Greedy algorithm

About: Greedy algorithm is a research topic. Over the lifetime, 15347 publications have been published within this topic receiving 393945 citations.

Strongly Polynomial Algorithms for the High Multiplicity Scheduling Problem

Abstract: A high multiplicity scheduling problem consists of many jobs which can be partitioned into relatively few groups, where all the jobs within each group are identical. Polynomial, and even strongly polynomial, algorithms for the standard scheduling problem, in which all jobs are assumed to be distinct, become exponential for the corresponding high multiplicity problem. In this paper, we study various high multiplicity problems of scheduling unit-time jobs on a single machine. We provide strongly polynomial algorithms for constructing optimal schedules with respect to several measures of efficiency (completion time, lateness, tardiness, the number of tardy jobs and their weighted counterparts). The algorithms require a number of operations that are polynomial in the number of groups rather than in the total number of jobs. As a by-product, we identify a new family of nxn transportation problems which are solvable in O(n log n) time by a simple greedy algorithm.

Online job-migration for reducing the electricity bill in the cloud

Abstract: Energy costs are becoming the fastest-growing element in datacenter operation costs. One basic approach to reduce these costs is to exploit the spatiotemporal variation in electricity prices by moving computation to datacenters in which energy is available at a cheaper price. However, injudicious job migration between datacenters might increase the overall operation cost due to the bandwidth costs of transferring application state and data over the wide-area network. To address this challenge, we propose novel online algorithms for migrating batch jobs between datacenters, which handle the fundamental tradeoff between energy and bandwidth costs. A distinctive feature of our algorithms is that they consider not only the current availability and cost of (possibly multiple) energy sources, but also the future variability and uncertainty thereof. Using the framework of competitive-analysis, we establish worst-case performance bounds for our basic online algorithm. We then propose a practical, easy-to-implement version of the basic algorithm, and evaluate it through simulations on real electricity pricing and job workload data. The simulation results indicate that our algorithm outperforms plausible greedy algorithms that ignore future outcomes. Notably, the actual performance of our approach is significantly better than the theoretical guarantees, within 6% of the optimal offline solution.

Sparse convex optimization methods for machine learning

Abstract: Convex optimization is at the core of many of today’s analysis tools for large datasets, and in particular machine learning methods. In this thesis we will study the general setting of optimizing (minimizing) a convex function over a compact convex domain. In the first part of this thesis, we study a simple iterative approximation algorithm for that class of optimization problems, based on the classical method by Frank & Wolfe. The algorithm only relies on supporting hyperplanes to the function that we need to optimize. In each iteration, we move slightly towards a point which (approximately) minimizes the linear function given by the supporting hyperplane at the current point, where the minimum is taken over the original optimization domain. In contrast to gradient-descent-type methods, this algorithm does not need any projection steps in order to stay inside the optimization domain. Our framework generalizes the sparse greedy algorithm of Frank & Wolfe and its recent primal-dual analysis by Clarkson (and the low-rank SDP approach by Hazan) to arbitrary compact convex domains. Analogously, we give a convergence proof guaranteeing e-small error — which in our context is the duality gap — after O( 1e ) iterations. This method allows us to understand the sparsity of approximate solutions for any `1-regularized convex optimization problem (and for optimization over the simplex), expressed as a function of the approximation quality. Here we obtain matching upper and lower bounds of Θ ( 1 e ) for the sparsity. The same bounds apply to low-rank semidefinite optimization with bounded trace, showing that rank O ( 1 e ) is best possible here as well. For some classes of geometric optimization problems, our algorithm has a simple geometric interpretation, which is also known as the coreset concept. Here we will study linear classifiers such as support vector machines (SVM) and perceptrons, as well as general distance computations between convex hulls (or polytopes). Here the framework will allow us to understand the sparsity of SVM solutions, here being the number of support vectors, in terms of the required approximation quality.

Trichromatic Online Matching in Real-Time Spatial Crowdsourcing

Abstract: The prevalence of mobile Internet techniques and Online-To-Offline (O2O) business models has led the emergence of various spatial crowdsourcing (SC) platforms in our daily life. A core issue of SC is to assign real-time tasks to suitable crowd workers. Existing approaches usually focus on the matching of two types of objects, tasks and workers, or assume the static offline scenarios, where the spatio-temporal information of all the tasks and workers is known in advance. Recently, some new emerging O2O applications incur new challenges: SC platforms need to assign three types of objects, tasks, workers and workplaces, and support dynamic real-time online scenarios, where the existing solutions cannot handle. In this paper, based on the aforementioned challenges, we formally define a novel dynamic online task assignment problem, called the trichromatic online matching in real-time spatial crowdsourcing (TOM) problem, which is proven to be NP-hard. Thus, we first devise an efficient greedy online algorithm. However, the greedy algorithm can be trapped into local optimal solutions easily. We then present a threshold-based randomized algorithm that not only guarantees a tighter competitive ratio but also includes an adaptive optimization technique, which can quickly learn the optimal threshold for the randomized algorithm. Finally, we verify the effectiveness and efficiency of the proposed methods through extensive experiments on real and synthetic datasets.

A Least-Squares Method for Sparse Low Rank Approximation of Multivariate Functions

Abstract: In this paper, we propose a low rank approximation method based on discrete least-squares for the approximation of a multivariate function from random, noise-free observations. Sparsity inducing regularization techniques are used within classical algorithms for low rank approximation in order to exploit the possible sparsity of low rank approximations. Sparse low rank approximations are constructed with a robust updated greedy algorithm, which includes an optimal selection of regularization parameters and approximation ranks using cross validation techniques. Numerical examples demonstrate the capability of approximating functions of many variables even when very few function evaluations are available, thus proving the interest of the proposed algorithm for the propagation of uncertainties through complex computational models.