{"type":"ScholarlyArticle","title":"Where the Really Hard Problems Are","authors":[{"name":"Peter C. Cheeseman","slug":"peter-c-cheeseman","given":"Peter C.","family":"Cheeseman","affiliation":""},{"name":"Bob Kanefsky","slug":"bob-kanefsky","given":"Bob","family":"Kanefsky","affiliation":""},{"name":"William M. Taylor","slug":"william-m-taylor","given":"William M.","family":"Taylor","affiliation":""}],"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 wellseparated 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.","doi":null,"url":"https://ijcai.org/proceedings/1991/50","pdf":"https://ijcai.org/proceedings/1991/0050.pdf","markdown":"https://ijcai.org/proceedings/1991/50.md","bibtex":"https://ijcai.org/proceedings/1991/bibtex/50","volume":{"slug":"1991","title":"Proceedings of the Twelfth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 1991","url":"https://ijcai.org/proceedings/1991/","year":1991,"location":"Sydney, New South Wales, Australia","dates":"24-30 August 1991","editors":"John Mylopoulos and Ray Reiter","isbn":"","publisher":"International Joint Conferences on Artificial Intelligence","doi_prefix":""},"section":"Automated Reasoning","track":"Automated Reasoning","number":50,"pages":{"first":331,"last":339},"keywords":[],"videos":[],"references":[],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}