max number of visited cities

1 people  travels  from D.C to San Francisco and gets back. He sets
off from D.C, and return to D.C later. When he takes flight from DC to
SF , he can only travel east to west, meanwhile , when he takes flight
from SF to DC , he can only travel from west to east.
   Constraint :  He can visit cities on the way once, except D.C. To
utilize his budget K to the utmost , he wants to visit max number of
cities possible. And he is given a list of cities served by airline as
well as a list of direct flights between pairs of cities.

  Design an algorithm which determines the max number of cities that
can be visited ,  and analyze the running time.

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