I think your definition of Integer programming is not correct. Integer programming or Mixed Integer Programming assumes that the solution has the all or some variables, integers. Most of the Integer programming models are usually using binary variable to indicate decisions.
Further, I think you tried to describe Linear programming in the beginning of your post. But Linear programming searches the vertices of the polytope and not the edges.
The biggest issue with Integer programming is the lack of convexity of the solution space when compared to Linear programming. So the approach for solving them is very different compared to Linear programming. If I made any mistakes in my post, I welcome to be corrected.
Further, I think you tried to describe Linear programming in the beginning of your post. But Linear programming searches the vertices of the polytope and not the edges.
The biggest issue with Integer programming is the lack of convexity of the solution space when compared to Linear programming. So the approach for solving them is very different compared to Linear programming. If I made any mistakes in my post, I welcome to be corrected.