Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In the Edge Clique Cover (ECC) problem, given an undirected graph $G$ and an integer $k$,  ask whether one can choose $k$ cliques in $G$ such that each edge of $G$ is contained in at least one of the chosen cliques. Gramm et al. [ACM J. Exp. Algorithmics, 13 (2008)] have shown a set of simple rules that reduce the number of vertices of $G$ to $2^k$ while preserving the ansr to the instance at hand, that is, they have shown a kernel for the problem with at most $2^k$ vertices. No algorithm is known with significantly better running time bound than a brute-force search on this kernel. ,  show that the approach of Gramm et al. is essentially optimal:  present a polynomial-time algorithm that reduces an arbitrary instance of $3$-CNF-SAT with $n$ variables and $m$ clauses to an equivalent ECC instance $(G,k)$ with $k = \mathcal{O}(\log n)$ and $|V(G)| = \mathcal{O}(n + m)$. Consequently, there is no $2^{2^{o(k)}}{\rm poly}(n)$ time algorithm for the ECC problem, unless the Exponential Time...
['Marek Cygan', 'Marcin Pilipczuk', 'Michał Pilipczuk']
I am writing a research paper on a idea Known Algorithms for Edge Clique Cover are Probably Optimal
[{"from": "human", "value": "I am writing a research paper on a idea Known Algorithms for Edge Clique Cover are Probably Optimal"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In the Edge Clique Cover (ECC) problem, given an undirected graph $G$ and an integer $k$,  ask whether one can choose $k$ cliques in $G$ such that each edge of $G$ is contained in at least one of the chosen cliques. Gramm et al. [ACM J. Exp. Algorithmics, 13 (2008)] have shown a set of simple rules that reduce the number of vertices of $G$ to $2^k$ while preserving the ansr to the instance at hand, that is, they have shown a kernel for the problem with at most $2^k$ vertices. No algorithm is known with significantly better running time bound than a brute-force search on this kernel. ,  show that the approach of Gramm et al. is essentially optimal:  present a polynomial-time algorithm that reduces an arbitrary instance of $3$-CNF-SAT with $n$ variables and $m$ clauses to an equivalent ECC instance $(G,k)$ with $k = \\mathcal{O}(\\log n)$ and $|V(G)| = \\mathcal{O}(n + m)$. Consequently, there is no $2^{2^{o(k)}}{\\rm poly}(n)$ time algorithm for the ECC problem, unless the Exponential Time..."}]