The council shall have authority to lay out, open, widen, extend, straighten, alter, close, vacate or abolish any highway, street or alley in the city, whenever they shall deem the same a public improvement; and if in so doing it shall be necessary to take or use private property, the same may be taken in the manner in this act provided for taking private property for public use. The expense of such improvement may be paid by special assessments upon the property adjacent to or benefited by such improvement, in the manner in this act provided for levying and collecting special assessments; or in the discretion of the council, a portion of such costs and expenses may be paid by special assessments as aforesaid, and the balance from the general street fund.