Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  prove the first non-trivial performance ratios strictly above 0.5 for ighted versions of the oblivious matching problem.#R##N#Even for the unighted version, since Aronson, Dyer, Frieze, and Suen first proved a non-trivial ratio above 0.5 in the mid-1990s, during the next tnty years several attempts have been made to improve this ratio, until Chan, Chen, Wu and Zhao successfully achieved a significant ratio of 0.523 very recently (SODA 2014). To the best of our knowledge, our work is the first in the literature that considers the node-ighted and edge-ighted versions of the problem in arbitrary graphs (as opposed to bipartite graphs).#R##N##R##N#(1) For arbitrary node ights,  prove that a ighted version of the Ranking algorithm has ratio strictly above 0.5.  have discovered a new structural property of the ranking algorithm: if a node has two unmatched neighbors at the end of algorithm, then it will still be matched even when its rank is demoted to the bottom.  This property allows us to form LP constraints for both the node-ighted and the unighted oblivious matching problems. As a result,  prove that the ratio for the node-ighted case is at least 0.501512. Interestingly via the structural property,  can also improve slightly the ratio for the unighted case to 0.526823 (from the previous best 0.523166 in SODA 2014).#R##N##R##N#(2) For a bounded number of distinct edge ights,  show that ratio strictly above 0.5 can be achieved by partitioning edges carefully according to the ights, and running the (unighted) Ranking algorithm on each part.  Our analysis is based on a new primal-dual framework known as \emph{matching coverage}, in which dual feasibility is bypassed.  Instead, only dual constraints corresponding to edges in an optimal matching are satisfied.#R##N#Using this framework  also design and analyze an algorithm for the edge-ighted online bipartite matching problem with free disposal.  prove that for the case of bounded online degrees, the ratio is strictly above 0.5.
['Melika Abolhassani', 'T.-H. Hubert Chan', 'Fei Chen', 'Hossein Esfandiari', 'MohammadTaghi Hajiaghayi', 'Hamid Mahini', 'Xiaowei Wu']
I am writing a research paper on a idea Beating Ratio 0.5 for Weighted Oblivious Matching Problems
[{"from": "human", "value": "I am writing a research paper on a idea Beating Ratio 0.5 for Weighted Oblivious Matching Problems"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  prove the first non-trivial performance ratios strictly above 0.5 for ighted versions of the oblivious matching problem.#R##N#Even for the unighted version, since Aronson, Dyer, Frieze, and Suen first proved a non-trivial ratio above 0.5 in the mid-1990s, during the next tnty years several attempts have been made to improve this ratio, until Chan, Chen, Wu and Zhao successfully achieved a significant ratio of 0.523 very recently (SODA 2014). To the best of our knowledge, our work is the first in the literature that considers the node-ighted and edge-ighted versions of the problem in arbitrary graphs (as opposed to bipartite graphs).#R##N##R##N#(1) For arbitrary node ights,  prove that a ighted version of the Ranking algorithm has ratio strictly above 0.5.  have discovered a new structural property of the ranking algorithm: if a node has two unmatched neighbors at the end of algorithm, then it will still be matched even when its rank is demoted to the bottom.  This property allows us to form LP constraints for both the node-ighted and the unighted oblivious matching problems. As a result,  prove that the ratio for the node-ighted case is at least 0.501512. Interestingly via the structural property,  can also improve slightly the ratio for the unighted case to 0.526823 (from the previous best 0.523166 in SODA 2014).#R##N##R##N#(2) For a bounded number of distinct edge ights,  show that ratio strictly above 0.5 can be achieved by partitioning edges carefully according to the ights, and running the (unighted) Ranking algorithm on each part.  Our analysis is based on a new primal-dual framework known as \\emph{matching coverage}, in which dual feasibility is bypassed.  Instead, only dual constraints corresponding to edges in an optimal matching are satisfied.#R##N#Using this framework  also design and analyze an algorithm for the edge-ighted online bipartite matching problem with free disposal.  prove that for the case of bounded online degrees, the ratio is strictly above 0.5."}]