Kernelization, Max-Cut Above Tight Bounds

Years and Authors of Summarized Original Work

• 2012; Crowston, Jones, Mnich

Problem Definition

In the problem Max Cut, we are given a graph G with n vertices and m edges, and asked to find a bipartite subgraph of G with the maximum number of edges.

In 1973, Edwards [5] proved that if G is connected, then G contains a bipartite subgraph with at least $\frac{m} {2} + \frac{n-1} {4}$ edges, proving a conjecture of Erdős. This lower bound on the size of a bipartite subgraph is known as the Edwards-Erdős bound. The bound is tight – for example, it is an upper bound when G is a clique with odd number of vertices. Thus, it is natural to consider parameterized Max Cut above this bound, as follows (AEE stands for Above Edwards-Erdős).

Max Cut AEE

Instance: :

A connected graphG with n vertices andm edges, and a nonnegative integer k.

Parameter:

Parameter: k.

Question: :

DoesG have a bipartite subgraph with at least $\frac{m} {2} + \frac{n-1} {4} + k$ edges?

Mahajan and Raman [6], in their...