It is a result of Zaw Win (paywall) that optimal Windy Postman tours in eulerian digraphs can be calculated in polynomial time.
Bodo Manthey has shown that directed 3-Cycle Covers are APX hard to approximate.
It is a trivial observation that weighted digraphs can be interpreted as instances of the Windy Postman Problem by combining pairs of antiparallel arcs into an undirected edge, whose weight depends on the direction of traversal.
The advent of undesirable 2-cyles can then be ruled out by taking the 3-Cycle Cover of the optimal Windy Postman tour instead of on the complete digraph with its antiprallel arcs as a compromise.
For the purpose of finding good(?) directed 3-Cycle Covers efficiently any non-eulerian complete instance can be turned into an eulerian one via a variant of the 17 camels trick by adding $3$ vertices $x,y,z$ and setting the weights of all adjazent edges to $INF$ and the weight of each edge on the subtour $(x,y,z,x)$ to $0$ if the cover of minimum weight is sought. The so modified instance will replicate the original cycle cover plus $(x,y,z,x)$, so the original solution can be easily recovered.
Question:
what is can be said about the quality of 3-Cycle Covers obtained by removing antiparallel arcs via the polynomial-time eulerian Windy Postman algorithm of Zaw Win
for arcweights retricted to $\lbrace 0,1\rbrace$ or drawn uniformly distributed in $[0,1]$ assuming $n=2k+1$?
