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Tur\'{a}n's Theorem Through Algorithmic Lens
The fundamental theorem of Tur\'{a}n from Extremal Graph Theory determines
the exact bound on the number of edges in an -vertex graph that
does not contain a clique of size . We establish an interesting link
between Extremal Graph Theory and Algorithms by providing a simple compression
algorithm that in linear time reduces the problem of finding a clique of size
in an -vertex graph with edges, where , to the problem of finding a maximum clique in a graph on at most
vertices. This also gives us an algorithm deciding in time whether has a clique of size . As a byproduct of the new
compression algorithm, we give an algorithm that in time decides whether a graph contains an independent set of size at least
. Here is the average vertex degree of the graph . The
multivariate complexity analysis based on ETH indicates that the asymptotical
dependence on several parameters in the running times of our algorithms is
tight
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