Is a person who has damaged another's property absolved if the owner tells him: "It's okay, it's not a problem," or is it necessary for him to say: "I forgive you"?

For the validity of an absolution (إبراء), it is not necessary to pronounce a specific word. Rather, any expression that signifies absolution or reduction (حط) is sufficient, such as forfeiture (إسقاط), gifting (هبة), relinquishment (ترك), rendering permissible (إحلال), declaring lawful (تحليل), pardon (عفو), or waiver (وضع).