The board may contract with the federal government (which term as used in this section shall include any agency thereof) whereby the federal government will pay the whole or any part of the cost of the project and/or will perform the whole or any part of the work connected therewith, which contract may include any specific terms required by act of congress or federal regulation as a condition for such participation on the part of the federal government. The board may also contract with any private corporation or with any public corporation (which term as used in this section shall include any agency thereof) in respect to any matter connected with the construction and/or maintenance of any project. Such a contract may provide that any payments made or work done by such a public corporation shall relieve it in whole or in part from assessment for the cost of the project. No construction work shall be undertaken until the board has advertised for and received bids for the performance of such work and a contract let to the lowest responsible bidder: Provided, That this provision shall not apply to work to be performed by the federal government or a public corporation.