Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The tree cover (TC) problem is to compute a minimum ight connected edge set, given a connected and edge-ighted graph G, such that its vertex set forms a vertex cover for G. Unlike related problems of vertex cover or edge dominating set, ighted TC is not yet known to be approximable in polynomial time as ll as the unighted version is. Moreover, the best approximation algorithm known so far for ighted TC is far from practical in its efficiency.   consider a restricted version of ighted TC, as a first step towards better approximation of general TC, where only two edge ights differing by at least a factor of 2 are available. It will be shown that a factor 2 approximation can be attained efficiently (in the complexity of max flow) in this case by a primal-dual method. Even under the limited ights as such, the primal-dual arguments used will be seen to be quite involved, having a nontrivial style of dual assignments as an essential part, unlike the case of uniform ights.
['Takashi Doi', 'Toshihiro Fujito']
I am writing a research paper on a idea A primal-dual method for approximating tree cover with two weights
[{"from": "human", "value": "I am writing a research paper on a idea A primal-dual method for approximating tree cover with two weights"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The tree cover (TC) problem is to compute a minimum ight connected edge set, given a connected and edge-ighted graph G, such that its vertex set forms a vertex cover for G. Unlike related problems of vertex cover or edge dominating set, ighted TC is not yet known to be approximable in polynomial time as ll as the unighted version is. Moreover, the best approximation algorithm known so far for ighted TC is far from practical in its efficiency.   consider a restricted version of ighted TC, as a first step towards better approximation of general TC, where only two edge ights differing by at least a factor of 2 are available. It will be shown that a factor 2 approximation can be attained efficiently (in the complexity of max flow) in this case by a primal-dual method. Even under the limited ights as such, the primal-dual arguments used will be seen to be quite involved, having a nontrivial style of dual assignments as an essential part, unlike the case of uniform ights."}]