Probabilistic analysis of Online Bin Coloring algorithms via Stochastic Comparison

This paper proposes a new method for probabilistic analysis of online algorithms that is based on the notion of stochastic dominance. We develop the method for the Online Bin Coloring problem introduced by Krumke et al. Using methods for the stochastic comparison of Markov chains we establish the strong result that the performance of the online algorithm GreedyFit is stochastically dominated by the performance of the algorithm OneBin for any number of items processed. This result gives a more realistic picture than competitive analysis and explains the behavior observed in simulations.mathematical applications;

Probabilistic alternatives for competitive analysis

In the last 20 years competitive analysis has become the main tool for analyzing the quality of online algorithms. Despite of this, competitive analysis has also been criticized: it sometimes cannot discriminate between algorithms that exhibit significantly different empirical behavior or it even favors an algorithm that is worse from an empirical point of view. Therefore, there have been several approaches to circumvent these drawbacks. In this survey, we discuss probabilistic alternatives for competitive analysis.operations research and management science;

Simple optimality proofs for Least Recently Used in the presence of locality of reference

It is well known that competitive analysis yields results that do not reflect the observed performance of online paging algorithms. Many deterministic paging algorithms achieve the same competitive ratio, ranging from inefficient strategies as flush-when-full to the well-performing least-recently-used (LRU). In this paper, we study this fundamental online problem from the viewpoint of stochastic dominance. We give simple proofs that whensequences are drawn from distributions modelling locality of reference, LRU stochastically dominates any other online paging algorithm. As a byproduct, we obtain simple proofs of some earlier results.operations research and management science;

Topics in Packing and Scheduling

Packing and scheduling models include some of the most fundamental problems in operations research and computer science. These broad classes include a wide range of models with applications including logistics, production planning, wireless network design, circuit design, and cloud computing, to name a few. In this thesis we study three such models: dynamic node packing, interval scheduling with economies of scale, and temporal bin packing with half-capacity jobs; each extends on a well-known problem in packing and scheduling. While the problems are generally distinct, this research was broadly inspired by applications to cloud computing. Specifically, this thesis is motivated by problems cloud service providers face when servicing requests for virtual machines. In Chapter 2, we propose a dynamic version of the node packing problem. In this model, instead of being given the edges upfront, we model them as Bernoulli random variables. At each step, the decision maker selects an available node and then observes edges adjacent to this node. The goal is a policy that maximizes the expected value of the resulting packing. We model the problem as a Markov decision problem and conduct a polyhedral study of the problem's achievable probabilities polytope. We develop a variety of valid inequalities based on paths, cycles, and cliques. In Chapter 3, we study interval scheduling problems exhibiting economies of scale. An instance is given by a set of interval jobs and a cost function. Specifically, we focus on the max-weight function and non-negative, non-decreasing concave functions of total schedule weight. The goal is a partition of the jobs minimizing the total cost with the constraint that jobs within the same schedule cannot overlap. We propose a set covering formulation and a column generation algorithm to solve its linear relaxation, providing efficient pricing algorithms for the studied cases. To obtain integer solutions, we extend the column generation approach using branch-and-price. In Chapter 4, we study a different model with interval jobs. In this problem, interval jobs are partitioned into bins such that at most two jobs in a bin overlap at once. The decision maker is tasked with minimizing the time-average number of bins required to pack all jobs. We call this problem temporal bin packing with half-capacity jobs; it is a special case of the general temporal bin packing problem with bounded parallelism. We study the worst-case performance of a well-known static lower bound, and, motivated by this analysis, we introduce a novel lower bound and integer programming formulation based on formulating the problem as a series of matching problems. We provide theoretical guarantees on the relative strengths of the static bound, the matching-based bound, and various linear programming bounds.Ph.D