Nature 435, 759-764 (9 June 2005) | doi:10.1038/nature03602; Received 16 December 2004; Accepted 31 March 2005

Rigorous location of phase transitions in hard optimization problems

Dimitris Achlioptas1, Assaf Naor1 & Yuval Peres2

  1. Microsoft Research, One Microsoft Way, Redmond, Washington 98052, USA
  2. Department of Statistics, University of California, Berkeley, California 94720, USA

Correspondence to: Dimitris Achlioptas1 Correspondence and requests for materials should be addressed to D.A. (Email:


It is widely believed that for many optimization problems, no algorithm is substantially more efficient than exhaustive search. This means that finding optimal solutions for many practical problems is completely beyond any current or projected computational capacity. To understand the origin of this extreme 'hardness', computer scientists, mathematicians and physicists have been investigating for two decades a connection between computational complexity and phase transitions in random instances of constraint satisfaction problems. Here we present a mathematically rigorous method for locating such phase transitions. Our method works by analysing the distribution of distances between pairs of solutions as constraints are added. By identifying critical behaviour in the evolution of this distribution, we can pinpoint the threshold location for a number of problems, including the two most-studied ones: random k-SAT and random graph colouring. Our results prove that the heuristic predictions of statistical physics in this context are essentially correct. Moreover, we establish that random instances of constraint satisfaction problems have solutions well beyond the reach of any analysed algorithm.


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