Max-Cut Parameterized Above the Edwards-Erd\H{o}s Bound

TL;DR

This paper proves that the Max-Cut problem above the Edwards-Erdős bound is fixed-parameter tractable, providing an efficient algorithm and polynomial kernel, resolving a 15-year open question in parameterized complexity.

Contribution

It introduces the first fixed-parameter algorithm for Max-Cut above the Edwards-Erdős bound, with optimal complexity under ETH and a polynomial kernel.

Findings

Algorithm finds cuts of size m/2 + (n-1)/4 + k in 2^O(k)n^4 time.

Decides the existence of such cuts or confirms their absence.

Provides a polynomial kernel of polynomial size.

Abstract

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.