Multi-Agent Submodular Optimization

Recent years have seen many algorithmic advances in the area of submodular optimization: (SO) min/max~f(S): S in F, where F is a given family of feasible sets over a ground set V and f:2^V - > R is submodular. This progress has been coupled with a wealth of new applications for these models. Our focus is on a more general class of multi-agent submodular optimization (MASO) min/max Sum_{i=1}^{k} f_i(S_i): S_1 u+ S_2 u+ ... u+ S_k in F. Here we use u+ to denote disjoint union and hence this model is attractive where resources are being allocated across k agents, each with its own submodular cost function f_i(). This was introduced in the minimization setting by Goel et al. In this paper we explore the extent to which the approximability of the multi-agent problems are linked to their single-agent versions, referred to informally as the multi-agent gap. We present different reductions that transform a multi-agent problem into a single-agent one. For minimization, we show that (MASO) has an O(alpha * min{k, log^2 (n)})-approximation whenever (SO) admits an alpha-approximation over the convex formulation. In addition, we discuss the class of "bounded blocker" families where there is a provably tight O(log n) multi-agent gap between (MASO) and (SO). For maximization, we show that monotone (resp. nonmonotone) (MASO) admits an alpha (1-1/e) (resp. alpha * 0.385) approximation whenever monotone (resp. nonmonotone) (SO) admits an alpha-approximation over the multilinear formulation; and the 1-1/e multi-agent gap for monotone objectives is tight. We also discuss several families (such as spanning trees, matroids, and p-systems) that have an (optimal) multi-agent gap of 1. These results substantially expand the family of tractable models for submodular maximization

Basic calculation proficiency and mathematics achievement in elementary school children

The relation between skill in simple addition and subtraction and more general math achievement in elementary school is well established but not understood. Both the intrinsic importance of skill in simple calculation for math and the influence of conceptual knowledge and cognitive factors (working memory, processing speed, oral language) on simple calculation and math are plausible. The authors investigated the development of basic calculation fluency and its relations to math achievement and other factors by tracking a group of 259 United Kingdom English children from second to third grade. In both grades the group did not retrieve the solutions to most problems, but their math achievement was typical. Improvement in basic calculation proficiency was partially predicted by conceptual knowledge and cognitive factors. These factors only partially mediated the relation between basic calculation and math achievement. The relation between reading and math was wholly mediated by number measures and cognitive factors

Factors Affecting the Internationalization of Manufacturing SMEs in Zimbabwe

The study investigated the factors affecting the internationalization of manufacturing SMEs in Zimbabwe. A triangulated research approach involving explanatory and descriptive designs was used to collect and analyze data from  302 manufacturing SMEs in Zimbabwe. The findings are that SME internationalization is positively influenced by availability of funds, management attitudes, and knowledge of the market, risk perception, international networks and  intensity of competition. We also observed that contrary to contemporary literature, age and size of the SME as well as technical ability of managers do not influence internationalization. We therefore argue that unless the  government and trade associations devise means to cushion the SMEs financially and expose them to the internationalization process, the number of SMEs that do business internationally will continue to dwindle

Alcohol-free and low-strength drinks: understanding their role in reducing alcohol-related harms.

This report examines the market for alcohol-free and low-strength beers, ciders, wines and spirits – “NoLo drinks” - in the UK. Key findings: • The market for NoLo drinks in the UK is seeing significant growth. • However, this growth is from a very low base and must be put into context. • Achieving a significant market share for NoLo drinks over the next decade would thus require consistent, strong sales growth. • The relatively small size of the NoLo drinks market thus needs to be taken into account when assessing the current health benefits from the use of such products at a population level

A Family of Calabi-Yau Varieties and Potential Automorphy

Mathematic