Small ℓ-edge-covers in k-connected graphs

Let G=(V,E) be a k-edge-connected graph with edge-costs {c(e):e ε E} and minimum degree d. We show by a simple and short proof, that for any integer ℓ with dk≤ℓ≤d(1-1k), G contains an ℓ-edge cover I such that: c(I)≤ℓdc(E) if G is bipartite, or if ℓ|V| is even, or if |E|≥d|V|2+d2ℓ; otherwise, c(I)≤(ℓd+1d|V|)c(E). The particular case d=k=ℓ+1 and unit costs already includes a result of Cheriyan and Thurimella (2000) [1], that G contains a (k-1)-edge-cover of size |E|-|V|/2. Using our result, we slightly improve the approximation ratios for the k-Connected Subgraph problem (the node-connectivity version) with uniform and β-metric costs. We then consider the dual problem of finding a spanning subgraph of maximum connectivity k* with a prescribed number of edges. We give an algorithm that computes a (k*-1)-connected subgraph, which is tight, since the problem is NP-hard.