We present techniques for decreasing the error probability of randomized algorithms and for converting randomized algorithms to deterministic (nonuniform) algorithms. Unlike most existing techniques that involve repetition of the randomized algorithm and hence a slowdown, our techniques produce algorithms with a similar run-time to the original randomized algorithms. The amplification technique is related to a certain stochastic multi-armed bandit problem. The derandomization technique - which is the main contribution of this work - points to an intriguing connection between derandomization and sketching/sparsification. We demonstrate the techniques by showing algorithms for approximating free games (constraint satisfaction problems on dense bipartite graphs).
{"title":"Amplification and Derandomization without Slowdown","authors":"O. Grossman, Dana Moshkovitz","doi":"10.1109/FOCS.2016.87","DOIUrl":"https://doi.org/10.1109/FOCS.2016.87","url":null,"abstract":"We present techniques for decreasing the error probability of randomized algorithms and for converting randomized algorithms to deterministic (nonuniform) algorithms. Unlike most existing techniques that involve repetition of the randomized algorithm and hence a slowdown, our techniques produce algorithms with a similar run-time to the original randomized algorithms. The amplification technique is related to a certain stochastic multi-armed bandit problem. The derandomization technique - which is the main contribution of this work - points to an intriguing connection between derandomization and sketching/sparsification. We demonstrate the techniques by showing algorithms for approximating free games (constraint satisfaction problems on dense bipartite graphs).","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132420419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A wide variety of problems in machine learning, including exemplar clustering, document summarization, and sensor placement, can be cast as constrained submodular maximization problems. A lot of recent effort has been devoted to developing distributed algorithms for these problems. However, these results suffer from high number of rounds, suboptimal approximation ratios, or both. We develop a framework for bringing existing algorithms in the sequential setting to the distributed setting, achieving near optimal approximation ratios for many settings in only a constant number of MapReduce rounds. Our techniques also give a fast sequential algorithm for non-monotone maximization subject to a matroid constraint.
{"title":"A New Framework for Distributed Submodular Maximization","authors":"R. Barbosa, Alina Ene, Huy L. Nguyen, Justin Ward","doi":"10.1109/FOCS.2016.74","DOIUrl":"https://doi.org/10.1109/FOCS.2016.74","url":null,"abstract":"A wide variety of problems in machine learning, including exemplar clustering, document summarization, and sensor placement, can be cast as constrained submodular maximization problems. A lot of recent effort has been devoted to developing distributed algorithms for these problems. However, these results suffer from high number of rounds, suboptimal approximation ratios, or both. We develop a framework for bringing existing algorithms in the sequential setting to the distributed setting, achieving near optimal approximation ratios for many settings in only a constant number of MapReduce rounds. Our techniques also give a fast sequential algorithm for non-monotone maximization subject to a matroid constraint.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115456392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the recent formulation of the Algorithmic Lovász Local Lemma [1], [2] for finding objects that avoid "bad features", or "flaws". It extends the Moser-Tardos resampling algorithm [3] to more general discrete spaces. At each step the method picks a flaw present in the current state and "resamples" it using a "resampling oracle" provided by the user. However, it is less flexible than the Moser-Tardos method since [1], [2] require a specific flaw selection rule, whereas [3] allows an arbitrary rule (and thus can potentially be implemented more efficiently). We formulate a new "commutativity" condition, and prove that it is sufficient for an arbitrary rule to work. It also enables an efficient parallelization under an additional assumption. We then show that existing resampling oracles for perfect matchings and permutations do satisfy this condition. Finally, we generalize the precondition in [2] (in the case of symmetric potential causality graphs). This unifies special cases that previously were treated separately.
{"title":"Commutativity in the Algorithmic Lovász Local Lemma","authors":"V. Kolmogorov","doi":"10.1109/FOCS.2016.88","DOIUrl":"https://doi.org/10.1109/FOCS.2016.88","url":null,"abstract":"We consider the recent formulation of the Algorithmic Lovász Local Lemma [1], [2] for finding objects that avoid \"bad features\", or \"flaws\". It extends the Moser-Tardos resampling algorithm [3] to more general discrete spaces. At each step the method picks a flaw present in the current state and \"resamples\" it using a \"resampling oracle\" provided by the user. However, it is less flexible than the Moser-Tardos method since [1], [2] require a specific flaw selection rule, whereas [3] allows an arbitrary rule (and thus can potentially be implemented more efficiently). We formulate a new \"commutativity\" condition, and prove that it is sufficient for an arbitrary rule to work. It also enables an efficient parallelization under an additional assumption. We then show that existing resampling oracles for perfect matchings and permutations do satisfy this condition. Finally, we generalize the precondition in [2] (in the case of symmetric potential causality graphs). This unifies special cases that previously were treated separately.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134306816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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