Edge covering problems with budget constrains

In the Maximum cost m′-subgraph problem we are given an undirected simple graph G(V,E) with weights w(v) on the vertices. The goal is to select a set U ⊆ V so that the number of edges with at least one endpoint in U is at most m′ with maximum weight W ∗. In [11] a ratio 3 approximation is given for problem. We improve this result by giving a 2 + approximation for the problem, for every constant > 0, that runs in time n ) . We give the same ratio to the Minimum edges, cost W , subgraph problem: given a weight bound on the subgraph, the goal is to choose a subset U of the vertices with sum of weights at most W and minimize the number m′ of edges touching U . The unweighted variant of Maximum weight m′-subgraph and Minimum Edges weight W subgraph admits a 2 ratio [11]. We show that if Minimum Edges weight W subgraph or Maximum weight m′-subgraph admit a better than ratio 2, a variant of the Dense ksubgraph must admit a constant approximation, while the best known approximation for that variant is O(n). This implies a 2 approximation threshold for approximating Maximum weight m′-subgraph and Minimum Edges weight W subgraph (up to low order terms) is highly likely. We break the ratio of 2 improving [11] in two special cases. If m∗ = c · k for some constant c, or if k = c · n for some constant c. The Degrees density augmentation problem we are given an undirected graph G = (V,A,B) and a set U ⊆ V . The objective is to find a set W so that (e(W ) + e(U,W ))/deg(W ) is maximum. We provide a polynomial time algorithm for this problem which is closely related to the Minimum Edges weight W subgraph problem.