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DOI: 10.4230/LIPIcs.ITCS.2023.95

An Improved Lower Bound for Matroid Intersection Prophet Inequalities

Authors: Raghuvansh R. Saxena, Santhoshini Velusamy, and S. Matthew Weinberg

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract

We consider prophet inequalities subject to feasibility constraints that are the intersection of q matroids. The best-known algorithms achieve a Θ(q)-approximation, even when restricted to instances that are the intersection of q partition matroids, and with i.i.d. Bernoulli random variables [José R. Correa et al., 2022; Moran Feldman et al., 2016; Marek Adamczyk and Michal Wlodarczyk, 2018]. The previous best-known lower bound is Θ(√q) due to a simple construction of [Robert Kleinberg and S. Matthew Weinberg, 2012] (which uses i.i.d. Bernoulli random variables, and writes the construction as the intersection of partition matroids). We establish an improved lower bound of q^{1/2+Ω(1/log log q)} by writing the construction of [Robert Kleinberg and S. Matthew Weinberg, 2012] as the intersection of asymptotically fewer partition matroids. We accomplish this via an improved upper bound on the product dimension of a graph with p^p disjoint cliques of size p, using recent techniques developed in [Noga Alon and Ryan Alweiss, 2020].

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Raghuvansh R. Saxena, Santhoshini Velusamy, and S. Matthew Weinberg. An Improved Lower Bound for Matroid Intersection Prophet Inequalities. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 95:1-95:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{saxena_et_al:LIPIcs.ITCS.2023.95,
  author =	{Saxena, Raghuvansh R. and Velusamy, Santhoshini and Weinberg, S. Matthew},
  title =	{{An Improved Lower Bound for Matroid Intersection Prophet Inequalities}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{95:1--95:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.95},
  URN =		{urn:nbn:de:0030-drops-175986},
  doi =		{10.4230/LIPIcs.ITCS.2023.95},
  annote =	{Keywords: Prophet Inequalities, Intersection of Matroids}
}

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DOI: 10.4230/LIPIcs.STACS.2021.14

A Nearly Optimal Deterministic Online Algorithm for Non-Metric Facility Location

Authors: Marcin Bienkowski, Björn Feldkord, and Paweł Schmidt

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract

In the online non-metric variant of the facility location problem, there is a given graph consisting of a set F of facilities (each with a certain opening cost), a set C of potential clients, and weighted connections between them. The online part of the input is a sequence of clients from C, and in response to any requested client, an online algorithm may open an additional subset of facilities and must connect the given client to an open facility. We give an online, polynomial-time deterministic algorithm for this problem, with a competitive ratio of O(log |F| ⋅ (log |C| + log log |F|)). The result is optimal up to loglog factors. Our algorithm improves over the O((log |C| + log |F|) ⋅ (log |C| + log log |F|))-competitive construction that first reduces the facility location instance to a set cover one and then later solves such instance using the deterministic algorithm by Alon et al. [TALG 2006]. This is an asymptotic improvement in a typical scenario where |F| ≪ |C|. We achieve this by a more direct approach: we design an algorithm for a fractional relaxation of the non-metric facility location problem with clustered facilities. To handle the constraints of such non-covering LP, we combine the dual fitting and multiplicative weight updates approach. By maintaining certain additional monotonicity properties of the created fractional solution, we can handle the dependencies between facilities and connections in a rounding routine. Our result, combined with the algorithm by Naor et al. [FOCS 2011] yields the first deterministic algorithm for the online node-weighted Steiner tree problem. The resulting competitive ratio is O(log k ⋅ log² 𝓁) on graphs of 𝓁 nodes and k terminals.

Cite as

Marcin Bienkowski, Björn Feldkord, and Paweł Schmidt. A Nearly Optimal Deterministic Online Algorithm for Non-Metric Facility Location. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bienkowski_et_al:LIPIcs.STACS.2021.14,
  author =	{Bienkowski, Marcin and Feldkord, Bj\"{o}rn and Schmidt, Pawe{\l}},
  title =	{{A Nearly Optimal Deterministic Online Algorithm for Non-Metric Facility Location}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{14:1--14:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.14},
  URN =		{urn:nbn:de:0030-drops-136598},
  doi =		{10.4230/LIPIcs.STACS.2021.14},
  annote =	{Keywords: Online algorithms, deterministic rounding, linear programming, facility location, set cover}
}

