[xkcd386]

> most real-world problems are far from worst case scenarios, therefore improving the worst case currently has little practical use.

This statement is probably mostly correct, but I think that in one way it could be misleading: I would not want to imply that real-world problem instances are somehow easier than the worst-case, in terms of computational complexity. They still very much exhibit exponential increase in computational cost as you scale them up.

Instead, most read-world instances have structure. Some of that structure is well understood (for example, 99% of optimization problems involve extremely sparse matrices), some is not. But sometimes, we can exploit structure even without understanding it fully (some algorithmic techniques work wonder on some instances, and we don't fully know why).

It could be argued that by exploiting structure, it is the constant factor in the big-O computational complexity that gets dramatically decreased. If that is the case, the theory and practice do not really contradict each other. It is just that in practice, we are willing to accept a larger exponent in exchange for a smaller constant factor. Asymptotically it is a losing bargain. But for a given instance, it could be extremely beneficial.