There are some API standards for background checks between businesses and agencies, so that could be a good place to start. You could offer authenticated profiles, where people can't lie on their resume about where they worked.
Of course, LI could just implement this if you became too competitive to them.
Can you list some or are you referring to LinkedIn?
It might be what Checkr is sitting on top of.
Can you explain this? I can add any company and whatever job title and list of responsibilities I want without any verification.
It also let's you make better judgements on likelihood they're lying. It's not impossible to create an authentic-looking fake profile, but generally speaking if you see someone claiming to work for X company and connected to lots of employees from X company it's a safe bet that they worked there.
Disclaimer: I'm the Co-Founder of Resumonk.