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assign (something) to
To designate someone as responsible for doing something, or as responsible for having done something. Oh, I already assigned that project to Nicole. Since I was driving when we got into the accident, they assigned the blame to me.
assign someone or something to someone or something
to designate someone or something as belonging to someone or something else. They assigned the new car to Roger. They assigned the new worker to the mail room.
assign something to someone
to attribute something to someone; to blame something on someone. We were forced to assign the blame to Robert. They assigned the responsibility for the accident to the driver of the car.