Online Bin Stretching: Algorithms and Computer Lower Bounds Author: Martin Böhm Abstract: We investigate a problem in semi-online algorithm design, called Online Bin Stretching. The problem can be understood as an online repacking problem: the goal of the algorithm is to repack items of various sizes into m containers of identical size R > 1. The input items arrive one by one and the algorithm must assign an item to a container before the next item arrives. A specialty of this problem is that there is a specific guarantee made to the algorithm: the algorithm learns at the start of the input that there exists a packing of all input items into m containers of capacity 1. Our goal is to design algorithms for this problem which successfully pack the entire incoming sequence one by one while requiring the lowest container capacity R possible. In this thesis, we show several new results about Online Bin Stretching: First, we design an algorithm that is able to pack the entire input into m containers of capacity 1.5 regardless of what the vale of m will be. Second, we show a specialized algorithm for the setting of just 3 containers; this algorithm is able to pack into 3 bins of capacity 1.375. Finally, we design and implement an involved search algorithm which is able to find lower bounds for Online Bin...

This thesis is concerned with caching algorithms. We investigate three variations of the caching problem: web caching in the Torng framework, relative competitiveness and caching with request reordering. <br /><br /> In the first variation we define different cost models involving page sizes and page costs. We also present the Torng cost framework introduced by Torng in [29]. Next we analyze the competitive ratio of online deterministic marking algorithms in the BIT cost model combined with the Torng framework. We show that given some specific restrictions on the set of possible request sequences, any marking algorithm is 2-competitive. <br /><br /> The second variation consists in using the relative competitiveness ratio on an access graph as a complexity measure. We use the concept of access graphs introduced by Borodin [11] to define our own concept of relative competitive ratio. We demonstrate results regarding the relative competitiveness of two cache eviction policies in both the basic and the Torng framework combined with the CLASSICAL cost model. <br /><br /> The third variation is caching with request reordering. Two reordering models are defined. We prove some important results about the value of a move and number of orderings, then demonstrate results about the approximation factor and competitive ratio of offline and online reordering schemes, respectively.

Allocation problems have been central to the development of the theory of algorithms and also find applications in several realms of computer science and economics. In this thesis we initiate a systematic study of these problems in situations with limited information. Towards this end we explore several modes by which data may be obfuscated from the algorithm. We begin by investigating temporal constraints where data is revealed to the algorithm over time. Concretely, we consider the online bipartite matching problem in the unknown distribution model and present the first algorithm that breaches the 1-1/e barrier for this problem. Next we study issues arising from data acquisition costs that are prevalent in ad-systems and kidney exchanges. Motivated by these constraints we introduce the query-commit model and present constant factor algorithms for the maximum matching and the adwords problem in this model. Finally we assess the approximability of several classical allocation problems with multiple agents having complex non-linear cost functions. This presents an additional obstacle since the support for the cost functions may be extremely large entailing oracle access. We show tight information theoretic lower bounds for the general class of submodular functions and also extend these results to get lower bounds for a subclass of succinctly representable non-linear cost functions.

Thesis (Ph. D.)--University of Hong Kong, 2005. / Title proper from title frame. Also available in printed format.

In various business and military settings, there is an expectation of on-demand delivery of supplies and services. Typically, several delivery vehicles (also called servers) carry these supplies. Requests arrive one at a time and when a request arrives, a server is assigned to this request at a cost that is proportional to the distance between the server and the request. Bad assignments will not only lead to larger costs but will also create bottlenecks by increasing delivery time. There is, therefore, a need to design decision-making algorithms that produce cost-effective assignments of servers to requests in real-time. In this thesis, we consider the online bipartite matching problem where each server can serve exactly one request. In the online minimum metric bipartite matching problem, we are provided with a set of server locations in a metric space. Requests arrive one at a time that have to be immediately and irrevocably matched to a free server. The total cost of matching all the requests to servers, also known as the online matching is the sum of the cost of all the edges in the matching. There are many well-studied models for request generation. We study the problem in the adversarial model where an adversary who knows the decisions made by the algorithm generates a request sequence to maximize ratio of the cost of the online matching and the minimum-cost matching (also called the competitive ratio). An algorithm is a-competitive if the cost of online matching is at most 'a' times the minimum cost. A recently discovered robust and deterministic online algorithm (we refer to this as the robust matching or the RM-Algorithm) was shown to have optimal competitive ratios in the adversarial model and a relatively weaker random arrival model. We extend the analysis of the RM-Algorithm in the adversarial model and show that the competitive ratio of the algorithm is sensitive to the input, i.e., for "nice" input metric spaces or "nice" server placements, the performance guarantees of the RM-Algorithm is significantly better. In fact, we show that the performance is almost optimal for any fixed metric space and server locations. / Master of Science

