All-Pairs kth Mincuts: Combinatorial and Structural Results

Abstract

Let G be an undirected graph on a set V of n vertices. For any non-empty subset A⊊ V, cut defined by A is the ordered pair (A,V\A). Suppose each cut is assigned a value, which is any arbitrary real number. Let u,v ∈ V be any pair of vertices. A cut is said to be a (u,v)-cut if it separates u and v. A (u,v)-cut of the minimum value is called a (u,v)-mincut. A 2nd (u,v)-mincut is a (u,v)-cut of second minimum value. We can define k-th mincut accordingly. We present the following results for the all-pairs k-th mincuts.
(1) Distinct Values of all-pairs k-th Mincuts: There exist k spanning trees on V such that for any pair (u,v), the value of k-th (u,v)-mincut is equal to the capacity of an edge on the (u,v)-path in one of the k trees. We also show a matching lower bound of Ω(min{kn,n²}). Our result generalizes the well-known result by Gomory and Hu [JSIAM 1961] stating that there are at most n-1 distinct values of all-pairs mincuts.
(2) Ancestor Trees for all-pairs k-th Mincuts: In 1991, Cheng and Hu [AOR 1991] invented a rooted binary tree, called ancestor tree, whose leaves are the vertices of the graph and each internal node stores a cut with the following property. For any pair (u,v), the cut stored at their lowest common ancestor (LCA) is a (u,v)-mincut. We introduce a tree called gen-ancestor tree, that generalizes the ancestor tree for k-th mincuts, and achieve the following result. There exists a set of 𝒪(klog n) gen-ancestor trees such that, for any pair (u,v), a k-th (u,v)-mincut is stored at the LCA of u and v in at least one of these trees.
(3) Data Structures: We present the following data structures for all-pairs k-th mincuts. (i) There exists an 𝒪(nlog n) space data structure that can report the value of 2nd (u,v)-mincut in 𝒪(log n) time for any given pair (u,v). We generalize this data structure for k-th mincuts with a factor of k² in the space and query time. (ii) There exists an 𝒪(kn² log n) space data structure that can report a k-th (u,v)-mincut (A,V\A) in 𝒪(|A|) time for any given pair (u,v). For any constant k, the bounds stated above match, up to a logarithmic factor, the best-known bounds guaranteed by the data structure for all-pairs mincuts (Cheng and Hu [AOR 1991]).