Where the Really Hard Problems Are

It is well known that for many NP-complete problems, such as K-Sat, etc., typical cases are easy to solve; so that computationally hard cases must be rare (assuming P = NP). This paper shows that NP-complete problems can be summarized by at least one "order parameter", and that the hard problems occur at a cri t ical value of such a parameter. This cri t ical value separates two regions of characteristically different properties. For example, for K-colorabil ity, the crit ical value separates overconstrained from underconstrained random graphs, and it marks the value at which the probabi l i ty of a solution changes abruptly from near 0 to near 1. It is the high density of wellseparated almost solutions (local minima) at this boundary that cause search algorithms to " thrash". This boundary is a type of phase transit ion and we show that it is preserved under mappings between problems. We show that for some P problems either there is no phase transit ion or it occurs for bounded N (and so bounds the cost). These results suggest a way of deciding if a problem is in P or NP and why they are different.