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Finding the exact integrality gap $\alpha$ for the LP relaxation of the2-edge-connected spanning multigraph problem (2EC) is closely related to thesame problem for the Held-Karp relaxation of the metric traveling salesmanproblem (TSP). While the former problem seems easier than the latter, since itis less constrained, currently the upper bounds on the respective integralitygaps for the two problems are the same.  An approach to proving integrality gaps for both of these problems is toconsider fundamental classes of extreme points. For 2EC, better bounds on theintegrality gap are known for certain important special cases of thesefundamental points. For example, for half-integer square points, theintegrality gap is between $\frac{6}{5}$ and $\frac{4}{3}$. Our main result isto improve the approximation factor to $\frac{9}{7}$ for 2EC for these points.Our approach is based on constructing convex combinations and our key tool isthe top-down coloring framework for tree augmentation, whose flexibility weemploy to exploit beneficial properties in both the initial spanning tree andin the input graph. We also show how these tools can be tailored to the closelyrelated problem of uniform covers for which the proofs of the best-known boundsdo not yield polynomial-time algorithms. Another key ingredient is to use arainbow spanning tree decomposition, which allows us to obtain a convexcombination of spanning trees with particular properties

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