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Register a new userLocal Algorithms for Estimating Effective Resistance
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Pan Peng
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Effective resistance is an important metric that measures the similarity of two vertices in a graph. It has found applications in graph clustering, recommendation systems and network reliability, among others. In spite of the importance of the effective resistances, we still lack efficient algorithms to exactly compute or approximate them on massive graphs. In this work, we design several emph{local algorithms} for estimating effective resistances, which are algorithms that only read a small portion of the input while still having provable performance guarantees. To illustrate, our main algorithm approximates the effective resistance between any vertex pair $s,t$ with an arbitrarily small additive error $varepsilon$ in time $O(mathrm{poly}(log n/varepsilon))$, whenever the underlying graph has bounded mixing time. We perform an extensive empirical study on several benchmark datasets, validating the performance of our algorithms.
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For providing quick and accurate search results, a search engine maintains a local snapshot of the entire web. And, to keep this local cache fresh, it employs a crawler for tracking changes across various web pages. It would have been ideal if the crawler managed to update the local snapshot as soon as a page changed on the web. However, finite bandwidth availability and server restrictions mean that there is a bound on how frequently the different pages can be crawled. This then brings forth the following optimisation problem: maximise the freshness of the local cache subject to the crawling frequency being within the prescribed bounds. Recently, tractable algorithms have been proposed to solve this optimisation problem under different cost criteria. However, these assume the knowledge of exact page change rates, which is unrealistic in practice. We address this issue here. Specifically, we provide three novel schemes for online estimation of page change rates. All these schemes only need partial information about the page change process, i.e., they only need to know if the page has changed or not since the last crawl instance. Our first scheme is based on the law of large numbers, the second on the theory of stochastic approximation, while the third is an extension of the second and involves an additional momentum term. For all of these schemes, we prove convergence and, also, provide their convergence rates. As far as we know, the results concerning the third estimator is quite novel. Specifically, this is the first convergence type result for a stochastic approximation algorithm with momentum. Finally, we provide some numerical experiments (on real as well as synthetic data) to compare the performance of our proposed estimators with the existing ones (e.g., MLE).
We consider the task of designing Local Computation Algorithms (LCA) for applications of the Lov{a}sz Local Lemma (LLL). LCA is a class of sublinear algorithms proposed by Rubinfeld et al.~cite{Ronitt} that have received a lot of attention in recent years. The LLL is an existential, sufficient condition for a collection of sets to have non-empty intersection (in applications, often, each set comprises all objects having a certain property). The ground-breaking algorithm of Moser and Tardos~cite{MT} made the LLL fully constructive, following earlier results by Beck~cite{beck_lll} and Alon~cite{alon_lll} giving algorithms under significantly stronger LLL-like conditions. LCAs under those stronger conditions were given in~cite{Ronitt}, where it was asked if the Moser-Tardos algorithm can be used to design LCAs under the standard LLL condition. The main contribution of this paper is to answer this question affirmatively. In fact, our techniques yield LCAs for settings beyond the standard LLL condition.
We consider the revenue maximization problem in social advertising, where a social network platform owner needs to select seed users for a group of advertisers, each with a payment budget, such that the total expected revenue that the owner gains from the advertisers by propagating their ads in the network is maximized. Previous studies on this problem show that it is intractable and present approximation algorithms. We revisit this problem from a fresh perspective and develop novel efficient approximation algorithms, both under the setting where an exact influence oracle is assumed and under one where this assumption is relaxed. Our approximation ratios significantly improve upon the previous ones. Furthermore, we empirically show, using extensive experiments on four datasets, that our algorithms considerably outperform the existing methods on both the solution quality and computation efficiency.
We present an $(e^{O(p)} frac{log ell}{loglogell})$-approximation algorithm for socially fair clustering with the $ell_p$-objective. In this problem, we are given a set of points in a metric space. Each point belongs to one (or several) of $ell$ groups. The goal is to find a $k$-medians, $k$-means, or, more generally, $ell_p$-clustering that is simultaneously good for all of the groups. More precisely, we need to find a set of $k$ centers $C$ so as to minimize the maximum over all groups $j$ of $sum_{u text{ in group }j} d(u,C)^p$. The socially fair clustering problem was independently proposed by Ghadiri, Samadi, and Vempala [2021] and Abbasi, Bhaskara, and Venkatasubramanian [2021]. Our algorithm improves and generalizes their $O(ell)$-approximation algorithms for the problem. The natural LP relaxation for the problem has an integrality gap of $Omega(ell)$. In order to obtain our result, we introduce a strengthened LP relaxation and show that it has an integrality gap of $Theta(frac{log ell}{loglogell})$ for a fixed $p$. Additionally, we present a bicriteria approximation algorithm, which generalizes the bicriteria approximation of Abbasi et al. [2021].
Submodular optimization has numerous applications such as crowdsourcing and viral marketing. In this paper, we study the fundamental problem of non-negative submodular function maximization subject to a $k$-system constraint, which generalizes many other important constraints in submodular optimization such as cardinality constraint, matroid constraint, and $k$-extendible system constraint. The existing approaches for this problem achieve the best-known approximation ratio of $k+2sqrt{k+2}+3$ (for a general submodular function) based on deterministic algorithmic frameworks. We propose several randomized algorithms that improve upon the state-of-the-art algorithms in terms of approximation ratio and time complexity, both under the non-adaptive setting and the adaptive setting. The empirical performance of our algorithms is extensively evaluated in several applications related to data mining and social computing, and the experimental results demonstrate the superiorities of our algorithms in terms of both utility and efficiency.
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