  We revisit the a priori TSP (with independent activation) and prove stronger
approximation guarantees than were previously known. In the a priori TSP, we
are given a metric space $(V,c)$ and an activation probability $p(v)$ for each
customer $v\in V$. We ask for a TSP tour $T$ for $V$ that minimizes the
expected length after cutting $T$ short by skipping the inactive customers. All
known approximation algorithms select a nonempty subset $S$ of the customers
and construct a master route solution, consisting of a TSP tour for $S$ and two
edges connecting every customer $v\in V\setminus S$ to a nearest customer in
$S$. We address the following questions. If we randomly sample the subset $S$,
what should be the sampling probabilities? How much worse than the optimum can
the best master route solution be? The answers to these questions (we provide
almost matching lower and upper bounds) lead to improved approximation
guarantees: less than 3.1 with randomized sampling, and less than 5.9 with a
deterministic polynomial-time algorithm.
