Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The vertex ranking problem is closely related to the problem of finding the elimination tree of minimum height for a given graph. This implies that the problem has applications in the parallel Cholesky factorization of matrices.  describe the connection beten this model of graph coloring and the matrix factorization.  also present a polynomial time algorithm for finding edge ranking of complete bipartite graphs.  use it to design an O(m 2+d ) algorithm for edge ranking of graphs obtained by removing O(log m) edges from a complete bipartite graph, where d is a fixed number. Then  extend our results to complete k-partite graphs for any fixed k > 2. In this way  give a new class of matrix factorization instances that can be optimally solved in polynomial time.
['Dariusz Dereniowski', 'Marek Kubale']
I am writing a research paper on a idea Cholesky Factorization of Matrices in Parallel and Ranking of Graphs
[{"from": "human", "value": "I am writing a research paper on a idea Cholesky Factorization of Matrices in Parallel and Ranking of Graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The vertex ranking problem is closely related to the problem of finding the elimination tree of minimum height for a given graph. This implies that the problem has applications in the parallel Cholesky factorization of matrices.  describe the connection beten this model of graph coloring and the matrix factorization.  also present a polynomial time algorithm for finding edge ranking of complete bipartite graphs.  use it to design an O(m 2+d ) algorithm for edge ranking of graphs obtained by removing O(log m) edges from a complete bipartite graph, where d is a fixed number. Then  extend our results to complete k-partite graphs for any fixed k > 2. In this way  give a new class of matrix factorization instances that can be optimally solved in polynomial time."}]