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APPROX

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

We give improved multi-pass streaming algorithms for the problem of maximizing a monotone or arbitrary non-negative submodular function subject to a general p-matchoid constraint in the model in which elements of the ground set arrive one at a time in a stream. The family of constraints we consider generalizes both the intersection of p arbitrary matroid constraints and p-uniform hypergraph matching. For monotone submodular functions, our algorithm attains a guarantee of p+1+ε using O(p/ε)-passes and requires storing only O(k) elements, where k is the maximum size of feasible solution. This immediately gives an O(1/ε)-pass (2+ε)-approximation for monotone submodular maximization in a matroid and (3+ε)-approximation for monotone submodular matching. Our algorithm is oblivious to the choice ε and can be stopped after any number of passes, delivering the appropriate guarantee. We extend our techniques to obtain the first multi-pass streaming algorithms for general, non-negative submodular functions subject to a p-matchoid constraint. We show that a randomized O(p/ε)-pass algorithm storing O(p³klog(k)/ε³) elements gives a (p+1+γ+O(ε))-approximation, where γ is the guarantee of the best-known offline algorithm for the same problem.

Chien-Chung Huang, Theophile Thiery, and Justin Ward. Improved Multi-Pass Streaming Algorithms for Submodular Maximization with Matroid Constraints. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 62:1-62:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{huang_et_al:LIPIcs.APPROX/RANDOM.2020.62, author = {Huang, Chien-Chung and Thiery, Theophile and Ward, Justin}, title = {{Improved Multi-Pass Streaming Algorithms for Submodular Maximization with Matroid Constraints}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {62:1--62:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.62}, URN = {urn:nbn:de:0030-drops-126657}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.62}, annote = {Keywords: submodular maximization, streaming algorithms, matroid, matchoid} }

Document

**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

We consider the classical k-means clustering problem in the setting of bi-criteria approximation, in which an algorithm is allowed to output beta*k > k clusters, and must produce a clustering with cost at most alpha times the to the cost of the optimal set of k clusters. We argue that this approach is natural in many settings, for which the exact number of clusters is a priori unknown, or unimportant up to a constant factor.
We give new bi-criteria approximation algorithms, based on linear programming and local search, respectively, which attain a guarantee alpha(beta) depending on the number beta*k of clusters that may be opened. Our guarantee alpha(beta) is always at most 9 + epsilon and improves rapidly with beta (for example: alpha(2) < 2.59, and alpha(3) < 1.4). Moreover, our algorithms have only polynomial dependence on the dimension of the input data, and so are applicable in high-dimensional settings.

Konstantin Makarychev, Yury Makarychev, Maxim Sviridenko, and Justin Ward. A Bi-Criteria Approximation Algorithm for k-Means. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 14:1-14:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{makarychev_et_al:LIPIcs.APPROX-RANDOM.2016.14, author = {Makarychev, Konstantin and Makarychev, Yury and Sviridenko, Maxim and Ward, Justin}, title = {{A Bi-Criteria Approximation Algorithm for k-Means}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {14:1--14:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.14}, URN = {urn:nbn:de:0030-drops-66370}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.14}, annote = {Keywords: k-means clustering, bicriteria approximation algorithms, linear programming, local search} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

In a stochastic probing problem we are given a universe E, where each element e in E is active independently with probability p in [0,1], and only a probe of e can tell us whether it is active or not. On this universe we execute a process that one by one probes elements - if a probed element is active, then we have to include it in the solution, which we gradually construct. Throughout the process we need to obey inner constraints on the set of elements taken into the solution, and outer constraints on the set of all probed elements. This abstract model was presented in [Gupta and Nagaraja, IPCO 2013], and provides a unified view of a number of problems. Thus far all the results in this general framework pertain only to the case in which we are maximizing a linear objective function of the successfully probed elements. In this paper we generalize the stochastic probing problem by considering a monotone submodular objective function. We give a (1-1/e)/(k_in+k_out+1)-approximation algorithm for the case in which we are given k_in greater than 0 matroids as inner constraints and k_out greater than 1 matroids as outer constraints. There are two main ingredients behind this result. First is a previously unpublished stronger bound on the continuous greedy algorithm due to Vondrak. Second is a rounding procedure that also allows us to obtain an improved 1/(k_in+k_out)-approximation for linear objective functions.

