The owners of an interest in any real estate abutting upon any street or alley who may desire to vacate the street or alley, or any part thereof, may petition the legislative authority to make vacation, giving a description of the property to be vacated, or the legislative authority may itself initiate by resolution such vacation procedure. The petition or resolution shall be filed with the city or town clerk, and, if the petition is signed by the owners of more than two-thirds of the property abutting upon the part of such street or alley sought to be vacated, legislative authority by resolution shall fix a time when the petition will be heard and determined by such authority or a committee thereof, which time shall not be more than sixty days nor less than twenty days after the date of the passage of such resolution.