  We study the boundary of tractability for the Max-Cut problem in graphs. Our
main result shows that Max-Cut above the Edwards-Erd\H{o}s bound is
fixed-parameter tractable: we give an algorithm that for any connected graph
with n vertices and m edges finds a cut of size m/2 + (n-1)/4 + k in time
2^O(k)n^4, or decides that no such cut exists. This answers a long-standing
open question from parameterized complexity that has been posed several times
over the past 15 years. Our algorithm is asymptotically optimal, under the
Exponential Time Hypothesis, and is strengthened by a polynomial-time
computable kernel of polynomial size.
