Given a metric (V,d) and a root∈V, the classic
\textsf{k-TSP} problem is to find a tour originating at the root
of minimum length that visits at least k nodes in V. In this work,
motivated by applications where the input to an optimization problem is
uncertain, we study two stochastic versions of \textsf{k-TSP}.
In Stoch-Reward k-TSP, originally defined by Ene-Nagarajan-Saket [ENS17],
each vertex v in the given metric (V,d) contains a stochastic reward Rv.
The goal is to adaptively find a tour of minimum expected length that collects
at least reward k; here "adaptively" means our next decision may depend on
previous outcomes. Ene et al. give an O(logk)-approximation adaptive
algorithm for this problem, and left open if there is an O(1)-approximation
algorithm. We totally resolve their open question and even give an
O(1)-approximation \emph{non-adaptive} algorithm for this problem.
We also introduce and obtain similar results for the Stoch-Cost k-TSP
problem. In this problem each vertex v has a stochastic cost Cv, and the
goal is to visit and select at least k vertices to minimize the expected
\emph{sum} of tour length and cost of selected vertices. This problem
generalizes the Price of Information framework [Singla18] from deterministic
probing costs to metric probing costs.
Our techniques are based on two crucial ideas: "repetitions" and "critical
scaling". We show using Freedman's and Jogdeo-Samuels' inequalities that for
our problems, if we truncate the random variables at an ideal threshold and
repeat, then their expected values form a good surrogate. Unfortunately, this
ideal threshold is adaptive as it depends on how far we are from achieving our
target k, so we truncate at various different scales and identify a
"critical" scale.Comment: ITCS 202