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Optimal Output Sensitive Fault Tolerant Cuts Banerjee, Niranka Raman, Venkatesh Saurabh, Saket Fault tolerant minimum cuts upper bound lower bound 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Niranka Banerjee and Venkatesh Raman and Saket Saurabh 2020 Is Part Of LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.FSTTCS.2020.10 urn:nbn:de:0030-drops-132515 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.10 eng https://creativecommons.org/licenses/by/3.0/legalcode