Mincut Sensitivity Data Structures for the Insertion of an Edge

Abstract

Let \(G=(V,E)\) be an undirected graph on n vertices with non-negative capacities on its edges. The mincut sensitivity problem for the insertion of an edge is defined as follows. Build a compact data structure for G and a given set \(S\subseteq V\) of vertices that, on receiving any edge \((x,y)\in S\times S\) of positive capacity as query input, can efficiently report the set of all pairs from \(S\times S\) whose mincut value increases upon insertion of the edge (x, y) to G. The only result that exists for this problem is for a single pair of vertices (Picard and Queyranne, in: Rayward-Smith (ed) Combinatorial optimization II. Mathematical programming Studies, vol 13, no 1. Springer, Berlin, pp 8–16, 1980. https://doi.org/10.1007/BFb0120902, and dates back to 1980. We present the following results for the single source and the all-pairs versions of this problem. 1. Single source Given any designated source vertex s, there exists a data structure of size \({\mathcal {O}}(|S|)\) (Data structure sizes are in words unless specified otherwise, where a word occupies \(\Theta (\log n)\) bits.) that can output all those vertices from S whose mincut value to s increases upon insertion of any given edge. The time taken by the data structure to answer any query is \({\mathcal {O}}(|S|)\). 2. All-pairs There exists an \({\mathcal {O}}(|S|^2)\) size data structure that can output all those pairs of vertices from \(S\times S\) whose mincut value increases upon insertion of any given edge. The time taken by the data structure to answer any query is \({\mathcal {O}}(k)\), where k is the number of pairs of vertices whose mincut value increases. For both these versions, we also address the problem of reporting the values of the mincuts upon insertion of any given edge. To derive our results, we use interesting insights into the nearest and the farthest mincuts for a pair of vertices. In addition, a crucial result, that we establish and use in our data structures, is that there exists a directed acyclic graph of \({\mathcal {O}}(n)\) size that compactly stores the farthest mincuts from all vertices of V to a designated vertex s in the graph. We believe that this result is of independent interest, especially, because it also complements a previously existing result (Hariharan et al., in: Proceedings of the 39th annual ACM symposium on theory of computing, San Diego, California, USA, June 11–13, 2007, pp 605–614, 2007. https://doi.org/10.1145/1250790) that the nearest mincuts from all vertices of V to s is a laminar family, and hence, can be stored compactly in the form of a rooted tree of \({\mathcal {O}}(n)\) size.