Online Mixed Packing and Covering

In many problems, the inputs arrive over time, and must be dealt with irrevocably when they arrive. Such problems are online problems. A common method of solving online problems is to first solve the corresponding linear program, and then round the fractional solution online to obtain an integral solution. We give algorithms for solving linear programs with mixed packing and covering constraints online. We first consider mixed packing and covering linear programs, where packing constraints are given offline and covering constraints are received online. The objective is to minimize the maximum multiplicative factor by which any packing constraint is violated, while satisfying the covering constraints. No prior sublinear competitive algorithms are known for this problem. We give the first such --- a polylogarithmic-competitive algorithm for solving mixed packing and covering linear programs online. We also show a nearly tight lower bound. Our techniques for the upper bound use an exponential penalty function in conjunction with multiplicative updates. While exponential penalty functions are used previously to solve linear programs offline approximately, offline algorithms know the constraints beforehand and can optimize greedily. In contrast, when constraints arrive online, updates need to be more complex. We apply our techniques to solve two online fixed-charge problems with congestion. These problems are motivated by applications in machine scheduling and facility location. The linear program for these problems is more complicated than mixed packing and covering, and presents unique challenges. We show that our techniques combined with a randomized rounding procedure give polylogarithmic-competitive integral solutions. These problems generalize online set-cover, for which there is a polylogarithmic lower bound. Hence, our results are close to tight

Identifying customer satisfaction at New Lives Animal Rescue Opshop

New lives animal rescue is a registered charity which is rescuing and rehoming almost all kinds of animals. They collect donated goods from people and sell those items at second hand price through the Opshop. All money obtained by the Opshop goes to the shelter home. They have two Opshops, located at Hamilton and Cambridge. The research work was held at New Lives Animal Rescue Opshop, Grey Street, Hamilton. The research topic was to identify customer satisfaction at the Opshop, as well as, how to maintain customer satisfaction in the future. Competitors for this Opshop include the Salvation Army and New Zealand Red Cross. The quantitative method research methodology was used is choosing the survey method to identify customer satisfaction. The customer survey was held at the Opshop. Thirty customers participated in this survey. Every customer was given 10 minutes to complete the survey. The limitations of the research work were time and money. There were 23% male and 77% female customers participating in the customer survey. Almost 50% of the customers were more than 50 years old. Almost 80% of customers were satisfied with this Opshop. The customers have natural views about the quality of products. Regarding customer loyalty, almost 40% claimed that this was their first purchase. The other 60% people are very loyal to the shop. I found that most of the customers were very likely to recommend this shop to their friends and family members. Almost 77% of customers were fully satisfied with the store location. The customers however were not very satisfied with the price of the products available in the Opshop. The research has concluded with recommendations to be made for further customer satisfaction at New Lives Animal Rescue Opshop

The Complexity of Partial Function Extension for Coverage Functions

Coverage functions are an important subclass of submodular functions, finding applications in machine learning, game theory, social networks, and facility location. We study the complexity of partial function extension to coverage functions. That is, given a partial function consisting of a family of subsets of [m] and a value at each point, does there exist a coverage function defined on all subsets of [m] that extends this partial function? Partial function extension is previously studied for other function classes, including boolean functions and convex functions, and is useful in many fields, such as obtaining bounds on learning these function classes. We show that determining extendibility of a partial function to a coverage function is NP-complete, establishing in the process that there is a polynomial-sized certificate of extendibility. The hardness also gives us a lower bound for learning coverage functions. We then study two natural notions of approximate extension, to account for errors in the data set. The two notions correspond roughly to multiplicative point-wise approximation and additive L_1 approximation. We show upper and lower bounds for both notions of approximation. In the second case we obtain nearly tight bounds