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 two different top-down coloring algorithms. Both of our algorithms differ from known techniques for obtaining 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 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 2 1+α -approximation. We propose a new conjecture on extreme points of LP relaxations for the problem, which if true, will lead to a potentially constructive proof of an integrality gap of at most 3 2 for weighted TAP. In the second algorithm, we introduce simple extra valid constraints to the tree edge covering LP. In this algorithm, we focus on deficient edges, edges which get covered to an extent less than 4 3 in the fractional solution. We show that in the support of extreme points for this LP, deficient edges occurs 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 (triangle). We give a Θ(log n)-approximation algorithm for this problem in the weighted case and a 4-approximation in the unweighted case.

Shorter tours and longer detours: uniform covers and a bit beyond

“…The best-known approximation factor for this problem is 2 [FJ81], but when the solution is half-integral, there is a 4 3 -approximation [CJR99]. This latter result has been generalized by Iglesias and Ravi [IR17].…”

“…Theorem 6 (Iglesias and Ravi [IR17]). If y ∈ Cover(G, F ) and y e ≥ α or y e = 0 for all e ∈ E, then there is an algorithm, whose running time is polynomial in the size of G, that writes the vector 2 1+α • y as a convex combination of 1-covers C 1 , .…”

An optimal rounding for half-integral weighted minimum strongly connected spanning subgraph

“…A recently fruitful such special case has been the assumption that solutions to the relevant linear program (LP) are half-integral-that is, each coordinate of an optimal solution is assumed to lie in {0, 1 2 , 1}. Notably, Cheriyan et al [4] showed that WTAP admits a 4 3-approximation if the relevant LP is half-integral and Iglesias and Ravi [15] generalized this by showing that a 2 (1 + f )-approximation is possible for WTAP if non-zero LP values are assumed to be at least f > 0. Similarly, a recent breakthrough of Karlin et al [17] showed that a ≈ 1.49993 approximation is possible for TSP if the LP solution is assumed to be half-integral.…”