  One of the most studied extensions of the famous Traveling Salesperson
Problem (TSP) is the {\sc Multiple TSP}: a set of $m\geq 1$ salespersons
collectively traverses a set of $n$ cities by $m$ non-trivial tours, to
minimize the total length of their tours.
  This problem can also be considered to be a variant of {\sc Uncapacitated
Vehicle Routing} where the objective function is the sum of all tour lengths.
  When all $m$ tours start from a single common \emph{depot} $v_0$, then the
metric {\sc Multiple TSP} can be approximated equally well as the standard
metric TSP, as shown by Frieze (1983).
  The {\sc Multiple TSP} becomes significantly harder to approximate when there
is a \emph{set} $D$ of $d \geq 1$ depots that form the starting and end points
of the $m$ tours.
  For this case only a $(2-1/d)$-approximation in polynomial time is known, as
well as a $3/2$-approximation for \emph{constant} $d$ which requires a
prohibitive run time of $n^{\Theta(d)}$ (Xu and Rodrigues, \emph{INFORMS J.
Comput.}, 2015).
  A recent work of Traub, Vygen and Zenklusen (STOC 2020) gives another
approximation algorithm for {\sc Multiple TSP} running in time $n^{\Theta(d)}$
and reducing the problem to approximating TSP.
  In this paper we overcome the $n^{\Theta(d)}$ time barrier: we give the first
efficient approximation algorithm for {\sc Multiple TSP} with a \emph{variable}
number $d$ of depots that yields a better-than-2 approximation.
  Our algorithm runs in time $(1/\varepsilon)^{\mathcal O(d\log d)}\cdot
n^{\mathcal O(1)}$, and produces a $(3/2+\varepsilon)$-approximation with
constant probability.
  For the graphic case, we obtain a deterministic $3/2$-approximation in time
$2^d\cdot n^{\mathcal O(1)}$.ithm for metric {\sc Multiple TSP} with run time
$n^{\Theta(d)}$, which reduces the problem to approximating metric TSP.
