Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider the average case behavior of a linear optimization problem on various series of combinatorially interesting polytopes. From general results of integral geometry it follows that for all but an asymptotically negligible fraction of linear functions a polytope can be replaced by a pair of concentric balls with asymptotically equal radii so that the optimal value of the linear function on the polytope is in the interval beten the optimal values of the linear function on these balls. In particular,  show that the average case behavior of the assignment problem, traveling salesman problem, and, generally speaking, of any optimization problem on a polynomial fraction of all permutations is the same.
['Alexander I. Barvinok']
I am writing a research paper on a idea Integral geometry of higher-dimensional polytopes and the average case in combinatorial optimization
[{"from": "human", "value": "I am writing a research paper on a idea Integral geometry of higher-dimensional polytopes and the average case in combinatorial optimization"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider the average case behavior of a linear optimization problem on various series of combinatorially interesting polytopes. From general results of integral geometry it follows that for all but an asymptotically negligible fraction of linear functions a polytope can be replaced by a pair of concentric balls with asymptotically equal radii so that the optimal value of the linear function on the polytope is in the interval beten the optimal values of the linear function on these balls. In particular,  show that the average case behavior of the assignment problem, traveling salesman problem, and, generally speaking, of any optimization problem on a polynomial fraction of all permutations is the same."}]