Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The Adwords and Online Bipartite Matching problems have enjoyed a rened attention over the past decade due to their connection to Internet advertising. Our community has contributed, among other things, new models (notably stochastic) and extensions to the classical formulations to address the issues that arise from practical needs. ,  propose a new generalization based on matroids and show that many of the previous results extend to this more general setting. Because of the rich structures and expressive por of matroids, our new setting is potentially of interest both in theory and in practice.   In the classical version of the problem, the offline side of a bipartite graph is known initially while vertices from the online side arrive one at a time along with their incident edges. The objective is to maintain a decent approximate matching from which no edge can be removed. Our generalization, called Matroid Online Bipartite Matching, additionally requires that the set of matched offline vertices be independent in a given matroid. In particular, the case of partition matroids corresponds to the natural scenario where each advertiser manages multiple ads with a fixed total budget.    Our algorithms attain the same performance as the classical version of the problems considered, which are often provably the best possible.  present $1-1/e$-competitive algorithms for Matroid Online Bipartite Matching under the small bid assumption, as ll as a $1-1/e$-competitive algorithm for Matroid Online Bipartite Matching in the random arrival model. A key technical ingredient of our results is a carefully designed primal-dual waterfilling procedure that accommodates for matroid constraints. This is inspired by the extension of our recent charging scheme for Online Bipartite Vertex Cover. Finally, given that only few online problems re studied in the submodular fashion, the techniques introduced in this paper for tackling submodularity in the online setting may be of independent interest.
['Yajun Wang', 'Sam Chiu-wai Wong']
I am writing a research paper on a idea Matroid Online Bipartite Matching and Vertex Cover
[{"from": "human", "value": "I am writing a research paper on a idea Matroid Online Bipartite Matching and Vertex Cover"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The Adwords and Online Bipartite Matching problems have enjoyed a rened attention over the past decade due to their connection to Internet advertising. Our community has contributed, among other things, new models (notably stochastic) and extensions to the classical formulations to address the issues that arise from practical needs. ,  propose a new generalization based on matroids and show that many of the previous results extend to this more general setting. Because of the rich structures and expressive por of matroids, our new setting is potentially of interest both in theory and in practice.   In the classical version of the problem, the offline side of a bipartite graph is known initially while vertices from the online side arrive one at a time along with their incident edges. The objective is to maintain a decent approximate matching from which no edge can be removed. Our generalization, called Matroid Online Bipartite Matching, additionally requires that the set of matched offline vertices be independent in a given matroid. In particular, the case of partition matroids corresponds to the natural scenario where each advertiser manages multiple ads with a fixed total budget.    Our algorithms attain the same performance as the classical version of the problems considered, which are often provably the best possible.  present $1-1/e$-competitive algorithms for Matroid Online Bipartite Matching under the small bid assumption, as ll as a $1-1/e$-competitive algorithm for Matroid Online Bipartite Matching in the random arrival model. A key technical ingredient of our results is a carefully designed primal-dual waterfilling procedure that accommodates for matroid constraints. This is inspired by the extension of our recent charging scheme for Online Bipartite Vertex Cover. Finally, given that only few online problems re studied in the submodular fashion, the techniques introduced in this paper for tackling submodularity in the online setting may be of independent interest."}]