Probabilistic Analysis of Optimization Problems on Generalized Random Shortest Path Metrics

Simple heuristics often show a remarkable performance in practice for optimization problems. Worst-case analysis often falls short of explaining this performance. Because of this, "beyond worst-case analysis" of algorithms has recently gained a lot of attention, including probabilistic analysis of algorithms. The instances of many optimization problems are essentially a discrete metric space. Probabilistic analysis for such metric optimization problems has nevertheless mostly been conducted on instances drawn from Euclidean space, which provides a structure that is usually heavily exploited in the analysis. However, most instances from practice are not Euclidean. Little work has been done on metric instances drawn from other, more realistic, distributions. Some initial results have been obtained by Bringmann et al. (Algorithmica, 2013), who have used random shortest path metrics on complete graphs to analyze heuristics. The goal of this paper is to generalize these findings to non-complete graphs, especially Erd\H{o}s-R\'enyi random graphs. A random shortest path metric is constructed by drawing independent random edge weights for each edge in the graph and setting the distance between every pair of vertices to the length of a shortest path between them with respect to the drawn weights. For such instances, we prove that the greedy heuristic for the minimum distance maximum matching problem, the nearest neighbor and insertion heuristics for the traveling salesman problem, and a trivial heuristic for the $k$-median problem all achieve a constant expected approximation ratio. Additionally, we show a polynomial upper bound for the expected number of iterations of the 2-opt heuristic for the traveling salesman problem.Comment: An extended abstract appeared in the proceedings of WALCOM 201

Probabilistic Analysis of Facility Location on Random Shortest Path Metrics

The facility location problem is an NP-hard optimization problem. Therefore, approximation algorithms are often used to solve large instances. Such algorithms often perform much better than worst-case analysis suggests. Therefore, probabilistic analysis is a widely used tool to analyze such algorithms. Most research on probabilistic analysis of NP-hard optimization problems involving metric spaces, such as the facility location problem, has been focused on Euclidean instances, and also instances with independent (random) edge lengths, which are non-metric, have been researched. We would like to extend this knowledge to other, more general, metrics. We investigate the facility location problem using random shortest path metrics. We analyze some probabilistic properties for a simple greedy heuristic which gives a solution to the facility location problem: opening the $\kappa$ cheapest facilities (with $\kappa$ only depending on the facility opening costs). If the facility opening costs are such that $\kappa$ is not too large, then we show that this heuristic is asymptotically optimal. On the other hand, for large values of $\kappa$, the analysis becomes more difficult, and we provide a closed-form expression as upper bound for the expected approximation ratio. In the special case where all facility opening costs are equal this closed-form expression reduces to $O(\sqrt[4]{\ln(n)})$ or $O(1)$ or even $1+o(1)$ if the opening costs are sufficiently small.Comment: A preliminary version accepted to CiE 201

Self-avoiding walks and connective constants

The connective constant $\mu(G)$ of a quasi-transitive graph $G$ is the asymptotic growth rate of the number of self-avoiding walks (SAWs) on $G$ from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph $G$. $\bullet$ We present upper and lower bounds for $\mu$ in terms of the vertex-degree and girth of a transitive graph. $\bullet$ We discuss the question of whether $\mu\ge\phi$ for transitive cubic graphs (where $\phi$ denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles). $\bullet$ We present strict inequalities for the connective constants $\mu(G)$ of transitive graphs $G$, as $G$ varies. $\bullet$ As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a further generator. $\bullet$ We describe so-called graph height functions within an account of "bridges" for quasi-transitive graphs, and indicate that the bridge constant equals the connective constant when the graph has a unimodular graph height function. $\bullet$ A partial answer is given to the question of the locality of connective constants, based around the existence of unimodular graph height functions. $\bullet$ Examples are presented of Cayley graphs of finitely presented groups that possess graph height functions (that are, in addition, harmonic and unimodular), and that do not. $\bullet$ The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with arXiv:1304.721

Geometrical Insights for Implicit Generative Modeling

Learning algorithms for implicit generative models can optimize a variety of criteria that measure how the data distribution differs from the implicit model distribution, including the Wasserstein distance, the Energy distance, and the Maximum Mean Discrepancy criterion. A careful look at the geometries induced by these distances on the space of probability measures reveals interesting differences. In particular, we can establish surprising approximate global convergence guarantees for the $1$-Wasserstein distance,even when the parametric generator has a nonconvex parametrization.Comment: this version fixes a typo in a definitio

The influence of 'significant others' on persistent back pain and work participation: a qualitative exploration of illness perceptions

