Coloring Down: $3/2$-approximation for special cases of the weighted tree augmentation problem

In this paper, we investigate the weighted tree augmentation problem (TAP), where the goal is to augment a tree with a minimum cost set of edges such that the graph becomes two edge connected. First we show that in weighted TAP, we can restrict our attention to trees which are binary and where all the non-tree edges go between two leaves of the tree. We then give two different top-down coloring algorithms. Both algorithms differ from known techniques for a 3/2-approximation in unweighted TAP and current attempts to reach a 3/2-approximation for weighted TAP. The first algorithm we describe always gives a 2-approximation for any feasible fractional solution to the natural edge cover LP. When the fractional solution is such that all the edges with non-zero weight are at least $\alpha$, then this algorithm achieves a $2/(1+\alpha)$-approximation. We propose a new conjecture on extreme points of LP relaxations for the problem, which if true, will lead to a constructive proof of an integrality gap of at most 3/2 for weighted TAP. In the second algorithm, we introduce simple valid constraints to the edge cover LP. In this algorithm, we focus on deficient edges, edges covered to an extent less than 4/3 in the fractional solution. We show that for fractional feasible solutions, deficient edges occur in node-disjoint paths in the tree. When the number of such paths is at most two, we give a top-down coloring algorithm which decomposes 3/2 times the fractional solution into a convex combination of integer solutions. We believe our algorithms will be useful in eventually resolving the integrality gap of linear programming formulations for TAP. We also investigate a variant of TAP where each edge in the solution must be covered by a cycle of length three. We give a $\Theta(\log n)$-approximation algorithm for this problem in the weighted case and a 4-approximation in the unweighted case.