contains at most $2n-1$ edges, (ii) the support of $x$ is 2-vertex connected, and (iii) $x \leq 1$. We will refer to the set of values $\{x_e\}$ for $e \in E$ as $x$-values.

Let $\sum_{e \in E} x_e = OPT_{LP}(G) = (1+\eps)n$ for some $\eps$, where $0 \leq \eps \leq 1$. We will eventually make use of the following definitions.

The [*excess*]{} $x$-value $\eps(v)$ at a vertex $v$ is the amount by which the total value on the adjacent edges exceeds 2, i.e $\eps(v) =
x(\delta(v)) -2.$

A vertex $v \in V$ is called [*heavy*]{} if $\x(\delta(v)) > 2$.

The following fact will be useful in our analysis. If $OPT_{LP}(G) = (1+\eps)n$, then, $$\begin{aligned}
\sum_{v \in V} \x(\delta(v)) & = & \sum_{v \in V} (2 + \eps(v)) ~~ =
~~ 2 (1 + \eps)n.\end{aligned}$$ This implies, $$\begin{aligned}
\sum_{v \in V} \eps(v) & = & 2 \eps n.\label{eps-vertices}\end{aligned}$$

Spanning trees and circulations {#sec:stc}
-------------------------------

Let us recall some useful definitions from the approach of Mömke and Svensson [@DBLP:conf/focs/MomkeS11] that we use throughout this paper.

A [*greedy DFS tree*]{} is a spanning tree formed via a depth-first search of $G$. If there is a choice as to which edge to traverse next, 