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starve (someone or something) out
To deprive someone or something of food in order to force them to submit or surrender. The terrorists have indicated that they will not leave the building peacefully, so police are planning to simply starve them out. The empire's army began starving the rebel city out, its siege lasting nearly three months.
To force someone or something to surrender by depriving them of food, as in a siege: The police starved out the criminals, who had refused to leave their apartment. If they refuse to surrender the garrison, we'll just starve them out.