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Summary: Algorithmic Barriers from Phase Transitions
Dimitris Achlioptas
Amin Coja-Oghlan
Abstract
For many random Constraint Satisfaction Problems, by
now there exist asymptotically tight estimates of the largest
constraint density for which solutions exist. At the same
time, for many of these problems, all known polynomial-
time algorithms stop finding solutions at much smaller den-
sities. For example, it is well-known that it is easy to color a
random graph using twice as many colors as its chromatic
number. Indeed, some of the simplest possible coloring al-
gorithms achieve this goal. Given the simplicity of those
algorithms, one would expect room for improvement. Yet,
to date, no algorithm is known that uses (2 - ) colors, in
spite of efforts by numerous researchers over the years.
In view of the remarkable resilience of this factor of 2
against every algorithm hurled at it, we find it natural to
inquire into its origin. We do so by analyzing the evolu-
tion of the set of k-colorings of a random graph, viewed
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