Why keep all the fun to myself. I asked Copilot for a harder max-cut problem
to work on, just to make sure I am getting the notion right. Below:
mathematics; could I see a difficult max-cut problem?
Sure! Here's an example of a moathemare challenging Max-Cut problem:
Problem Description
Consider a graph with 8 vertices and 14 edges. The vertices are labeled from 0 to 7, and the edges are as follows:
(0, 1), (0, 2), (0, 3), (1, 2), (1, 4), (2, 5), (3, 6), (4, 5), (4, 7), (5, 6), (5, 7), (6, 7), (1, 6), (2, 7)
Objective
Partition the vertices into two sets, S and T, such that the number of edges between the sets is maximized.
Solution Approach
Graph Representation: Visualize the graph with the given vertices and edges.
Initial Partition: Start with an initial random partition of the vertices into two sets.
terative Improvement: Use an algorithm like the Kernighan-Lin algorithm to iteratively improve the partition by swapping vertices between the sets to increase the number of edges crossing the partition.
Maximize the Cut: Continue the process until no further improvement can be made.
Example Solution
After applying the algorithm, one possible partition could be:
Set S: {0, 1, 4, 5}
Set T: {2, 3, 6, 7}
In this partition, the edges crossing the partition are:
(0, 2), (0, 3), (1, 2), (1, 6), (2, 5), (3, 6), (4, 7), (5, 6), (5, 7), (2, 7)
This results in a maximum cut with 10 edges crossing the partition.
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