The Weight Function Lemma for graph pebbling

Graph pebbling is a network model for studying whether or not a given supply of discrete pebbles can satisfy a given demand via pebbling moves. A pebbling move across an edge of a graph takes two pebbles from one endpoint and places one pebble at the other endpoint; the other pebble is lost in transit as a toll. It has been shown that deciding whether a supply can meet a demand on a graph is NP-complete. The pebbling number of a graph is the smallest t such that every supply of t pebbles can satisfy every demand of one pebble byt a vertex. Deciding if the pebbling number is at most k is \(\Pi _2^\mathsf{P}\)-complete. In this paper we develop a tool, called the Weight Function Lemma, for computing upper bounds and sometimes exact values for pebbling numbers with the assistance of linear optimization. With this tool we are able to calculate the pebbling numbers of much larger graphs than in previous algorithms, and much more quickly as well. We also obtain results for many families of graphs, in many cases by hand, with much simpler and remarkably shorter proofs than given in previously existing arguments (certificates typically of size at most the number of vertices times the maximum degree), especially for highly symmetric graphs. Here we apply the Weight Function Lemma to several specific graphs, including the Petersen, Lemke, \(4\mathrm{th}\) weak Bruhat, and Lemke squared, as well as to a number of infinite families of graphs, such as trees, cycles, graph powers of cycles, cubes, and some generalized Petersen and Coxeter graphs. In doing so we partly answer a question of Pachter, et al., by computing the pebbling exponent of cycles to within an asymptotically small range. It is conceivable that this method yields an approximation algorithm for graph pebbling.     

GROUP INEQUALITY

We explore the combined effect of segregation in social networks, peer effects, and the relative size of a historically disadvantaged group on the incentives to invest in market‐rewarded skills and the dynamics of inequality between social groups. We identify conditions under which group inequality will persist in the absence of differences in ability, credit constraints, or labor market discrimination. Under these conditions, group inequality may be amplified even if initial group differences are negligible. Increases in social integration may destabilize an unequal state and make group equality possible, but the distributional and human capital effects of this depend on the demographic composition of the population. When the size of the initially disadvantaged group is sufficiently small, integration can lower the long‐run costs of human capital investment in both groups and result in an increase the aggregate skill share. In contrast, when the initially disadvantaged group is large, integration can induce a fall in the aggregate skill share as the costs of human capital investment rise in both groups. We consider applications to concrete cases and policy implications.     

Generic Assessment Endpoints Are Needed for Ecological Risk Assessment

This article presents arguments for the development of generic assessment endpoints for ecological risk assessment. Generic assessment endpoints would be ecological entities and attributes that are assumed to be worthy of protection in most contexts. The existence of generic assessment endpoints would neither create a requirement that they be used in every assessment nor preclude the use of other assessment endpoints. They would simply be a starting point in the process of identifying the assessment endpoints for a particular assessment. They are needed to meet legal mandates, to provide a floor for environmental degradation, to provide some consistency in environmental regulation, as exemplars for site- or project-specific assessment endpoints, to allow development of methods and models, to give risk managers the courage to act, for screening and site-independent assessments, to support environmental monitoring, to facilitate communication, and to avoid paralysis by analysis. Generic assessment endpoints should include not only a list of entities and attributes, but also explanations of each endpoint, guidance on their use and interpretation, and measures and models that could be used to estimate them.     

Mixed Levels of Uncertainty in Complex Policy Models

The characterization and treatment of uncertainty poses special challenges when modeling indeterminate or complex coupled systems such as those involved in the interactions between human activity, climate and the ecosystem. Uncertainty about model structure may become as, or more important than, uncertainty about parameter values. When uncertainty grows so large that prediction or optimization no longer makes sense, it may still be possible to use the model as a behavioral test bed to examine the relative robustness of alternative observational and behavioral strategies. When models must be run into portions of their phase space that are not well understood, different submodels may become unreliable at different rates. A common example involves running a time stepped model far into the future. Several strategies can be used to deal with such situations. The probability of model failure can be reported as a function of time. Possible alternative surprises can be assigned probabilities, modeled separately, and combined. Finally, through the use of subjective judgments, one may be able to combine, and over time shift between models, moving from more detailed to progressively simpler order-of-magnitude models, and perhaps ultimately, on to simple bounding analysis.     

