The Ising Model Is NP - Complete

In 1925, the German physicist Ernst Ising introduced a simple mathematical model of phase transitions, the abrupt changes of state that occur, for example, when water freezes or a cooling lump of iron becomes magnetic. In the 75 years since, the Ising model has been analyzed, generalized, and computerized—but never, except in special cases, solved. Researchers managed to get exact answers for physically unrealistic, two-dimensional systems, but have never been able to make the leap out of the plane. There could be a good reason: The Ising model, in its full, nonplanar glory, is NP-complete. The complexity result was announced in May by Sorin Istrail, a theoretical computer scientist at Sandia National Laboratories (who subsequently joined Celera Genomics in Rockville, Maryland). Extending earlier work of Francisco Barahona of the University of Chile, Istrail showed that essentially all versions of the Ising model are computationally intractable when the setting is three-dimensional. Moreover, the new results show that the computational barrier lies not so much in the extra dimension as in the nonplanarity of an essential underlying graph—which explains why physicists have been stymied even in certain two-dimensional generalizations of the Ising model. Although it doesn’t completely put the kibosh on the search for exact solutions (for one thing, the P-versusNP question is still famously open), Istrail’s work sheds new light on the likely limitations of techniques that, because of their success in the plane, had theorists chasing wild geese into the third dimension.