  A metro-line crossing minimization problem is to draw multiple lines on an
underlying graph that models stations and rail tracks so that the number of
crossings of lines becomes minimum. It has several variations by adding
restrictions on how lines are drawn. Among those, there is one with a
restriction that line terminals have to be drawn at a verge of a station, and
it is known to be NP-hard even when underlying graphs are paths. This paper
studies the problem in this setting, and propose new exact algorithms. We first
show that a problem to decide if lines can be drawn without crossings is solved
in polynomial time, and propose a fast exponential algorithm to solve a
crossing minimization problem. We then propose a fixed-parameter algorithm with
respect to the multiplicity of lines, which implies that the problem is FPT.
