My philosophy has always been if you are "in love" give it your BEST.
Odds are if you dumped them there was a (good) reason and if they dumped you then you will never really feel "secure" in the relationship again.
Either way once you have given something your BEST there is nothing more you can do the "second time around".
It makes little sense to me to have a yo-yo relationship filled with breakups and makeup sessions. Some people actually think it is "romantic". I believe your future lies ahead of you and not behind you.
The only exception I can imagine is if there was a gap of many years after the breakup which allowed both people to evolve into "different people". For example you broke up at age 21 and met up again at 41 or 51. Odds are you would have both learned a lot about yourselves as well as about life.