This is a project consisting of two questions. Each with a few components.
The first involves designing a turing machine (flow chart or table representation) which executes a particular behaviour
The second involves proofs regarding maximum flow and minimum cost in graphs subject to various constraints (not trivially max/flow min cut)
The algorithms in each are examined for efficiency etc
In order to complete this project you need a solid understanding of mathematical proof.
-- send message asking for details and a quick note as to why you feel qualified and I will send the pdf description.