Relationships between Sales Management Control, Territory Design, Salesforce Performance and Sales Organization Effectiveness

This research evaluates determinants of sales organization effectiveness in a sample of British companies, and contributes to an important new research stream by following recent empirical studies in the USA and Australia. We discuss a model of sales organization effectiveness determined by salesforce outcome performance and behavioural performance, as well as by the use of a behaviour-based control approach. Sales territory design is also considered as a particularly important managerial variable, which has received little analytical attention in the traditional literature, but which appears to be an important influence on the effectiveness of the sales operation. Our exploratory path analytical model suggests that sales territory design has a large effect on sales organization effectiveness both directly, and indirectly through its relationship with salesforce behavioural performance. These findings are somewhat different to those in similar studies in other countries, and suggest some important implications for managers as well as for researchers in this field.     

Proportional hazards regression with interval censored data using an inverse probability weight

The prevalence of interval censored data is increasing in medical studies due to the growing use of biomarkers to define a disease progression endpoint. Interval censoring results from periodic monitoring of the progression status. For example, disease progression is established in the interval between the clinic visit where progression is recorded and the prior clinic visit where there was no evidence of disease progression. A methodology is proposed for estimation and inference on the regression coefficients in the Cox proportional hazards model with interval censored data. The methodology is based on estimating equations and uses an inverse probability weight to select event time pairs where the ordering is unambiguous. Simulations are performed to examine the finite sample properties of the estimate and a colon cancer data set is used to demonstrate its performance relative to the conventional partial likelihood estimate that ignores the interval censoring.     

Integrative Family Therapy With Childhood Chronic Illness: An Ethics of Practice

A challenge for contemporary family therapists is negotiating differences between modern and postmodern frameworks in the practice context. Modernists espouse a systemic metaphor; use evidence‐based and interventive approaches, including strategic, structural‐ or solution‐focused techniques, and believe in the therapist's knowledge, expertise and power to influence individuals or families to change. On the other hand, postmodernists follow a social constructionist, dialogical or narrative paradigm, which identifies the main ingredient of therapy as language, conversation, understanding and the therapist's ‘not knowing’ stance in eliciting a person's expertise and story Yet many practitioners adopt a middle way between these paradigm polarities, one that is less theory‐driven and more pragmatic, flexible, integrative and practice‐based. This is consistent with evidence‐based practice and research demonstrating common factors across all therapies. The value of preserving systemic thinking in family therapy is recognised while reaching forward to a postmodern social constructionist and dialogical approach. The article describes this integrative stance in family therapy as paramodern based on an ethics of practice. This is illustrated by a detailed case study of integrative family therapy, which addresses anxiety, anger and sleeping issues associated with a chronic childhood illness called Perthe's disease.     

The residential segregation of occupational groups in central cities and suburbs

The impression of journalists and social critics in the 1950’s that post-war suburbia was uniformly middle-class has been generally rejected by social scientists, but there is a persisting belief in a high degree of residential segregation by social level in suburbia and in a high degree of socio-economic homogeneity within suburban neighborhoods. A comparison of eight central cities with their suburban zones in 1950 and in 1960 revealed, for both dates, (a) small differences in occupational distributions between the central cities and the suburban zones and (b) generally higher Index of Residential Dissimilarity values for pairs of occupational groups in the central cities. These findings indicate that suburban neighborhoods, at least in the eight suburban zones studied, were little, if any, more occupationally homogeneous than the central city neighborhoods. This suggests that the belief in homogeneous suburban neighborhoods should be added to the growing list of discredited “myths of suburbia. ”     

GAPS between MFT supervision assumptions and common practice: suggested best practices

This article summarizes gaps between assumptions that appear in the literature, research, and accepted standards for marital and family therapy (MFT) supervision, and the common practice of supervisors. Issues that stem from these gaps are highlighted and recommendations are made for closing them. In an effort to refine the standard of practice for MFT supervision, best practice recommendations are offered for MFT supervision.