The T-tour problem is a natural generalization of TSP and Path TSP. Given a
graph G=(V,E), edge cost c:E→R≥0​, and an even
cardinality set T⊆V, we want to compute a minimum-cost T-join
connecting all vertices of G (and possibly containing parallel edges).
In this paper we give an 711​-approximation for the T-tour
problem and show that the integrality ratio of the standard LP relaxation is at
most 711​. Despite much progress for the special case Path TSP, for
general T-tours this is the first improvement on Seb\H{o}'s analysis of the
Best-of-Many-Christofides algorithm (Seb\H{o} [2013])