Monster, Ijousha no Ai Recommendations
Both have a psychopath in direct relationship with the protagonist and that he does not want to kill.
Both begin because the protagonist was kind to the psychopath.
Both have an excellent maturation and decadence of the protagonist that takes up to years, no maturation from one day to the next.
In none can they go to the police due to the nature of the crimes.
In both the secondary characters end up having a lot of weight in the plot in a logical way.