To begin, I think it's best to see an attorney on this issue to make sure that things are done correctly. The analysis in situations such as these can be contingent upon a few different factors. That said, generally, the individual that you owe child support to can sign a satisfaction of judgment if they wish for your arrears to go away (short and best case scenario answer), however, if some of your arrears are owed to the state it might not be that simple.
If your ex has been on public assistance, some of the money that you owe in back child support may in fact be owed to the state. In that situation your ex does not have the authority to waive the arrears (back support). If you have been providing a great deal of your ex's actual support, the state may work with you to alleviate some or all of the money you owe, but if the state has been providing health care, food stamps, or some other state assistance you will need to work with them directly.
I hope this gives you some place to start and\or helps you to learn what you still need to investigate to find out what it will take to handle this situation. The reality is though that whether or not you are fully informed as to the nature of your back support, this situation will require a great deal of paperwork and you may want to consult a lawyer in your community to help you out.
Jeffrey K. TraylorAsk a similar question