Algorithms for Packing and Scheduling

This is the Second Year Progress Report for the “Algorithms for Packing and Scheduling” Ph.D. project. In this report, we give the aims of the project, a summary of the current results obtained, we address the questions raised in the Postgraduate Workshop, and we conclude with possible future work and a timetabled research plan. The project is concerned with the study, design, and analysis, of deterministic online and offline algorithms. We present the current results obtained for packing and scheduling algorithms. More specifically, for the online dynamic bin packing problem we first present a new lower bound of 8/3 ∼ 2.666 for the one-dimensional model. Secondly, we give algorithms for the twoand three-dimensional model when the input consists of unit fraction (lengths are 1 k , for some integer k ≥ 1) and power fraction (lengths are 1 2k , for some integer k ≥ 0) items. For scheduling algorithms, we present a polynomial time optimal offline algorithm for minimizing the electricity cost in Smart Grid. We consider the model in which the power requirement and the duration a request needs service are both unit-size. A possible future direction is to consider a generalization of this model. 1 Aims of the project This project is concerned with the study of algorithms for packing and scheduling. The aims of the project are the study of existing algorithms for packing and scheduling problems, design and analysis of improved algorithms for these problems, formulation of new problems for scheduling, and possibly experimental studies for some of the problems. In particular, for packing problems we look at the online dynamic bin packing problem, while for scheduling problems we consider an offline problem where we aim to minimize the total electricity cost in Smart Grid. 2 Summary of results In this section, we introduce the two types of problems we consider, for online packing algorithms and for an offline scheduling algorithm. For both types, we give an introduction to the problem and summarize the results obtained.