Max-Cut Parameterized above the Edwards-Erdős Bound

We study the boundary of tractability for the max-cut problem in graphs. Our main result shows that max-cut above the edwards-erdos 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 \frac{m}{2} + \frac{n-1}{4} + k \frac{m}{2} + \frac{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 a number of 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.