Algorithms for the Densest Subgraph with at Least k Vertices and with a Specified Subset.

The density of a subgraph in an undirected graph is the sum of the subgraph's edge weights divided by the number of the subgraph's vertices. Finding an induced subgraph of maximum density among all subgraphs with at leastk vertices is called as the densest at-least-k-subgraph problem DalkS. In this paper, we first present a polynomial time algorithms for DalkS when k is bounded by some constant c. For a graph of n vertices and m edges, our algorithm is of time complexity $$On^{c + 3} \\log n$$Onc+3logn, which improve previous best time complexity $$On^cn+m^{4.5}$$Oncn+m4.5. Second, we give a greedy approximation algorithm for the Densest Subgraph with a Specified Subset Problem. We show that the greedy algorithm is of approximation ratio $$2\\cdot 1+ \\frac{k}{3}$$2﾿1+k3, where k is the element number of the specified subset.