Many studies of constraint
satisfaction problems have demonstrated, both empirically and theoretically,
that easily computed structural parameters of these problems can predict, on
average, how hard the problems are to solve by a variety of search methods.
A major result of this work is that hard instances of NPcomplete problems are
concentrated near an abrupt transition between under and overconstrained
problems. This transition is analogous to phase transitions seen in some
physical systems.
For a collection of papers on phase transitions in search, see the Artificial Intelligence special issue on Frontiers in Problem Solving: Phase Transitions and Complexity.

Observations and Theory 


» T. Hogg, B. A. Huberman and C. Williams, Phase Transitions and the Search Problem, 1996
A tutorial on phase transitions in search
» C. P. Williams and T. Hogg, Exploiting the Deep Structure of Constraint Problems, 1994
A theory describing why the phase transitions occur and the approximate location of the transition point, with comparison to empirical observations and extensions to a variety of search problems
» T. Hogg and C. P. Williams, The Hardest Constraint Problems: A Double Phase Transition, 1994
Hard problems are concentrated around the phase transition, but there are also rare, extremely hard cases far from the transition region, indicating more complex behaviors
» T. Hogg, Applications of Statistical Mechanics to Combinatorial Search Problems, 1995
A review of the phase transition examining problem classes representing some more realistic types of constraint situations
» T. Hogg, Which Search Problems Are Random?, 1998
Analysis of the structure of hard problems using approximate entropy



Search Methods for Hard Problems 



Using Phase Transitions as a Search Heuristic 



Phase Transitions in General Heuristic Search 


Other types of phase transitions can also appear in more general searches.
» B. A. Huberman and T. Hogg, Phase Transitions in Artificial Intelligence Systems, 1987
Phase transitions in heuristic backtrack search
» C. P. Williams and T. Hogg, The Typicality of Phase Transitions in Search, 1993
An extended analysis of phase transitions in heuristic backtrack search
» T. Hogg and J. O. Kephart, Phase Transitions in HighDimensional Pattern Classification, 1990
Phase transitions can also appear in the behavior of database retrieval
» J. Shrager, T. Hogg and B. A. Huberman, Observation of Phase Transitions in Spreading Activation Networks, 1987
and in spreading activation networks
» T. Hogg and B. A. Huberman, Artificial Intelligence and Large Scale Computation: A Physics Perspective, 1987
A more general discussion of analogies from physics applying to search and other computational problems in artificial intelligence
» J. Shrager, T. Hogg and B. A. Huberman, A GraphDynamic Model of the Power Law of Practice and the ProblemSolving FanEffect, 1988
Statistical models also describe learning behavior (for people as well as sophisticated search methods) such as the power law of practice
These papers also contain numerous references to other studies of the phase transition behavior.



Quantum Computing 



Other Places to Visit 