<p>Scheduling a set of jobs over a collection of machines to optimize a certain quality-of-service measure is one of the most important research topics in both computer science theory and practice. In this thesis, we design algorithms that optimize {\em flow-time} (or delay) of jobs for scheduling problems that arise in a wide range of applications. We consider the classical model of unrelated machine scheduling and resolve several long standing open problems; we introduce new models that capture the novel algorithmic challenges in scheduling jobs in data centers or large clusters; we study the effect of selfish behavior in distributed and decentralized environments; we design algorithms that strive to balance the energy consumption and performance. </p><p>The technically interesting aspect of our work is the surprising connections we establish between approximation and online algorithms, economics, game theory, and queuing theory. It is the interplay of ideas from these different areas that lies at the heart of most of the algorithms presented in this thesis.</p><p>The main contributions of the thesis can be placed in one of the following categories.</p><p>1. Classical Unrelated Machine Scheduling: We give the first polygorithmic approximation algorithms for minimizing the average flow-time and minimizing the maximum flow-time in the offline setting. In the online and non-clairvoyant setting, we design the first non-clairvoyant algorithm for minimizing the weighted flow-time in the resource augmentation model. Our work introduces iterated rounding technique for the offline flow-time optimization, and gives the first framework to analyze non-clairvoyant algorithms for unrelated machines.</p><p>2. Polytope Scheduling Problem: To capture the multidimensional nature of the scheduling problems that arise in practice, we introduce Polytope Scheduling Problem (\psp). The \psp problem generalizes almost all classical scheduling models, and also captures hitherto unstudied scheduling problems such as routing multi-commodity flows, routing multicast (video-on-demand) trees, and multi-dimensional resource allocation. We design several competitive algorithms for the \psp problem and its variants for the objectives of minimizing the flow-time and completion time. Our work establishes many interesting connections between scheduling and market equilibrium concepts, fairness and non-clairvoyant scheduling, and queuing theoretic notion of stability and resource augmentation analysis.</p><p>3. Energy Efficient Scheduling: We give the first non-clairvoyant algorithm for minimizing the total flow-time + energy in the online and resource augmentation model for the most general setting of unrelated machines.</p><p>4. Selfish Scheduling: We study the effect of selfish behavior in scheduling and routing problems. We define a fairness index for scheduling policies called {\em bounded stretch}, and show that for the objective of minimizing the average (weighted) completion time, policies with small stretch lead to equilibrium outcomes with small price of anarchy. Our work gives the first linear/ convex programming duality based framework to bound the price of anarchy for general equilibrium concepts such as coarse correlated equilibrium.</p> / Dissertation

published_or_final_version / abstract / Computer Science / Doctoral / Doctor of Philosophy

This thesis proposes a novel experimental environment (non-linear stochastic gradient descent, NL-SGD), as well as a novel online learning algorithm (OL SVM), for solving a classic nonlinear Soft Margin L1 Support Vector Machine (SVM) problem using a Stochastic Gradient Descent (SGD) algorithm. The NL-SGD implementation has a unique method of random sampling and alpha calculations. The developed code produces a competitive accuracy and speed in comparison with the solutions of the Direct L2 SVM obtained by software for Minimal Norm SVM (MN-SVM) and Non-Negative Iterative Single Data Algorithm (NN-ISDA). The latter two algorithms have shown excellent performances on large datasets; which is why we chose to compare NL-SGD and OL SVM to them. All experiments have been done under strict double (nested) cross-validation, and the results are reported in terms of accuracy and CPU times. OL SVM has been implemented within MATLAB and is compared to the classic Sequential Minimal Optimization (SMO) algorithm implemented within MATLAB's solver, fitcsvm. The experiments with OL SVM have been done using k-fold cross-validation and the results reported in % error and % speedup of CPU Time.

In a search game, a seeker searches for a hider in some space. The seeker wishes to find the hider as quickly as possible, and the hider wishes to avoid capture as long as possible. In this paper, I will focus on the case where the search space is a star, and the only information the seeker has is the speed of the hider. I will provide algorithms for some cases where the seeker is guaranteed to find the hider and prove optimality for some of these cases. Also, I will look at some cases where the hider can avoid capture indefinitely. I will also present some results for searching on trees.

The online and data stream models of computation have recently attracted considerable research attention due to many real-world applications in various areas such as data mining, machine learning, distributed computing, and robotics. In both these models, input items arrive one at a time, and the algorithms must decide based on the partial data received so far, without any secure information about the data that will arrive in the future. In this thesis, we investigate efficient algorithms for a number of fundamental geometric optimization problems in the online and data stream models. The problems studied in this thesis can be divided into two major categories: geometric clustering and computing various extent measures of a set of points. In the online setting, we show that the basic unit clustering problem admits non-trivial algorithms even in the simplest one-dimensional case: we show that the naive upper bounds on the competitive ratio of algorithms for this problem can be beaten using randomization. In the data stream model, we propose a new streaming algorithm for maintaining "core-sets" of a set of points in fixed dimensions, and also, introduce a new simple framework for transforming a class of offline algorithms to their equivalents in the data stream model. These results together lead to improved streaming approximation algorithms for a wide variety of geometric optimization problems in fixed dimensions, including diameter, width, k-center, smallest enclosing ball, minimum-volume bounding box, minimum enclosing cylinder, minimum-width enclosing spherical shell/annulus, etc. In high-dimensional data streams, where the dimension is not a constant, we propose a simple streaming algorithm for the minimum enclosing ball (the 1-center) problem with an improved approximation factor.