Yes, Mueller was authorized by DOJ to investigate alleged Manafort collusion with Russia

Deputy Attorney General Rod Rosenstein told special counsel Robert Mueller in a classified August 2, 2017, memo that he should investigate allegations that President Donald Trump’s former campaign chairman Paul Manafort was “colluding with Russian government officials” to interfere in the 2016 presidential election, prosecutors in the Russia probe revealed late Monday night…

The revelation of the August 2 memo comes amid a broader court filing from Mueller’s prosecutors that offers a full-throated defense of their investigative powers and indictments thus far. In the filing, the special counsel’s office argues that a federal judge should not throw out Manafort’s case. Manafort has sought to have the case dismissed, arguing that the charges against him are outside of Mueller’s authority.

The filing Monday night crystallizes the extent to which Rosenstein, who has come under fire by President Donald Trump and others, has backed the investigation’s actions. (Rosenstein oversees Mueller’s investigation following Attorney General Jeff Sessions’ recusal.)