Interest-based news readers collect and present news according to topics (as opposed to, say, RSS feed collection). The bestest of all the interest-based readers was Zite. More specifically, Zite 1.2. (The update they issued after being acquired by Flipboard was, visually, poo.) But Zite has a date with the hangperson. The service will be shut down Dec 7, of this the year 2015. Now, Flipboard, the supposed market leader for interest-based news consumption, aside from not working in China, is not a real option. The interface is claustrophobic and from the point of view of actual interest-based consumption, counterproductive. Focus on one or two stories at a time? That is data paucity, not density. So anyway, one is seeking a replacement. Those that I know of, for Android, include:
News360 - not bad. Not unlike Flipboard in appearance but making so much better use of screen real-estate that it feels open and welcoming, and switching between topics is relatively easy.
Prismatic - just started using this today and found it odd to set up, and slow in China, but once it got going, it could turn out to be good; reading topics is difficult, but the main feed where everything is collected works well.
Others? Feedly is not interest-based, no matter what they say. It's all about the RSS feeds. Beyond that... any suggestions?