Greedy algorithm

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

Full-View Area Coverage in Camera Sensor Networks: Dimension Reduction and Near-Optimal Solutions

Abstract: We study the problem of minimum-number full-view area coverage in camera sensor networks, i.e., how to select the minimum number of camera sensors to guarantee the full-view coverage of a given region. Full-view area coverage is challenging because the full-view coverage of a 2-D continuous domain has to be considered. To tackle this challenge, we first study the intrinsic geometric relationship between the full-view area coverage and the full-view point coverage and prove that the full-view area coverage can be guaranteed, as long as a selected full-view ensuring set of points is full-view covered. This leads to a significant dimension reduction for the full-view area coverage problem. Next, we prove that the minimum-number full-view point coverage is NP-hard and propose two approximation algorithms to solve it from two different perspectives, respectively: 1) By introducing a full-view coverage ratio function, we quantify the “contribution” of each camera sensor to the full-view coverage through which we transform the full-view point coverage into a submodular set cover problem and propose a greedy algorithm (GA); and 2) by studying the geometric relationship between the full-view coverage and the traditional coverage, we propose a set-cover-based algorithm (SCA). We analyze the performance of these two approximation algorithms and characterize their approximation ratios. Furthermore, we devise two distributed algorithms that obtain the same approximation ratios as GA and SCA, respectively. Finally, we provide extensive simulation results to validate our analysis.

Text Summarization Model Based on Maximum Coverage Problem and its Variant

Abstract: We discuss text summarization in terms of maximum coverage problem and its variant. We explore some decoding algorithms including the ones never used in this summarization formulation, such as a greedy algorithm with performance guarantee, a randomized algorithm, and a branch-and-bound method. On the basis of the results of comparative experiments, we also augment the summarization model so that it takes into account the relevance to the document cluster. Through experiments, we showed that the augmented model is superior to the best-performing method of DUC'04 on ROUGE-1 without stopwords.

Efficient Spare Allocation in Reconfigurable Arrays

Abstract: The issue of yield degradation due to physical failures in large memory and processor arrays is of significant importance to semiconductor manufacturers. One method of increasing the yield for iterated arrays of memory cells or processing elements is by incorporating spare rows and columns in the die or wafer which can be programmed into the array. This paper addresses the issue of computer-aided design approaches to optimal reconfiguration of such arrays. The paper presents the first formal analysis of the problem. The complexity of optimal reconfiguration is shown to be NP-complete for rectangular arrays utilizing spare rows and columns. In contrast to previously proposed exhaustive search and greedy algorithms, this paper develops a heuristic branch and bound approach based on the complexity analysis, which allows for flexible and highly efficient reconfiguration. Initial screening is performed by a bipartite graph matching algorithm.

On minimizing budget and time in influence propagation over social networks

Abstract: In recent years, study of influence propagation in social networks has gained tremendous attention. In this context, we can identify three orthogonal dimensions—the number of seed nodes activated at the beginning (known as budget), the expected number of activated nodes at the end of the propagation (known as expected spread or coverage), and the time taken for the propagation. We can constrain one or two of these and try to optimize the third. In their seminal paper, Kempe et al. constrained the budget, left time unconstrained, and maximized the coverage: this problem is known as Influence Maximization (or MAXINF for short). In this paper, we study alternative optimization problems which are naturally motivated by resource and time constraints on viral marketing campaigns. In the first problem, termed minimum target set selection (or MINTSS for short), a coverage threshold η is given and the task is to find the minimum size seed set such that by activating it, at least η nodes are eventually activated in the expected sense. This naturally captures the problem of deploying a viral campaign on a budget. In the second problem, termed MINTIME, the goal is to minimize the time in which a predefined coverage is achieved. More precisely, in MINTIME, a coverage threshold η and a budget threshold k are given, and the task is to find a seed set of size at most k such that by activating it, at least η nodes are activated in the expected sense, in the minimum possible time. This problem addresses the issue of timing when deploying viral campaigns. Both these problems are NP-hard, which motivates our interest in their approximation. For MINTSS, we develop a simple greedy algorithm and show that it provides a bicriteria approximation. We also establish a generic hardness result suggesting that improving this bicriteria approximation is likely to be hard. For MINTIME, we show that even bicriteria and tricriteria approximations are hard under several conditions. We show, however, that if we allow the budget for number of seeds k to be boosted by a logarithmic factor and allow the coverage to fall short, then the problem can be solved exactly in PTIME, i.e., we can achieve the required coverage within the time achieved by the optimal solution to MINTIME with budget k and coverage threshold η. Finally, we establish the value of the approximation algorithms, by conducting an experimental evaluation, comparing their quality against that achieved by various heuristics.

Convergence Rates of the POD–Greedy Method

Abstract: Iterative approximation algorithms are successfully applied in parametric approximation tasks. In particular, reduced basis methods make use of the so-called Greedy algorithm for approxi- mating solution sets of parametrized partial differential equations. Recently, ap rioriconvergence rate statements for this algorithm have been given (Buffa et al. 2009, Binev et al. 2010). The goal of the current study is the extension to time-dependent problems, which are typically approximated using the POD-Greedy algorithm (Haasdonk and Ohlberger 2008). In this algorithm, each greedy step is invoking a temporal compression step by performing a proper orthogonal decomposition (POD). Using a suitable coefficient representation of the POD-Greedy algorithm, we show that the existing conver- gence rate results of the Greedy algorithm can be extended. In particular, exponential or algebraic convergence rates of the Kolmogorov n-widths are maintained by the POD-Greedy algorithm.