Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Many known optimal NP-hardness of approximation results are reductions from a problem called L abel -C over . The input is a bipartite graph  G  = ( L, R, E ) and each edge  e  = ( x, y ) ∈  E  carries a projection  π e   that maps labels to  x  to labels to  y . The objective is to find a labeling of the vertices that satisfies as many of the projections as possible. It is believed that the best approximation ratio efficiently achievable for L abel -C over  is of the form  N −c   where  N  =  nk, n  is the number of vertices,  k  is the number of labels, and 0  c   some  constant.   Inspired by a framework originally developed for D ensest   k -S ubgraph ,  propose a "log density threshold" for the approximability of Label-Cover. Specifically,  suggest the possibility that the Label-Cover approximation problem undergoes a computational phase transition at the same threshold at which local algorithms for its random counterpart fail. This threshold is [EQUATION].  then design, for any  e  > 0, a polynomial-time approximation algorithm for  semi-random  L abel -C over  whose approximation ratio is [EQUATION]. In our semi-random model, the input graph is random (or even just expanding), and the projections on the edges are arbitrary.   For  worst-case  L abel -C over   show a polynomial-time algorithm whose approximation ratio is roughly  N  −0.233 . The previous best efficient approximation ratio was  N  −0.25 .  present some evidence towards an  N −c   threshold by constructing integrality gaps for  N  Ω(1)  rounds of the Sum-of-squares/Lasserre hierarchy of the natural relaxation of Label Cover. For general 2CSP the "log density threshold" is  N  −0.25 , and  give a polynomial-time algorithm in the semi-random model whose approximation ratio is  N  −0.25+ e   for any  e  > 0.
['Eden Chlamtac', 'Pasin Manurangsi', 'Dana Moshkovitz', 'Aravindan Vijayaraghavan']
I am writing a research paper on a idea Approximation algorithms for label cover and the log-density threshold
[{"from": "human", "value": "I am writing a research paper on a idea Approximation algorithms for label cover and the log-density threshold"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Many known optimal NP-hardness of approximation results are reductions from a problem called L abel -C over . The input is a bipartite graph  G  = ( L, R, E ) and each edge  e  = ( x, y ) \u2208  E  carries a projection  \u03c0 e   that maps labels to  x  to labels to  y . The objective is to find a labeling of the vertices that satisfies as many of the projections as possible. It is believed that the best approximation ratio efficiently achievable for L abel -C over  is of the form  N \u2212c   where  N  =  nk, n  is the number of vertices,  k  is the number of labels, and 0  c   some  constant.   Inspired by a framework originally developed for D ensest   k -S ubgraph ,  propose a \"log density threshold\" for the approximability of Label-Cover. Specifically,  suggest the possibility that the Label-Cover approximation problem undergoes a computational phase transition at the same threshold at which local algorithms for its random counterpart fail. This threshold is [EQUATION].  then design, for any  e  > 0, a polynomial-time approximation algorithm for  semi-random  L abel -C over  whose approximation ratio is [EQUATION]. In our semi-random model, the input graph is random (or even just expanding), and the projections on the edges are arbitrary.   For  worst-case  L abel -C over   show a polynomial-time algorithm whose approximation ratio is roughly  N  \u22120.233 . The previous best efficient approximation ratio was  N  \u22120.25 .  present some evidence towards an  N \u2212c   threshold by constructing integrality gaps for  N  \u03a9(1)  rounds of the Sum-of-squares/Lasserre hierarchy of the natural relaxation of Label Cover. For general 2CSP the \"log density threshold\" is  N  \u22120.25 , and  give a polynomial-time algorithm in the semi-random model whose approximation ratio is  N  \u22120.25+ e   for any  e  > 0."}]