3 Search Results for "Rothvoss, Thomas"


Document
Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing

Authors: Arindam Khan, Eklavya Sharma, and K. V. N. Sreenivas

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)


Abstract
We study the generalized multidimensional bin packing problem (GVBP) that generalizes both geometric packing and vector packing. Here, we are given n rectangular items where the i-th item has width w(i), height h(i), and d nonnegative weights v₁(i), v₂(i), …, v_d(i). Our goal is to get an axis-parallel non-overlapping packing of the items into square bins so that for all j ∈ [d], the sum of the j-th weight of items in each bin is at most 1. This is a natural problem arising in logistics, resource allocation, and scheduling. Despite being well-studied in practice, approximation algorithms for this problem have rarely been explored. We first obtain two simple algorithms for GVBP having asymptotic approximation ratios 6(d+1) and 3(1 + ln(d+1) + ε). We then extend the Round-and-Approx (R&A) framework [Bansal et al., 2009; Bansal and Khan, 2014] to wider classes of algorithms, and show how it can be adapted to GVBP. Using more sophisticated techniques, we obtain better approximation algorithms for GVBP, and we get further improvement by combining them with the R&A framework. This gives us an asymptotic approximation ratio of 2(1 + ln((d+4)/2)) + ε for GVBP, which improves to 2.919+ε for the special case of d = 1. We obtain further improvement when the items are allowed to be rotated. We also present algorithms for a generalization of GVBP where the items are high dimensional cuboids.

Cite as

Arindam Khan, Eklavya Sharma, and K. V. N. Sreenivas. Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 23:1-23:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{khan_et_al:LIPIcs.FSTTCS.2022.23,
  author =	{Khan, Arindam and Sharma, Eklavya and Sreenivas, K. V. N.},
  title =	{{Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{23:1--23:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.23},
  URN =		{urn:nbn:de:0030-drops-174151},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.23},
  annote =	{Keywords: Bin packing, rectangle packing, multidimensional packing, approximation algorithms}
}
Document
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-dev.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
Recent Hardness Results for Periodic Uni-processor Scheduling

Authors: Friedrich Eisenbrand and Thomas Rothvoss

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


Abstract
Consider a set of $n$ periodic tasks $ au_1,ldots, au_n$ where $ au_i$ is described by an execution time $c_i$, a (relative) deadline $d_i$ and a period $p_i$. We assume that jobs are released synchronously (i.e. at each multiple of $p_i$) and consider pre-emptive, uni-processor schedules. We show that computing the response time of a task $ au_n$ in a Rate-monotonic schedule i.e. computing [ minleft{ r geq mid c_n + sum_{i=1}^{n-1} leftlceil frac{r}{p_i} ight ceil c_i leq r ight} ] is (weakly) $mathbf{NP}$-hard (where $ au_n$ has the lowest priority and the deadlines are implicit, i.e. $d_i = p_i$). Furthermore we obtain that verifying EDF-schedulability, i.e. [ forall Q geq 0: sum_{i=1}^n left( leftlfloor frac{Q-d_i}{p_i} ight floor +1 ight)cdot c_i leq Q ] for constrained-deadline tasks ($d_i leq p_i$) is weakly $mathbf{coNP}$-hard.

Cite as

Friedrich Eisenbrand and Thomas Rothvoss. Recent Hardness Results for Periodic Uni-processor Scheduling. In Scheduling. Dagstuhl Seminar Proceedings, Volume 10071, pp. 1-7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)


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@InProceedings{eisenbrand_et_al:DagSemProc.10071.10,
  author =	{Eisenbrand, Friedrich and Rothvoss, Thomas},
  title =	{{Recent Hardness Results for Periodic Uni-processor Scheduling}},
  booktitle =	{Scheduling},
  pages =	{1--7},
  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-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.10071.10},
  URN =		{urn:nbn:de:0030-drops-25458},
  doi =		{10.4230/DagSemProc.10071.10},
  annote =	{Keywords: Hardness, periodic scheduling, uni-processor scheduling}
}
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