There are two scenarios in all broken relationships. You left, or they left. The factor being evaluated here is I think Value/Worth, yours and/or theirs.
If you left, why did you leave in the first place? If you wanted to fix the relationship, what changed? Did you change? Are you okay with what you weren't okay with before? Did they change and are now no longer crossing the line that they used to cross? Or are you just desperate and want all the good things you had before?
If they left, they gave up on you. Are you going to let someone that gave up on you come back? Are they just desperate to have someone again because they're at the bottom of the barrel? Are they truly sorry for leaving? Will they leave again?
If I chose to leave, I probably have no reason to reconnect. If they chose to leave (keyword - choose) then it would depend on how sincere they are when they want to reconnect.