Average Case NP-Complete Graph Coloring Problem

Title: Average Case NP-Complete Graph Coloring Problem

Research Question: Can we find an average case NP-complete problem for graph coloring that is not just hard in the worst case, but also on typical instances?

Methodology: The researchers used a random graph model to generate instances of the graph coloring problem. They then designed a reduction that takes an instance of the graph coloring problem and transforms it into another graph coloring problem with the same number of colors. This reduction must preserve the hardness of the original problem, meaning that if the original problem is hard on average, then the transformed problem will also be hard on average.

Results: The researchers were able to find an average case NP-complete graph coloring problem. They showed that this problem is hard on average under a certain distribution of instances, meaning that there is no efficient algorithm that can solve this problem for all instances.

Implications: This result has implications for the field of computer science and complexity theory. It shows that there are problems that are hard on average, not just in the worst case. This could have implications for the design of algorithms and the understanding of computational complexity. It also has implications for cryptography, as it provides evidence that there are problems that are hard to solve on average, which could be useful for creating secure cryptographic systems.

In conclusion, the researchers have shown that there is an average case NP-complete graph coloring problem, which is hard on average under a certain distribution of instances. This result has important implications for the field of computer science and complexity theory.

Link to Article: https://arxiv.org/abs/0112001v10 Authors: arXiv ID: 0112001v10