 
Summary: Rigorous location of phase transitions in
hard optimization problems
Dimitris Achlioptas1
, Assaf Naor1
& Yuval Peres2
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 moststudied ones: random kSAT
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.
Constraint satisfaction problems are at the heart of statistical physics,
information theory and computer science. Typically, they involve
a large set of variables, each taking values in a small domain, such
as {0, 1}, and a collection of constraints, each binding a few of
