Where the really hard problems are

  • Authors:

  • Peter Cheeseman;Bob Kanefsky;William M. Taylor

  • Affiliations:

  • Research Institute for Advanced Computer Science, NASA Ames Research Center, Moffett Field, CA;Sterling Software, NASA Ames Research Center, Moffett Field, CA;Sterling Software, NASA Ames Research Center, Moffett Field, CA

  • Venue:
  • IJCAI'91 Proceedings of the 12th international joint conference on Artificial intelligence - Volume 1
  • Year:
  • 1991

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Abstract

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 critical value of such a parameter. This critical value separates two regions of characteristically different properties. For example, for K-colorability, the critical value separates overconstrained from underconstrained random graphs, and it marks the value at which the probability of a solution changes abruptly from near 0 to near 1. It is the high density of well-separated almost solutions (local minima) at this boundary that cause search algorithms to "thrash". This boundary is a type of phase transition and we show that it is preserved under mappings between problems. We show that for some P problems either there is no phase transition 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.