I am working on a relationship management app (finding new people to follow, following back, and unfollowing). Currently only the Twitter API has been implemented, and I wanted to add Instagram support. However, the Instagram Platform Policy makes it seem impossible, especially the line (19): "Don't use follower information for anything other than analytics without our prior permission. For example, don't display these relationships in your app." Is relationship management completely unsupported, or are there guidelines and rules that would make such an app acceptable?
Yes, completely unsupported. The three use cases are very explicit and line 19 is very explicit as well. Unless you can frame your application's use case to fit into one of the three use cases AND comply with the policies, you're out of luck.
Instagram has threatened to revoke API access for a popular application that was trying to do what you're trying to do. My advice is to not waste time trying beat the system. Figure out how your application can succeed while complying with Instagram's API policies. Here is a case study for you: Crowdfire decides to remove relationship management features.
Notice how they remove the features completely from their application and replace them with a "report card." The report card, like the score card, is a fundamental analytics technique to provide actionable insights. While the application can not automatically manage relationships, there is nothing stopping them from providing actionable insights since this falls into "analytics." Make sense?
I hope this helps you. IMO, when developing a third-party app for Instagram, you want to fit into their use cases. Instagram's smart and they set these policies for good reasons. As the platform evolves, applications that don't fit into the use cases and violate the policies will be banned. Users will not use them because they won't want to get in trouble and lose their accounts. The accounts I work with that have > 100k followings already know this and they're very sensitive to the applications they're willing to use because of this. So, even if you cheat the system, you won't get any users. You see what I'm getting at?