Such improvements may be ordered by the governing body and the cost thereof apportioned in pursuance of an agreement between the governing body and the abutting landowners, and, in the absence of such an agreement, the cost of improvements which is to be defrayed in whole or in part by such local tax or assessment, may in cities and towns be ordered on a petition from the owners of not less than three-fourths of the parcels to be affected thereby, or in counties on a petition from not less than sixty percent of the landowners to be affected thereby or by a two-thirds vote of all the members elected to the governing body. Notice shall be given to the abutting landowners, notifying them when and where they may appear before the governing body, or some committee thereof, or the administrative board or other similar board of the locality to whom the matter may be referred, to be heard in favor of or against such improvements.