Background Individual illness perceptions have been highlighted as important influences on clinical outcomes for back pain. However, the illness perceptions of 'significant others' (spouse/partner/close family member) are rarely explored, particularly in relation to persistent back pain and work participation. The aim of this study was to initiate qualitative research in this area in order to further understand these wider influences on outcome. Methods Semi-structured interviews based on the chronic pain version of the Illness Perceptions Questionnaire-Revised were conducted with a convenience sample of UK disability benefit claimants, along with their significant others (n=5 dyads). Data were analysed using template analysis. Results Significant others shared, and perhaps further reinforced, claimants' unhelpful illness beliefs including fear of pain/re-injury associated with certain types of work and activity, and pessimism about the likelihood of return to work. In some cases, significant others appeared more resigned to the permanence and negative inevitable consequences of the claimant's back pain condition on work participation, and were more sceptical about the availability of suitable work and sympathy from employers. In their pursuit of authenticity, claimants were keen to stress their desire to work whilst emphasising how the severity and physical limitations of their condition prevented them from doing so. In this vein, and seemingly based on their perceptions of what makes a 'good' significant other, significant others acted as a 'witness to pain', supporting claimants' self-limiting behaviour and statements of incapacity, often responding with empathy and assistance. The beliefs and responses of significant others may also have been influenced by their own experience of chronic illness, thus participants lives were often intertwined and defined by illness. Conclusions The findings from this exploratory study reveal how others and wider social circumstances might contribute both to the propensity of persistent back pain and to its consequences. This is an area that has received little attention to date, and wider support of these findings may usefully inform the design of future intervention programmes aimed at restoring work participation

Degenerate Stars and Gravitational Collapse in AdS/CFT

We construct composite CFT operators from a large number of fermionic primary fields corresponding to states that are holographically dual to a zero temperature Fermi gas in AdS space. We identify a large N regime in which the fermions behave as free particles. In the hydrodynamic limit the Fermi gas forms a degenerate star with a radius determined by the Fermi level, and a mass and angular momentum that exactly matches the boundary calculations. Next we consider an interacting regime, and calculate the effect of the gravitational back-reaction on the radius and the mass of the star using the Tolman-Oppenheimer-Volkoff equations. Ignoring other interactions, we determine the "Chandrasekhar limit" beyond which the degenerate star (presumably) undergoes gravitational collapse towards a black hole. This is interpreted on the boundary as a high density phase transition from a cold baryonic phase to a hot deconfined phase.Comment: 75 page

Holographic Geometry of Entanglement Renormalization in Quantum Field Theories

We study a conjectured connection between the AdS/CFT and a real-space quantum renormalization group scheme, the multi-scale entanglement renormalization ansatz (MERA). By making a close contact with the holographic formula of the entanglement entropy, we propose a general definition of the metric in the MERA in the extra holographic direction, which is formulated purely in terms of quantum field theoretical data. Using the continuum version of the MERA (cMERA), we calculate this emergent holographic metric explicitly for free scalar boson and free fermions theories, and check that the metric so computed has the properties expected from AdS/CFT. We also discuss the cMERA in a time-dependent background induced by quantum quench and estimate its corresponding metric.Comment: 42pages, 9figures, reference added, minor chang

Extracting Spacetimes using the AdS/CFT Conjecture: Part II

Motivated by the holographic principle, within the context of the AdS/CFT Correspondence in the large t'Hooft limit, we investigate how the geometry of certain highly symmetric bulk spacetimes can be recovered given information of physical quantities in the dual boundary CFT. In particular, we use holographic entanglement entropy proposal (relating the entanglement entropy of certain subsystems on the boundary to the area of static minimal surfaces) to recover the bulk metric using higher dimensional minimal surface probes within a class of static, planar symmetric, asymptotically AdS spacetimes. We find analytic and perturbative expressions for the metric function in terms of the entanglement entropy of straight belt and circular disk subsystems of the boundary theory respectively. Finally, we discuss how such extractions can be generalised

Why do women invest in pre-pregnancy health and care? A qualitative investigation with women attending maternity services

Background Despite the importance attributed to good pre-pregnancy care and its potential to improve pregnancy and child health outcomes, relatively little is known about why women invest in pre-pregnancy health and care. We sought to gain insight into why women invested in pre-pregnancy health and care. Methods We carried out 20 qualitative in-depth interviews with pregnant or recently pregnant women who were drawn from a survey of antenatal clinic attendees in London, UK. Interviewees were purposively sampled to include high and low investors in pre-pregnancy health and care, with variation in age, partnership status, ethnicity and pre-existing medical conditions. Data analysis was conducted using the Framework method. Results We identified three groups in relation to pre-pregnancy health and care: 1) The “prepared” group, who had high levels of pregnancy planning and mostly positive attitudes to micronutrient supplementation outside of pregnancy, carried out pre-pregnancy activities such as taking folic acid and making changes to diet and lifestyle. 2) The “poor knowledge” group, who also had high levels of pregnancy planning, did not carry out pre-pregnancy activities and described themselves as having poor knowledge. Elsewhere in their interviews they expressed a strong dislike of micronutrient supplementation. 3) The “absent pre-pregnancy period” group, had the lowest levels of pregnancy planning and also expressed anti-supplement views. Even discussing the pre-pregnancy period with this group was difficult as responses to questions quickly shifted to focus on pregnancy itself. Knowledge of folic acid was poor in all groups. Conclusion Different pre-pregnancy care approaches are likely to be needed for each of the groups. Among the “prepared” group, who were proactive and receptive to health messages, greater availability of information and better response from health professionals could improve the range of pre-pregnancy activities carried out. Among the “poor knowledge” group, better response from health professionals might yield greater uptake of pre-pregnancy information. A different, general health strategy might be more appropriate for the “absent pre-pregnancy period” group. The fact that general attitudes to micronutrient supplementation were closely related to whether or not women invested in pre-pregnancy health and care was an unanticipated finding and warrants further investigation.This report is independent research commissioned and funded by the Department of Health Policy Research Programme Pre-Pregnancy Health and Care in England: Exploring Implementation and Public Health Impact, 006/0068