does not improve on the best-known guarantee of $7/5$. However, as we finally show in Section \[sec:combine\], if we take the best of the two circulations, we can show that at least one of the circulations will lead to an approximation guarantee of at most $46/33$.

We remark that our notation differs from that in [@DBLP:conf/focs/MomkeS11] and [@DBLP:conf/stacs/Mucha12], even though we are using exactly the same circulation network and we use their approach for obtaining the feasible circulation described in Section \[sec:x-circ\]. This different notation allows us to more easily analyze the tradeoff between the two circulations.

Preliminaries: Notation and definitions {#sec:prelim}
=======================================

Throughout this paper, we make use of the following well-studied linear programming relaxation for graph-TSP.

Linear program for graph-TSP {#sec:LP}
----------------------------

For a graph $G=(V,E)$, the following linear program is a relaxation of graph-TSP. We refer to Section 2 of [@DBLP:conf/focs/MomkeS11] for a discussion of its derivation and history. $$\begin{aligned}
& \min & \sum_{e \in E} y_e \\
y(\delta(S)) & \geq &  2  ~~ {\text{for }} \emptyset \neq S \subset V,   \\
y & \geq & 0.\end{aligned}$$ We denote this linear program by $LP(G)$ and we denote the value of an optimal solution for $LP(G)$ by $OPT_{LP}(G)$. 