instruments and what forms of recruitment methods they are subjected to, and on the other hand, how you can make them continue to play once they have begun. To acquire this information, I have interviewed five music teachers and two principals at municipal music- and culture schools in Värmland. The questions treated, among other things, what affects children’s instrument preferences, recruitment methods and what you can do to keep interest for the chosen instrument alive. The result of the study shows some apparent tendencies. Even though some of the instruments have trouble recruiting students and other instruments have trouble keeping them, the old methods, which obviously do not work in a satisfactory manner, are still being used. Instead, increased commitment at the occasion of recruitment, more group teaching which strengthens the social bonds through role models and a will to have the courage to change the organization when it does not work, is required