Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study Fisher markets and the problem of maximizing the Nash social lfare (NSW), and show several closely related new results. In particular,  obtain: #R##N#-- A new integer program for the NSW maximization problem whose fractional relaxation has a bounded integrality gap. In contrast, the natural integer program has an unbounded integrality gap. #R##N#-- An improved, and tight, factor 2 analysis of the algorithm of [7]; in turn showing that the integrality gap of the above relaxation is at most 2. The approximation factor shown by [7] was $2e^{1/e} \approx 2.89$. #R##N#-- A lor bound of $e^{1/e}\approx 1.44$ on the integrality gap of this relaxation. #R##N#-- New convex programs for natural generalizations of linear Fisher markets and proofs that these markets admit rational equilibria. #R##N#These results re obtained by establishing connections beten previously known disparate results, and they help uncover their mathematical underpinnings.  show a formal connection beten the convex programs of Eisenberg and Gale and that of Shmyrev, namely that their duals are equivalent up to a change of variables. Both programs capture equilibria of linear Fisher markets. By adding suitable constraints to Shmyrev's program,  obtain a convex program that captures equilibria of the spending-restricted market model defined by [7] in the context of the NSW maximization problem. Further, adding certain integral constraints to this program  get the integer program for the NSW mentioned above. #R##N#The basic tool  use is convex programming duality. In the special case of convex programs with linear constraints (but convex objectives),  show a particularly simple way of obtaining dual programs, putting it almost at par with linear program duality. This simple way of finding duals has been used subsequently for many other applications.
['Richard Cole', 'Nikhil R. Devanur', 'Vasilis Gkatzelis', 'Kamal Jain', 'Tung Mai', 'Vijay V. Vazirani', 'Sadra Yazdanbod']
I am writing a research paper on a idea Convex Program Duality, Fisher Markets, and Nash Social Welfare
[{"from": "human", "value": "I am writing a research paper on a idea Convex Program Duality, Fisher Markets, and Nash Social Welfare"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study Fisher markets and the problem of maximizing the Nash social lfare (NSW), and show several closely related new results. In particular,  obtain: #R##N#-- A new integer program for the NSW maximization problem whose fractional relaxation has a bounded integrality gap. In contrast, the natural integer program has an unbounded integrality gap. #R##N#-- An improved, and tight, factor 2 analysis of the algorithm of [7]; in turn showing that the integrality gap of the above relaxation is at most 2. The approximation factor shown by [7] was $2e^{1/e} \\approx 2.89$. #R##N#-- A lor bound of $e^{1/e}\\approx 1.44$ on the integrality gap of this relaxation. #R##N#-- New convex programs for natural generalizations of linear Fisher markets and proofs that these markets admit rational equilibria. #R##N#These results re obtained by establishing connections beten previously known disparate results, and they help uncover their mathematical underpinnings.  show a formal connection beten the convex programs of Eisenberg and Gale and that of Shmyrev, namely that their duals are equivalent up to a change of variables. Both programs capture equilibria of linear Fisher markets. By adding suitable constraints to Shmyrev's program,  obtain a convex program that captures equilibria of the spending-restricted market model defined by [7] in the context of the NSW maximization problem. Further, adding certain integral constraints to this program  get the integer program for the NSW mentioned above. #R##N#The basic tool  use is convex programming duality. In the special case of convex programs with linear constraints (but convex objectives),  show a particularly simple way of obtaining dual programs, putting it almost at par with linear program duality. This simple way of finding duals has been used subsequently for many other applications."}]