eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-12-04
10:1
10:19
10.4230/LIPIcs.FSTTCS.2020.10
article
Optimal Output Sensitive Fault Tolerant Cuts
Banerjee, Niranka
1
Raman, Venkatesh
1
Saurabh, Saket
1
2
The Institute of Mathematical Sciences, HBNI, Chennai, India
University of Bergen, Norway
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)+𝓁} 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol182-fsttcs2020/LIPIcs.FSTTCS.2020.10/LIPIcs.FSTTCS.2020.10.pdf
Fault tolerant
minimum cuts
upper bound
lower bound