Document

DOI: 10.4230/LIPIcs.IPEC.2020.14

FPT Approximation for Constrained Metric k-Median/Means

Authors: Dishant Goyal, Ragesh Jaiswal, and Amit Kumar

Published in: LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

Abstract

The Metric k-median problem over a metric space (𝒳, d) is defined as follows: given a set L ⊆ 𝒳 of facility locations and a set C ⊆ 𝒳 of clients, open a set F ⊆ L of k facilities such that the total service cost, defined as Φ(F, C) := ∑_{x ∈ C} min_{f ∈ F} d(x, f), is minimised. The metric k-means problem is defined similarly using squared distances (i.e., d²(., .) instead of d(., .)). In many applications there are additional constraints that any solution needs to satisfy. For example, to balance the load among the facilities in resource allocation problems, a capacity u is imposed on every facility. That is, no more than u clients can be assigned to any facility. This problem is known as the capacitated k-means/k-median problem. Likewise, various other applications have different constraints, which give rise to different constrained versions of the problem such as r-gather, fault-tolerant, outlier k-means/k-median problem. Surprisingly, for many of these constrained problems, no constant-approximation algorithm is known. Moreover, the unconstrained problem itself is known [Marek Adamczyk et al., 2019] to be W[2]-hard when parameterized by k. We give FPT algorithms with constant approximation guarantee for a range of constrained k-median/means problems. For some of the constrained problems, ours is the first constant factor approximation algorithm whereas for others, we improve or match the approximation guarantee of previous works. We work within the unified framework of Ding and Xu [Ding and Xu, 2015] that allows us to simultaneously obtain algorithms for a range of constrained problems. In particular, we obtain a (3+ε)-approximation and (9+ε)-approximation for the constrained versions of the k-median and k-means problem respectively in FPT time. In many practical settings of the k-median/means problem, one is allowed to open a facility at any client location, i.e., C ⊆ L. For this special case, our algorithm gives a (2+ε)-approximation and (4+ε)-approximation for the constrained versions of k-median and k-means problem respectively in FPT time. Since our algorithm is based on simple sampling technique, it can also be converted to a constant-pass log-space streaming algorithm. In particular, here are some of the main highlights of this work: 1) For the uniform capacitated k-median/means problems our results matches previously known results of Addad et al. [Vincent Cohen-Addad and Jason Li, 2019]. 2) For the r-gather k-median/means problem (clustering with lower bound on the size of clusters), our FPT approximation bounds are better than what was previously known. 3) Our approximation bounds for the fault-tolerant, outlier, and uncertain versions is better than all previously known results, albeit in FPT time. 4) For certain constrained settings such as chromatic, l-diversity, and semi-supervised k-median/means, we obtain the first constant factor approximation algorithms to the best of our knowledge. 5) Since our algorithms are based on a simple sampling based approach, we also obtain constant-pass log-space streaming algorithms for most of the above-mentioned problems.

Cite as

Dishant Goyal, Ragesh Jaiswal, and Amit Kumar. FPT Approximation for Constrained Metric k-Median/Means. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{goyal_et_al:LIPIcs.IPEC.2020.14,
  author =	{Goyal, Dishant and Jaiswal, Ragesh and Kumar, Amit},
  title =	{{FPT Approximation for Constrained Metric k-Median/Means}},
  booktitle =	{15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
  pages =	{14:1--14:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-172-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{180},
  editor =	{Cao, Yixin and Pilipczuk, Marcin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.14},
  URN =		{urn:nbn:de:0030-drops-133170},
  doi =		{10.4230/LIPIcs.IPEC.2020.14},
  annote =	{Keywords: k-means, k-median, approximation algorithms, parameterised algorithms}
}

Document

DOI: 10.4230/LIPIcs.ESA.2019.1

Constant-Factor FPT Approximation for Capacitated k-Median

Authors: Marek Adamczyk, Jarosław Byrka, Jan Marcinkowski, Syed M. Meesum, and Michał Włodarczyk

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract

Capacitated k-median is one of the few outstanding optimization problems for which the existence of a polynomial time constant factor approximation algorithm remains an open problem. In a series of recent papers algorithms producing solutions violating either the number of facilities or the capacity by a multiplicative factor were obtained. However, to produce solutions without violations appears to be hard and potentially requires different algorithmic techniques. Notably, if parameterized by the number of facilities k, the problem is also W[2] hard, making the existence of an exact FPT algorithm unlikely. In this work we provide an FPT-time constant factor approximation algorithm preserving both cardinality and capacity of the facilities. The algorithm runs in time 2^O(k log k) n^O(1) and achieves an approximation ratio of 7+epsilon.

