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

Abstract

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 a top-down coloring algorithm that differs from known techniques for approximating TAP.The algorithm we describe always gives a 2-approximation starting from any feasible fractional solution to the natural tree cut covering LP. When the structure of the fractional solution is such that all the edges with non-zero weight are at least α, then this algorithm achieves a 21+α-approximation.We also investigate a variant of TAP where every tree edge must belong to a cycle of length three (triangle) in the solution. We give a Θ(log⁡n)-approximation algorithm for this problem in the weighted case in n-node graphs and a 4-approximation in the unweighted case.