There are probably a lot of different ways to look at it. One perspective is that easy problems usually involve convex sets (or minimizing convex functions, wh

There are probably a lot of different ways to look at it. One perspective is that easy problems usually involve convex sets (or minimizing convex functions, which is pretty much the same thing since the solution set is guaranteed to be convex). Discrete sets, such as the vertex set of a hypercube, are trivially non-convex (unless they only contain a single element). You can also think about it geometrically. Imagine you have a plane that passes through a cube. How do you determine if it intersects one of the cube vertices (this is basically equivalent to the subset sum problem) ? Of course it's easy to find a method that will work in 3D, but these methods all end up scaling exponentially with the dimension of the space.