With the upcoming departure of Louis A. DePasquale in July, the search for the next Cambridge city manager — the most influential government post in the city — is well underway. The Initial Screening Committee, composed of four City Councilors and 15 Cambridge residents, has narrowed down its list of potential candidates to four finalists. The City Council will publicly interview each of the candidates on June 1 and will vote on the next city manager during its meeting on June 6.