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Rigorous location of phase transitions in hard optimization problems


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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Figure 1: The Satisfiability Threshold Conjecture.
Figure 2: Our results for random Max k -SAT.
Figure 3
Figure 4: Plots of entropy, correlation, and their product for the vanilla second moment method.
Figure 5: Plots of the correlation function and its product with entropy for the weighted second moment method, when the weighing is given by equation (3).


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We thank S. Kirkpatrick, J. Kleinberg and S. Mertens for feedback on the presentation of the results.

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Correspondence to Dimitris Achlioptas.

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Achlioptas, D., Naor, A. & Peres, Y. Rigorous location of phase transitions in hard optimization problems. Nature 435, 759–764 (2005).

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