Cite as

Marek Adamczyk, Jarosław Byrka, Jan Marcinkowski, Syed M. Meesum, and Michał Włodarczyk. Constant-Factor FPT Approximation for Capacitated k-Median. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 1:1-1:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{adamczyk_et_al:LIPIcs.ESA.2019.1,
  author =	{Adamczyk, Marek and Byrka, Jaros{\l}aw and Marcinkowski, Jan and Meesum, Syed M. and W{\l}odarczyk, Micha{\l}},
  title =	{{Constant-Factor FPT Approximation for Capacitated k-Median}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{1:1--1:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.1},
  URN =		{urn:nbn:de:0030-drops-111225},
  doi =		{10.4230/LIPIcs.ESA.2019.1},
  annote =	{Keywords: K-median, Clustering, Approximation Algorithms, Fixed Parameter Tractability}
}

Document

DOI: 10.4230/LIPIcs.ICALP.2017.35

When the Optimum is also Blind: a New Perspective on Universal Optimization

Authors: Marek Adamczyk, Fabrizio Grandoni, Stefano Leonardi, and Michal Wlodarczyk

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Abstract

Consider the following variant of the set cover problem. We are given a universe U={1,...,n} and a collection of subsets C = {S_1,...,S_m} where each S_i is a subset of U. For every element u from U we need to find a set phi(u) from collection C such that u belongs to phi(u). Once we construct and fix the mapping phi from U to C a subset X from the universe U is revealed, and we need to cover all elements from X with exactly phi(X), that is {phi(u)}_{all u from X}. The goal is to find a mapping such that the cover phi(X) is as cheap as possible. This is an example of a universal problem where the solution has to be created before the actual instance to deal with is revealed. Such problems appear naturally in some settings when we need to optimize under uncertainty and it may be actually too expensive to begin finding a good solution once the input starts being revealed. A rich body of work was devoted to investigate such problems under the regime of worst case analysis, i.e., when we measure how good the solution is by looking at the worst-case ratio: universal solution for a given instance vs optimum solution for the same instance. As the universal solution is significantly more constrained, it is typical that such a worst-case ratio is actually quite big. One way to give a viewpoint on the problem that would be less vulnerable to such extreme worst-cases is to assume that the instance, for which we will have to create a solution, will be drawn randomly from some probability distribution. In this case one wants to minimize the expected value of the ratio: universal solution vs optimum solution. Here the bounds obtained are indeed smaller than when we compare to the worst-case ratio. But even in this case we still compare apples to oranges as no universal solution is able to construct the optimum solution for every possible instance. What if we would compare our approximate universal solution against an optimal universal solution that obeys the same rules as we do? We show that under this viewpoint, but still in the stochastic variant, we can indeed obtain better bounds than in the expected ratio model. For example, for the set cover problem we obtain $H_n$ approximation which matches the approximation ratio from the classic deterministic setup. Moreover, we show this for all possible probability distributions over $U$ that have a polynomially large carrier, while all previous results pertained to a model in which elements were sampled independently. Our result is based on rounding a proper configuration IP that captures the optimal universal solution, and using tools from submodular optimization. The same basic approach leads to improved approximation algorithms for other related problems, including Vertex Cover, Edge Cover, Directed Steiner Tree, Multicut, and Facility Location.

Cite as

Marek Adamczyk, Fabrizio Grandoni, Stefano Leonardi, and Michal Wlodarczyk. When the Optimum is also Blind: a New Perspective on Universal Optimization. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{adamczyk_et_al:LIPIcs.ICALP.2017.35,
  author =	{Adamczyk, Marek and Grandoni, Fabrizio and Leonardi, Stefano and Wlodarczyk, Michal},
  title =	{{When the Optimum is also Blind: a New Perspective on Universal Optimization}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{35:1--35:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.35},
  URN =		{urn:nbn:de:0030-drops-74436},
  doi =		{10.4230/LIPIcs.ICALP.2017.35},
  annote =	{Keywords: approximation algorithms, stochastic optimization, submodularity}
}

Document

DOI: 10.4230/LIPIcs.STACS.2014.29

Submodular Stochastic Probing on Matroids

Authors: Marek Adamczyk, Maxim Sviridenko, and Justin Ward

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

Abstract

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.

Cite as

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-dev.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}
}

6 Search Results for "Adamczyk, Marek"

An Improved Lower Bound for Matroid Intersection Prophet Inequalities

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A Nearly Optimal Deterministic Online Algorithm for Non-Metric Facility Location

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FPT Approximation for Constrained Metric k-Median/Means

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Constant-Factor FPT Approximation for Capacitated k-Median

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When the Optimum is also Blind: a New Perspective on Universal Optimization

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Submodular Stochastic Probing on Matroids

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