  The $k-$traveling salesman problem ($k$-TSP) seeks a tour of minimal length
that visits a subset of $k\leq n$ points. The traveling repairman problem (TRP)
seeks a complete tour with minimal latency. This paper provides constant-factor
probabilistic approximations of both problems. We first show that the optimal
length of the $k$-TSP path grows at a rate of
$\Theta\left(k/n^{\frac{1}{2}\left(1+\frac{1}{k-1}\right)}\right)$. The proof
provides a constant-factor approximation scheme, which solves a TSP in a
high-concentration zone -- leveraging large deviations of local concentrations.
Then, we show that the optimal TRP latency grows at a rate of $\Theta(n\sqrt
n)$. This result extends the classical Beardwood-Halton-Hammersley theorem to
the TRP. Again, the proof provides a constant-factor approximation scheme,
which visits zones by decreasing order of probability density. We discuss
practical implications of this result in the design of transportation and
logistics systems. Finally, we propose dedicated notions of fairness --
randomized population-based fairness for the $k$-TSP and geographical fairness
for the TRP -- and give algorithms to balance efficiency and fairness.
