Dick Lipton of Georgia Tech has a very nice blog on the theory of computation (though it ranges broader than that). He has a fascinating post on “Mathematical Embarrassments”. A “Mathematical Embarrassment” does not refer to the mistaken proof I had in a real analysis course twenty five years ago (that still causes me to cringe) or a similar goof. Instead, it is defined as:
A mathematical embarrassment (ME) is a problem that should have been solved by now. An ME usually is easy to state, seems approachable, and yet resists all attempts at an attack.
This contrasts with a Mathematical Disease (MD), which is harder to define but has a number of characteristics:
1. A problem must be easy to state to be a MD. This is not sufficient, but is required. Thus, the Hodge-Conjecture will never be a disease. I have no clue what it is about.
2. A problem must seem to be accessible, even to an amateur. This is a key requirement. When you first hear the problem your reaction should be: that is open? The problem must seem to be easy.
3. A problem must also have been repeatedly “solved” to be a true MD. A good MD usually has been “proved” many times—often by the same person. If you see a paper in arXiv.org with many “updates” that’s a good sign that the problem is a MD.
So, in operations research we see many people catch the “P=NP (or not)” disease, but I would not call the problem an embarrassment.
Lipton has a number of good examples of mathematical embarrassments, some of which overlap with the skills/interests of more mathematically inclined operations researchers, but none are “really” operations research. For instance, it seems that proving that both the sum and difference of pi and e are transcendental would be a homework problem from centuries ago, but it is still open. That is an embarrassment!
So what would be an operations research embarrassment?
On the computational side, I think of the Traveling Tournament Problem as a bit of an embarrassment (embarrassment is too strong for this: perhaps this is an “operations research discomfiture”). It is a sports scheduling problem that is easy to state, but even very small instances seem beyond our current methods. It was a big deal when the eight team problem was solved!
Just this month, I received word from New Zealand that the three 10-team examples (NL10, CIRC10, and Galaxy10) have all been solved (by David Uthus, Patricia Riddle, and Hans Guesgen). This is huge for the couple of dozen people really into sports scheduling! Proving optimality for this size takes 120 processors about a week so I would say that the problem is still a discomfiture.
But a discomfiture is not an embarrassment. The problem, though simple to state, is too small to rank as an embarrassment. So what would an “operations research embarrassment” be? And are there other “operations research diseases” other than P=NP (though one could argue that by causing people to spend computer-years in solving Traveling Tournament Problems, I may have infected some with a mild disease).