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-Erdős 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 $$ \frac{m}{2} + \frac{n-1}{4} + k $$ in time 2O(k)·n4, 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.

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