A 4/3-approximation algorithm for the minimum 2-edge connected multisubgraph problem in the half-integral case

Given a connected undirected graph (G) over bar on n vertices and nonnegative edge costs c, the 2ECM problem is that of finding a 2-edge connected spanning multisubgraph of (G) over bar of minimum cost. The natural linear program (LP) for 2ECM, which coincides with the subtour LP for the traveling salesman problem on the metric closure of (G) over bar, gives a lower bound on the optimal cost. For instances where this LP is optimized by a half-integral solution x, Carr and Ravi (1998) showed that the integrality gap is at most 4/3: they show that the vector 4/3x dominates a convex combination of incidence vectors of 2-edge connected spanning multisubgraphs of (G) over bar . We present a simpler proof of the result due to Carr and Ravi by applying an extension of Lovasz's splitting-off theorem. Our proof naturally leads to a 4/3-approximation algorithm for half-integral instances. Given a half-integral solution x to the LP for 2ECM, we give an O(n(2))-time algorithm to obtain a 2-edge connected spanning multisubgraph of G with cost at most 4/3 c(T) x. We also consider a related problem of finding a cheap 2-edge connected spanning subgraph of a 3-regular, 3-edge connected graph G = (V, E) with arbitrary edge costs c. We give a polynomial-time Las Vegas algorithm that finds a random 2-edge connected spanning subgraph H of G whose expected cost, E [c(H)], is at most 4/5 c(E).