A 1.751.75 LP approximation for the Tree Augmentation Problem

In the Tree Augmentation Problem (TAP) the goal is to augment a tree $T$ by a minimum size edge set $F$ from a given edge set $E$ such that $T \cup F$ is $2$-edge-connected. The best approximation ratio known for TAP is $1.5$. In the more general Weighted TAP problem, $F$ should be of minimum weight. Weighted TAP admits several $2$-approximation algorithms w.r.t. to the standard cut LP-relaxation, but for all of them the performance ratio of $2$ is tight even for TAP. The problem is equivalent to the problem of covering a laminar set family. Laminar set families play an important role in the design of approximation algorithms for connectivity network design problems. In fact, Weighted TAP is the simplest connectivity network design problem for which a ratio better than $2$ is not known. Improving this "natural" ratio is a major open problem, which may have implications on many other network design problems. It seems that achieving this goal requires finding an LP-relaxation with integrality gap better than $2$, which is a long time open problem even for TAP. In this paper we introduce such an LP-relaxation and give an algorithm that computes a feasible solution for TAP of size at most $1.75$ times the optimal LP value. This gives some hope to break the ratio $2$ for the weighted case. Our algorithm computes some initial edge set by solving a partial system of constraints that form the integral edge-cover polytope, and then applies local search on $3$-leaf subtrees to exchange some of the edges and to add additional edges. Thus we do not need to solve the LP, and the algorithm runs roughly in time required to find a minimum weight edge-cover in a general graph.Comment: arXiv admin note: substantial text overlap with arXiv:1507.0279

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

Motivated by the well known four-thirds conjecture for the traveling salesman problem (TSP), we study the problem of {\em uniform covers}. A graph $G=(V,E)$ has an $\alpha$-uniform cover for TSP (2EC, respectively) if the everywhere $\alpha$ vector (i.e. $\{\alpha\}^{E}$) dominates a convex combination of incidence vectors of tours (2-edge-connected spanning multigraphs, respectively). The polyhedral analysis of Christofides' algorithm directly implies that a 3-edge-connected, cubic graph has a 1-uniform cover for TSP. Seb\H{o} asked if such graphs have $(1-\epsilon)$-uniform covers for TSP for some $\epsilon > 0$. Indeed, the four-thirds conjecture implies that such graphs have 8/9-uniform covers. We show that these graphs have 18/19-uniform covers for TSP. We also study uniform covers for 2EC and show that the everywhere 15/17 vector can be efficiently written as a convex combination of 2-edge-connected spanning multigraphs. For a weighted, 3-edge-connected, cubic graph, our results show that if the everywhere 2/3 vector is an optimal solution for the subtour linear programming relaxation, then a tour with weight at most 27/19 times that of an optimal tour can be found efficiently. Node-weighted, 3-edge-connected, cubic graphs fall into this category. In this special case, we can apply our tools to obtain an even better approximation guarantee. To extend our approach to input graphs that are 2-edge-connected, we present a procedure to decompose an optimal solution for the subtour relaxation for TSP into spanning, connected multigraphs that cover each 2-edge cut an even number of times. Using this decomposition, we obtain a 17/12-approximation algorithm for minimum weight 2-edge-connected spanning subgraphs on subcubic, node-weighted graphs