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.FSTTCS.2020.10
URN: urn:nbn:de:0030-drops-132515
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/13251/
Go to the corresponding LIPIcs Volume Portal

Banerjee, Niranka ; Raman, Venkatesh ; Saurabh, Saket

Optimal Output Sensitive Fault Tolerant Cuts

pdf-format:
LIPIcs-FSTTCS-2020-10.pdf (0.7 MB)

Abstract

In this paper we consider two classic cut-problems, Global Min-Cut and Min k-Cut, via the lens of fault tolerant network design. In particular, given a graph G on n vertices, and a positive integer f, our objective is to compute an upper bound on the size of the sparsest subgraph H of G that preserves edge connectivity of G (denoted by λ(G)) in the case of Global Min-Cut, and λ(G,k) (denotes the minimum number of edges whose removal would partition the graph into at least k connected components) in the case of Min k-Cut, upon failure of any f edges of G. The subgraph H corresponding to Global Min-Cut and Min k-Cut is called f-FTCS and f-FT-k-CS, respectively. We obtain the following results about the sizes of f-FTCS and f-FT-k-CS.
- There exists an f-FTCS with (n-1)(f+λ(G)) edges. We complement this upper bound with a matching lower bound, by constructing an infinite family of graphs where any f-FTCS must have at least ((n-λ(G)-1)(λ(G)+f-1))/2+(n-λ(G)-1)+/λ(G)(λ(G)+1))/2 edges.
- There exists an f-FT-k-CS with min{(2f+λ(G,k)-(k-1))(n-1), (f+λ(G,k))(n-k)+?} edges. We complement this upper bound with a lower bound, by constructing an infinite family of graphs where any f-FT-k-CS must have at least ((n-λ(G,k)-1)(λ(G,k)+f-k+1))/2)+n-λ(G,k)+k-3+((λ(G,k)-k+3)(λ(G,k)-k+2))/2 edges. Our upper bounds exploit the structural properties of k-connectivity certificates. On the other hand, for our lower bounds we construct an infinite family of graphs, such that for any graph in the family any f-FTCS (or f-FT-k-CS) must contain all its edges. We also add that our upper bounds are constructive. That is, there exist polynomial time algorithms that construct H with the aforementioned number of edges.

BibTeX - Entry

@InProceedings{banerjee_et_al:LIPIcs:2020:13251,
  author =	{Niranka Banerjee and Venkatesh Raman and Saket Saurabh},
  title =	{{Optimal Output Sensitive Fault Tolerant Cuts}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{10:1--10:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Nitin Saxena and Sunil Simon},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/13251},
  URN =		{urn:nbn:de:0030-drops-132515},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.10},
  annote =	{Keywords: Fault tolerant, minimum cuts, upper bound, lower bound}
}

Keywords: Fault tolerant, minimum cuts, upper bound, lower bound
Collection: 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)
Issue Date: 2020
Date of publication: 04.12.2020

DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI