Is there a Constrainedness Knife-edge?

Recent work on search has identified an intriguing feature dubbed the constrainedness knife-edge by Walsh (Proc. AAAI98, 406–411) whereby overconstrained problems seem to become on average even more constrained as the search goes deeper, while underconstrained ones become less constrained. The present paper shows that while the knife-edge phenomenon itself is real, many of the claims that have been made about it are incorrect. It is not always associated with criticality, it is not a function of the problem so much as of the algorithm used to solve it, and it does not help to explain the peculiar hardness of problem instances near phase transitions. Despite the negative findings, the upshot is that Walsh’s work has opened a fascinating line of research which will surely repay further investigation.