On size-constrained minimum s-t cut problems and size-constrained dense subgraph problems

On the other hand, we also study the densest at-least-k-subgraph problem (DalkS) and the densest at-most-k-subgraph problem (DamkS) introduced by Andersen and Chellapilla [1]. We present a polynomial time algorithm for DalkS when k is bounded by some constant c. We also present two approximation algorithms for DamkS. The first approximation algorithm for DamkS has an approximation ratio of n−1k−1, where n is the number of vertices in the input graph. The second approximation algorithm for DamkS has an approximation ratio of O(nδ), for some δ<1/3.