One approach would be to spin out the search index and algorithms out of Google, turning it into a B2B service provider that provides (for a fee) an API that everyone could use to provide a search engine. The "remaining" Google could keep working like before, just that they would no longer have their own search technology but would buy API access like anyone else could.

This would address the idea that Internet search has become a monopoly because Google is so dominant and building up your own index and algorithms has become so expensive that no one on the free market can challenge it. (This assumes that Bing is a huge loss maker for Microsoft, Yandex and Baidu can only exist because of government interference against Google in their home markets and all the others like DuckDuckGo are relying on API access provided by these three.)

I am not sure how true this second paragraph really is and I also think there would be quite big efficiency losses by such an approach (e.g. knowing how to search the web might also be helpful for searching your email etc. and such an API model would make it very difficult to use knowledge about the user to influence search results; which is something all the privacy conscious search engines such as DDG struggle with), but at least that's what I have heard people suggest when they want to see Google broken up.