Think carefully about why the relationship broke down (different working styles, clashing personalities, a crisis) then find a cure that fits the cause. Ill-considered fixes (flattering someone who is tired of your insincerity) won't work.
Make amends. If the breakdown was your fault, apologise for your behaviour and its impact. Build on 'I'm sorry I wasn't prepared for the meeting' with 'and that you lost your biggest client as a result'. Explain how you'll avoid a repeat offence.
Watch your step. When a relationship is at breaking point, the smallest misunderstanding can ruin all chance of repair. Avoid sarcasm, jokes and comments that could be misconstrued.
Find common ground. Forge new bonds by focusing your conversations on shared beliefs and goals. If you don't have any, highlight a common evil to unite against.
Be adults. We interact using three roles - parent, adult and child, says research. Relationships that trap us in one role tend to fail. Ready to cut ties with your childish direct report? Give him a project to lead, leave him to it and don't respond to his pleas for help. Achieve adult-to-adult interaction and watch your relationship rekindle.
Make the first move. Rifts caused by clashing personalities are rarely solved collaboratively. For a week, take responsibility for improving every interaction you have with the other person. Ignore insults, bite your tongue, smile. He'll soon follow suit.
Hang in there. A full relationship repair takes time, patience and consistency. If it's worth it, it's worth the wait.
Know when to quit. Sometimes, no matter how hard we try, a relationship is beyond repair. If you regularly ignore one another, don't want to make peace and don't admire each other, move on.