  This work analyzes the minimum tollbooth problem in atomic network congestion
games with unsplittable flows. The goal is to place tolls on edges, such that
there exists a pure Nash equilibrium in the tolled game that is a social
optimum in the untolled one. Additionally, we require the number of tolled
edges to be the minimum. This problem has been extensively studied in
non-atomic games, however, to the best of our knowledge, it as not been
considered for atomic games before. By a reduction from the weighted CNF SAT
problem, we show both the NP-hardness of the problem and the W[2]-hardness when
parameterizing the problem with the number of tolled edges. On the positive
side, we present a polynomial time algorithm for networks on series-parallel
graphs that turns any given state of the untolled game into a pure Nash
equilibrium of the tolled game with the minimum number of tolled edges.
