Efficient constructions of convex combinations for 2-edge-connected subgraphs on fundamental classes

We present coloring-based algorithms for tree augmentation and use them to construct convex combinations of 2-edge-connected subgraphs. This classic tool has been applied previously the problem, but our illustrate its flexibility, which – in coordination with choice spanning can be used obtain various properties (e.g., 2-vertex connectivity) that are useful applications. these coloring design approximation multigraph problem (2ECM) subgraph (2ECS) on two well-studied types LP solutions. The first type points, half-integer square belong a class fundamental extreme exhibit same integrality gap as general case. For 2ECM is known between 65 43. improve upper bound 97. second points we study uniform whose support 3-edge-connected graph each entry 23. Although best-known 2ECS less than 43, previous results do not yield an efficient algorithm. give algorithm ratio below 43 this points.