  We introduce the $L_p$ Traveling Salesman Problem ($L_p$-TSP), given by an
origin, a set of destinations, and underlying distances. The objective is to
schedule a destination visit sequence for a traveler of unit speed to minimize
the Minkowski $p$-norm of the resulting vector of visit/service times. For $p =
\infty$ the problem becomes a path variant of the TSP, and for $p = 1$ it
defines the Traveling Repairman Problem (TRP), both at the center of classical
combinatorial optimization. We provide an approximation preserving
polynomial-time reduction of $L_p$-TSP to the segmented-TSP Problem [Sitters
'14] and further study the case of $p = 2$, which we term the Traveling
Firefighter Problem (TFP), when the cost due to a delay in service is quadratic
in time.
  We also study the all-norm-TSP problem [Golovin et al. '08], in which the
objective is to find a route that is (approximately) optimal with respect to
the minimization of any norm of the visit times, and improve corresponding
(in)approximability bounds on metric spaces.
