License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2018.41
URN: urn:nbn:de:0030-drops-85255
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8525/
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Iwata, Yoichi ; Ogasawara, Tomoaki ; Ohsaka, Naoto

On the Power of Tree-Depth for Fully Polynomial FPT Algorithms

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LIPIcs-STACS-2018-41.pdf (0.6 MB)

Abstract

There are many classical problems in P whose time complexities have not been improved over the past decades.
Recent studies of "Hardness in P" have revealed that, for several of such problems, the current fastest algorithm is the best possible under some complexity assumptions.
To bypass this difficulty, the concept of "FPT inside P" has been introduced.
For a problem with the current best time complexity O(n^c), the goal is to design an algorithm running in k^{O(1)}n^{c'} time for a parameter k and a constant c'<c.

In this paper, we investigate the complexity of graph problems in P parameterized by tree-depth, a graph parameter related to tree-width.
We show that a simple divide-and-conquer method can solve many graph problems, including
Weighted Matching, Negative Cycle Detection, Minimum Weight Cycle, Replacement Paths, and 2-hop Cover,
in O(td m) time or O(td (m+nlog n)) time, where td is the tree-depth of the input graph.
Because any graph of tree-width tw has tree-depth at most (tw+1)log_2 n, our algorithms also run in O(tw mlog n) time or O(tw (m+nlog n)log n) time.
These results match or improve the previous best algorithms parameterized by tree-width.
Especially, we solve an open problem of fully polynomial FPT algorithm for Weighted Matching parameterized by tree-width posed by Fomin et al. (SODA 2017).

BibTeX - Entry

@InProceedings{iwata_et_al:LIPIcs:2018:8525,
  author =	{Yoichi Iwata and Tomoaki Ogasawara and Naoto Ohsaka},
  title =	{{On the Power of Tree-Depth for Fully Polynomial FPT Algorithms}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{41:1--41:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Rolf Niedermeier and Brigitte Vall{\'e}e},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8525},
  URN =		{urn:nbn:de:0030-drops-85255},
  doi =		{10.4230/LIPIcs.STACS.2018.41},
  annote =	{Keywords: Fully Polynomial FPT Algorithm, Tree-Depth, Divide-and-Conquer}
}

Keywords: Fully Polynomial FPT Algorithm, Tree-Depth, Divide-and-Conquer
Collection: 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)
Issue Date: 2018
Date of publication: 27.02.2018

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