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Documents authored by Hähnle, Nicolai

Document
DOI: 10.4230/LIPIcs.SOCG.2015.345
On the Shadow Simplex Method for Curved Polyhedra

Authors: Daniel Dadush and Nicolai Hähnle

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

Abstract
We study the simplex method over polyhedra satisfying certain "discrete curvature" lower bounds, which enforce that the boundary always meets vertices at sharp angles. Motivated by linear programs with totally unimodular constraint matrices, recent results of Bonifas et al. (SOCG 2012), Brunsch and Röglin (ICALP 2013), and Eisenbrand and Vempala (2014) have improved our understanding of such polyhedra. We develop a new type of dual analysis of the shadow simplex method which provides a clean and powerful tool for improving all previously mentioned results. Our methods are inspired by the recent work of Bonifas and the first named author, who analyzed a remarkably similar process as part of an algorithm for the Closest Vector Problem with Preprocessing. For our first result, we obtain a constructive diameter bound of O((n^2 / delta) ln (n / delta)) for n-dimensional polyhedra with curvature parameter delta in (0, 1]. For the class of polyhedra arising from totally unimodular constraint matrices, this implies a bound of O(n^3 ln n). For linear optimization, given an initial feasible vertex, we show that an optimal vertex can be found using an expected O((n^3 / delta) ln (n / delta)) simplex pivots, each requiring O(mn) time to compute. An initial feasible solution can be found using O((mn^3 / delta) ln (n / delta)) pivot steps.
Cite as

Daniel Dadush and Nicolai Hähnle. On the Shadow Simplex Method for Curved Polyhedra. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 345-359, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{dadush_et_al:LIPIcs.SOCG.2015.345,
  author =	{Dadush, Daniel and H\"{a}hnle, Nicolai},
  title =	{{On the Shadow Simplex Method for Curved Polyhedra}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{345--359},
  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.2015.345},
  URN =		{urn:nbn:de:0030-drops-51142},
  doi =		{10.4230/LIPIcs.SOCG.2015.345},
  annote =	{Keywords: Optimization, Linear Programming, Simplex Method, Diameter of Polyhedra}
}
Document
DOI: 10.4230/DagSemProc.10211.2
Diameter of Polyhedra: Limits of Abstraction

Authors: Friedrich Eisenbrand, Nicolai Hähnle, Alexander Razborov, and Thomas Rothvoß

Published in: Dagstuhl Seminar Proceedings, Volume 10211, Flexible Network Design (2010)

Abstract
We investigate the diameter of a natural abstraction of the $1$-skeleton of polyhedra. Even if this abstraction is more general than other abstractions previously studied in the literature, known upper bounds on the diameter of polyhedra continue to hold here. On the other hand, we show that this abstraction has its limits by providing an almost quadratic lower bound.
Cite as

Friedrich Eisenbrand, Nicolai Hähnle, Alexander Razborov, and Thomas Rothvoß. Diameter of Polyhedra: Limits of Abstraction. In Flexible Network Design. Dagstuhl Seminar Proceedings, Volume 10211, pp. 1-5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{eisenbrand_et_al:DagSemProc.10211.2,
  author =	{Eisenbrand, Friedrich and H\"{a}hnle, Nicolai and Razborov, Alexander and Rothvo{\ss}, Thomas},
  title =	{{Diameter of Polyhedra: Limits of Abstraction}},
  booktitle =	{Flexible Network Design},
  pages =	{1--5},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{10211},
  editor =	{Anupam Gupta and Stefano Leonardi and Berthold V\"{o}cking and Roger Wattenhofer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10211.2},
  URN =		{urn:nbn:de:0030-drops-27247},
  doi =		{10.4230/DagSemProc.10211.2},
  annote =	{Keywords: Polyhedra, Graphs}
}
Document
DOI: 10.4230/DagSemProc.10071.13
Scheduling periodic tasks in a hard real-time environment

Authors: Friedrich Eisenbrand, Nicolai Hähnle, Martin Niemeier, Martin Skutella, Jose Verschae, and Andreas Wiese

Published in: Dagstuhl Seminar Proceedings, Volume 10071, Scheduling (2010)

Abstract
We consider a real-time scheduling problem that occurs in the design of software-based aircraft control. The goal is to distribute tasks $ au_i=(c_i,p_i)$ on a minimum number of identical machines and to compute offsets $a_i$ for the tasks such that no collision occurs. A task $ au_i$ releases a job of running time $c_i$ at each time $a_i + kcdot p_i, , k in mathbb{N}_0$ and a collision occurs if two jobs are simultaneously active on the same machine. We shed some light on the complexity and approximability landscape of this problem. Although the problem cannot be approximated within a factor of $n^{1-varepsilon}$ for any $varepsilon>0$, an interesting restriction is much more tractable: If the periods are dividing (for each $i,j$ one has $p_i | p_j$ or $p_j | p_i$), the problem allows for a better structured representation of solutions, which leads to a 2-approximation. This result is tight, even asymptotically.
Cite as

Friedrich Eisenbrand, Nicolai Hähnle, Martin Niemeier, Martin Skutella, Jose Verschae, and Andreas Wiese. Scheduling periodic tasks in a hard real-time environment. In Scheduling. Dagstuhl Seminar Proceedings, Volume 10071, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{eisenbrand_et_al:DagSemProc.10071.13,
  author =	{Eisenbrand, Friedrich and H\"{a}hnle, Nicolai and Niemeier, Martin and Skutella, Martin and Verschae, Jose and Wiese, Andreas},
  title =	{{Scheduling periodic tasks in a hard real-time environment}},
  booktitle =	{Scheduling},
  pages =	{1--3},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{10071},
  editor =	{Susanne Albers and Sanjoy K. Baruah and Rolf H. M\"{o}hring and Kirk Pruhs},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10071.13},
  URN =		{urn:nbn:de:0030-drops-25348},
  doi =		{10.4230/DagSemProc.10071.13},
  annote =	{Keywords: Real-Time Scheduling, Periodic scheduling problem, Periodic maintenance problem, Approximation hardness, Approximation algorithm}
}
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