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Being the unbalanced version of the famous Min s – t Cut problem, the Min k -Size s – t Cut problem asks to find a k -size s – t cut with the minimum capacity, where a k -size s – t cut means an s – t cut with its s -side having size at most k . This problem is fundamental and has extensive applications, especially in community identification in social and information networks. It is known that the Min k -Size s – t Cut problem is NP-hard and can be approximated within O ( log ⁡ n ) , where n is the number of vertices in the input graph. In this paper, we give a new approximation algorithm for the Min k -Size s – t Cut problem based on the parametric flow technique. The algorithm is very simple and has only three lines to state. Its approximation ratio is k + 1 k + 1 − k ⁎ , where k ⁎ is the size of the s -side of an optimal solution. [ABSTRACT FROM AUTHOR]

We present an O(log² k)-approximation algorithm for the problem of finding a k-vertex connected spanning subgraph of minimum cost, where n is the number of vertices in an input graph, and k is a connectivity requirement. Our algorithm is the first that achieves a polylog-arithmic approximation ratio for all values of k and n, and it works for both directed and undirected graphs. As in previous works, we use the Frank-Tardos algorithm for finding k-outconnected subgraphs as a subroutine. However, with our structural lemmas, we are able to show that we need only partial solutions returned by the Frank-Tardos algorithm; thus, we can avoid paying the whole cost of an optimal solution every time the algorithm is applied. [ABSTRACT FROM AUTHOR]