Marek Adamczyk, Maxim Sviridenko, and Justin Ward. Submodular Stochastic Probing on Matroids. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 29-40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{adamczyk_et_al:LIPIcs.STACS.2014.29, author = {Adamczyk, Marek and Sviridenko, Maxim and Ward, Justin}, title = {{Submodular Stochastic Probing on Matroids}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {29--40}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.29}, URN = {urn:nbn:de:0030-drops-44445}, doi = {10.4230/LIPIcs.STACS.2014.29}, annote = {Keywords: approximation algorithms, stochastic optimization, submodular optimization, matroids, iterative rounding} }

Document

**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

We consider the monotone submodular k-set packing problem in the context of the more general problem of maximizing a monotone submodular function in a k-exchange system. These systems, introduced by Feldman et al. [Feldman,2011], generalize the matroid k-parity problem in a wide class of matroids and capture many other combinatorial optimization problems. We give a deterministic, non-oblivious local search algorithm that attains an approximation ratio of (k + 3)/2 + epsilon for the problem of maximizing a monotone submodular function in a k-exchange system, improving on the best known result of k+epsilon, and answering an open question posed by Feldman et al.

Justin Ward. A (k+3)/2-approximation algorithm for monotone submodular k-set packing and general k-exchange systems. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 42-53, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{ward:LIPIcs.STACS.2012.42, author = {Ward, Justin}, title = {{A (k+3)/2-approximation algorithm for monotone submodular k-set packing and general k-exchange systems}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {42--53}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.42}, URN = {urn:nbn:de:0030-drops-34315}, doi = {10.4230/LIPIcs.STACS.2012.42}, annote = {Keywords: k-set packing, k-exchange systems, submodular maximization, local search, approximation algorithms} }

Document

**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

We present an optimal, combinatorial 1-1/e approximation algorithm for Maximum Coverage over a matroid constraint, using non-oblivious local search. Calinescu, Chekuri, Pál and Vondrák have given an optimal 1-1/e approximation algorithm for the more general problem of monotone submodular maximization over a matroid constraint. The advantage of our algorithm is that it is entirely combinatorial, and in many circumstances also faster, as well as conceptually simpler.
Following previous work on satisfiability problems by Alimonti, as well as by Khanna, Motwani, Sudan and Vazirani, our local search algorithm is *non-oblivious*. That is, our algorithm uses an auxiliary linear objective function to evaluate solutions. This function gives more weight to elements covered multiple times. We show that the locality ratio of the resulting local search procedure is at least 1-1/e. Our local search procedure only considers improvements of size 1. In contrast, we show that oblivious local search, guided only by the problem's objective function, achieves an approximation ratio of only (n-1)/(2n-1-k) when improvements of size k are considered.
In general, our local search algorithm could take an exponential amount of time to converge to an *exact* local optimum. We address this situation by using a combination of *approximate* local search and the same partial enumeration techniques as Calinescu et al., resulting in a clean 1 - 1/e-approximation algorithm running in polynomial time.

Yuval Filmus and Justin Ward. The Power of Local Search: Maximum Coverage over a Matroid. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 601-612, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{filmus_et_al:LIPIcs.STACS.2012.601, author = {Filmus, Yuval and Ward, Justin}, title = {{The Power of Local Search: Maximum Coverage over a Matroid}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {601--612}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.601}, URN = {urn:nbn:de:0030-drops-33968}, doi = {10.4230/LIPIcs.STACS.2012.601}, annote = {Keywords: approximation algorithms; maximum coverage; matroids; local search} }

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