When the cold case unit was formed in Los Angeles, detectives faced a daunting backlog of 9,000 unsolved murder cases dating back four decades. But killings don't get any colder than the 1957 murder of two young cops during a routine traffic stop. Investigators had all but given up on the case -- one of the oldest unsolved homicides in the country -- when modern technology finally caught up with justice.
I wonder whether I remove " on" away and the phrase "given up the case " is correct. Are the meanings of the two same to each other?
e.g. I give up drinking