Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH scholarly article en Fomin, Fedor V.; Panolan, Fahad; Ramanujan, M. S.; Saurabh, Saket http://www.dagstuhl.de/lipics License
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URN: urn:nbn:de:0030-drops-94949
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On the Optimality of Pseudo-polynomial Algorithms for Integer Programming

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Abstract

In the classic Integer Programming (IP) problem, the objective is to decide whether, for a given m x n matrix A and an m-vector b=(b_1,..., b_m), there is a non-negative integer n-vector x such that Ax=b. Solving (IP) is an important step in numerous algorithms and it is important to obtain an understanding of the precise complexity of this problem as a function of natural parameters of the input. The classic pseudo-polynomial time algorithm of Papadimitriou [J. ACM 1981] for instances of (IP) with a constant number of constraints was only recently improved upon by Eisenbrand and Weismantel [SODA 2018] and Jansen and Rohwedder [ArXiv 2018]. We continue this line of work and show that under the Exponential Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal. We also show that when the matrix A is assumed to be non-negative, a component of Papadimitriou's original algorithm is already nearly optimal under ETH. This motivates us to pick up the line of research initiated by Cunningham and Geelen [IPCO 2007] who studied the complexity of solving (IP) with non-negative matrices in which the number of constraints may be unbounded, but the branch-width of the column-matroid corresponding to the constraint matrix is a constant. We prove a lower bound on the complexity of solving (IP) for such instances and obtain optimal results with respect to a closely related parameter, path-width. Specifically, we prove matching upper and lower bounds for (IP) when the path-width of the corresponding column-matroid is a constant.

BibTeX - Entry

@InProceedings{fomin_et_al:LIPIcs:2018:9494,
  author =	{Fedor V. Fomin and Fahad Panolan and M. S. Ramanujan and Saket Saurabh},
  title =	{{On the Optimality of Pseudo-polynomial Algorithms for Integer Programming}},
  booktitle =	{26th Annual European Symposium on Algorithms (ESA 2018)},
  pages =	{31:1--31:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-081-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{112},
  editor =	{Yossi Azar and Hannah Bast and Grzegorz Herman},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9494},
  URN =		{urn:nbn:de:0030-drops-94949},
  doi =		{10.4230/LIPIcs.ESA.2018.31},
  annote =	{Keywords: Integer Programming, Strong Exponential Time Hypothesis, Branch-width of a matrix, Fine-grained Complexity}
}

Keywords: Integer Programming, Strong Exponential Time Hypothesis, Branch-width of a matrix, Fine-grained Complexity
Seminar: 26th Annual European Symposium on Algorithms (ESA 2018)
Issue date: 2018
Date of publication: 2018


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