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DOI: 10.4230/LIPIcs.SoCG.2016.40

Random Sampling with Removal

Authors: Bernd Gärtner, Johannes Lengler, and May Szedlák

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Abstract

Random sampling is a classical tool in constrained optimization. Under favorable conditions, the optimal solution subject to a small subset of randomly chosen constraints violates only a small subset of the remaining constraints. Here we study the following variant that we call random sampling with removal: suppose that after sampling the subset, we remove a fixed number of constraints from the sample, according to an arbitrary rule. Is it still true that the optimal solution of the reduced sample violates only a small subset of the constraints? The question naturally comes up in situations where the solution subject to the sampled constraints is used as an approximate solution to the original problem. In this case, it makes sense to improve cost and volatility of the sample solution by removing some of the constraints that appear most restricting. At the same time, the approximation quality (measured in terms of violated constraints) should remain high. We study random sampling with removal in a generalized, completely abstract setting where we assign to each subset R of the constraints an arbitrary set V(R) of constraints disjoint from R; in applications, V(R) corresponds to the constraints violated by the optimal solution subject to only the constraints in R. Furthermore, our results are parametrized by the dimension d, i.e., we assume that every set R has a subset B of size at most d with the same set of violated constraints. This is the first time this generalized setting is studied. In this setting, we prove matching upper and lower bounds for the expected number of constraints violated by a random sample, after the removal of k elements. For a large range of values of k, the new upper bounds improve the previously best bounds for LP-type problems, which moreover had only been known in special cases. We show that this bound on special LP-type problems, can be derived in the much more general setting of violator spaces, and with very elementary proofs.

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Bernd Gärtner, Johannes Lengler, and May Szedlák. Random Sampling with Removal. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 40:1-40:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{gartner_et_al:LIPIcs.SoCG.2016.40,
  author =	{G\"{a}rtner, Bernd and Lengler, Johannes and Szedl\'{a}k, May},
  title =	{{Random Sampling with Removal}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{40:1--40:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.40},
  URN =		{urn:nbn:de:0030-drops-59328},
  doi =		{10.4230/LIPIcs.SoCG.2016.40},
  annote =	{Keywords: LP-type problem, violator space, random sampling, sampling with removal}
}

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DOI: 10.4230/LIPIcs.SOCG.2015.315

Combinatorial Redundancy Detection

Authors: Komei Fukuda, Bernd Gärtner, and May Szedlák

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract

The problem of detecting and removing redundant constraints is fundamental in optimization. We focus on the case of linear programs (LPs) in dictionary form, given by n equality constraints in n+d variables, where the variables are constrained to be nonnegative. A variable x_r is called redundant, if after removing its nonnegativity constraint the LP still has the same feasible region. The time needed to solve such an LP is denoted by LP(n,d). It is easy to see that solving n+d LPs of the above size is sufficient to detect all redundancies. The currently fastest practical method is the one by Clarkson: it solves n+d linear programs, but each of them has at most s variables, where s is the number of nonredundant constraints. In the first part we show that knowing all of the finitely many dictionaries of the LP is sufficient for the purpose of redundancy detection. A dictionary is a matrix that can be thought of as an enriched encoding of a vertex in the LP. Moreover - and this is the combinatorial aspect - it is enough to know only the signs of the entries, the actual values do not matter. Concretely we show that for any variable x_r one can find a dictionary, such that its sign pattern is either a redundancy or nonredundancy certificate for x_r. In the second part we show that considering only the sign patterns of the dictionary, there is an output sensitive algorithm of running time of order d (n+d) s^{d-1} LP(s,d) + d s^{d} LP(n,d) to detect all redundancies. In the case where all constraints are in general position, the running time is of order s LP(n,d) + (n+d) LP(s,d), which is essentially the running time of the Clarkson method. Our algorithm extends naturally to a more general setting of arrangements of oriented topological hyperplane arrangements.

Cite as

Komei Fukuda, Bernd Gärtner, and May Szedlák. Combinatorial Redundancy Detection. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 315-328, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{fukuda_et_al:LIPIcs.SOCG.2015.315,
  author =	{Fukuda, Komei and G\"{a}rtner, Bernd and Szedl\'{a}k, May},
  title =	{{Combinatorial Redundancy Detection}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{315--328},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.315},
  URN =		{urn:nbn:de:0030-drops-51434},
  doi =		{10.4230/LIPIcs.SOCG.2015.315},
  annote =	{Keywords: system of linear inequalities, redundancy removal, linear programming, output sensitive algorithm, Clarkson’s method}
}

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Random Sampling with Removal

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Combinatorial Redundancy Detection

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