Comparing Intersection Cut Closures using Simple Families of Lattice-Free Convex Sets

Mixed integer programs are a powerful mathematical tool, providing a general model for expressing both theoretically difficult and practically useful problems. One important subroutine of algorithms solving mixed integer programs is a cut generation procedure. The job of a cut generation procedure is to produce a linear inequality that separates a given infeasible point x* (usually a basic feasible solution of the linear programming relaxation) from the set of feasible solutions for the problem at hand. Early and well-known cut generation procedures rely on analyzing a single row of the simplex tableau for x*. Andersen et al. renewed interest in d-row cuts (i.e. cuts derived from d rows of the simplex tableau) by showing that these cuts afford some theoretical benefit. One lens from which to study d-row cuts is in the context of the intersection cuts of Balas and, in particular, intersection cuts obtained from lattice-free convex sets. The strongest d-row intersection cuts are obtained from maximal lattice-free convex sets in $R^d$ - all of which are polyhedra with at most $2^d$ facets. This thesis is concerned with theoretical comparison of the d-row cuts generated by different families of maximal lattice-free convex sets. We use the gauge measure to appraise the quality of the approximation. The main area of focus is 2-row cuts. The problem of generating 2-row cuts can be re-posed as generating valid inequalities for a mixed integer linear set F with two free integer variables and any number of non-negative continuous variables, where there are two defining equations. Every minimal valid inequality for the convex hull of F is an intersection cut generated by a maximal lattice-free split, triangle or quadrilateral. The family of maximal lattice-free triangles can be subdivided into the families of type 1, type 2, and type 3 triangles. Previous results of Basu et al. and Awate et al. compare how well the inequalities from one of these families approximates the convex hull of F (a.k.a. the corner polyhedron). In particular, the closure of all type 2 triangle inequalities is shown to be within a factor of 3/2 of the corner polyhedron. The authors also provide an instance where all type 2 triangles inequalities cannot approximate the corner polyhedron better than a factor of 9/8. The same bounds are shown for type 3 triangles and quadrilaterals. These results are obtained not by directly comparing the given closures to the convex hull of F, but rather to each other. In this thesis, we tighten one of the underlying bounds, showing that the closure of all type 2 triangle inequalities are within a factor of 5/4 of the closure of all quadrilateral inequalities. We also consider the sub-family of quadrilaterals where opposite edges have equal slope. We show that these parallelogram cuts can be approximated by all type 2 triangle inequalities within a factor of 9/8 and there exist instances where no better approximation is possible. In proving both these bounds, we use a subset of the family of type 2 triangles; we call the members of this sub-family ray-sliding triangles. A secondary area of focus in this thesis is d-row cuts for d >= 3. For d-row cuts in general, the underlying maximal lattice-free convex sets in $R^d$ are not easily classified. Absent a classification, Averkov et al, show that all inequalities generated by lattice-free convex sets with at most $i$ facets approximate the corner polyhedron within a finite factor only when $i > 2^{d-1}$. Here we take a different tact and try to prove analogues of 2-row cut results. We extend the proof techniques to obtain a constant factor approximation between two structured families of maximal lattice-free convex sets in $R^d$ for d >= 3

Single Commodity Flow Algorithms for Lifts of Graphic and Cographic Matroids

Consider a binary matroid M given by its matrix representation. We show that if M is a lift of a graphic or a cographic matroid, then in polynomial time we can either solve the single commodity flow problem for M or find an obstruction for which the Max-Flow Min-Cut relation does not hold. The key tool is an algorithmic version of Lehman's Theorem for the set covering polyhedron

Diagnosing Apraxia of Speech on the Basis of Eight Distinctive Signs

This paper reports the results of a study on the use of a fixed number of specific signs to differentially diagnose Apraxia of Speech (AoS) from aphasia or dysarthria. This was done with a diagnostic instrument for AoS that was developed in the Netherlands in 2012, the Diagnostic Instrument for Apraxia of Speech (DIAS; Feiken & Jonkers, 2012). There were 8 signs identified as specific to AoS, namely: inconsistency of errors, number of errors with consonants versus vowels, difference between sequencing and alternating diadochokinesis, groping, initiation problems, syllable segmentation, cluster segmentation, and articulatory complexity. The DIAS was administered to 30 individuals with AoS, 10 individuals with aphasia, 10 individuals with dysarthria, and 35 control individuals. Results showed that a differential diagnosis could be made in 88% of the cases using a minimum of 3 out of 8 specific signs of AoS as criteria. With the exception of 2 patients with aphasia no other group exhibited the presence of 3 or more signs of AoS. It was concluded that the presence of 3 signs is sufficient to differentially diagnose AoSfrom aphasia and dysarthria, despite the fact that there is a large amount of variability in the presence of signs of AoS itself in the different individuals

Central Sensitisation and functioning in patients with chronic low back pain: protocol for a cross-sectional and cohort study

Contains fulltext : 218544.pdf (publisher's version ) (Open Access

Effectiveness and feasibility of We12BFit!:improving physical fitness and lifestyle physical activity in children with developmental coordination disorder in a paediatric rehabilitation setting-a small sample field study

OBJECTIVES: To examine the effectiveness and feasibility of We12BFit!, a family-focused intervention aimed at increasing physical fitness (PF) and motivation for physical activity (PA) in 7-year-old to 12-year-old children with developmental coordination disorder (DCD). DESIGN: A single-arm mixed methods small sample field study. SETTING: Rehabilitation centres and schools for special education in The Netherlands. PARTICIPANTS: Twenty children with DCD diagnosis. INTERVENTIONS: We12BFit! consists of We12BFit!-PF and We12BFit!-Lifestyle PA. During We12BFit!-PF, cardiorespiratory fitness (CRF), muscle strength and anaerobic power were trained in small groups (10 weeks 2*60 min/week). We12Bfit!-Lifestyle PA, which addresses motivation for PA in children and parents, was added in week 6 of We12BFit!-PF and ended 12 weeks after We12BFit!-PF. OUTCOME MEASURES: The 20-Metre Shuttle Run Test (20mSRT), Muscle Power Sprint Test and Hand Held Dynamometry were performed before and after We12BFit!-PF and after We12BFit!-Lifestyle PA (T0-T1-T2). Parents and coaches were interviewed and trainers participated in a focus group to assess motivation for PA, perceived effectiveness, and feasibility of the intervention. RESULTS: Attendance rates of participants were 88% (We12BFit!-PF) and 89% (We12BFit!-Lifestyle PA). From T0 to T1, significant improvements were found in VO2peak, number of runs on the 20mSRT and mean anaerobic power. From T1 to T2, improvements were maintained. No changes were found after We12BFit!-Lifestyle PA in time spent on moderate to vigorous activity and metabolic equivalent of task; parents observed their child improved in qualitative aspects of activities and participation. Feasibility of We12Bfit! was confirmed, although some adaptations were recommended. CONCLUSIONS: We12BFit! resulted in significant improvements and maintenance of CRF and anaerobic power in a small group of children with DCD and seemed to improve motivation for PA. The group aspect of We12BFit!-PF, the high intensity and positive motivational climate of We12BFit!-PF may have improved children's self-efficacy. We12BFit! seems feasible to improve PF and PA in children with DCD. TRIAL REGISTRATION NUMBER: NTR6334