We consider the restricted case of the Santa Claus problem and improve the current best approximation ratio by presenting a polynomial time 12-approximation algorithm using linear programming and novel quadratic program that we consider . Our algorithm starts by solving the configuration LP and uses the optimum value to get a 12-gap instance. This is then followed by the well-known clustering technique of Bansal and Sviridenkobansal. We then apply the analysis of Asadpour et al. AFS,AFS2 to show that the clustered instance has an integer solution which is at least 1{6} times the best possible value, which was computed by solving the configuration LP.  From this solution we find a solution to a new quadratic program that we introduce to select one machine from each cluster, and then we show that the resulting instance has an Assignment LP fractional solution of value at least 1{6} times the optimum. We then use the well known rounding technique due to Bezakova and Dani bezakova on the 12-gap instance to get our 12-approximate solution. %DIFDELCMD < {\em %%% More recently, we show that our technique gives a 4-approximation algorithm. We consider the 4-gap instance and the solution to the clustered instance obtained by Bansal and Sviridenko \mbox{%DIFAUXCMD bansal0pt%DIFAUXCMD . We show that the soltuion can be used to select a single machine from each cluster so that the resulting assignment LP instance has a fractional solution of value at least the 1{2} times the optimum. Using the rounding technique in Bezakova and Dani \mbox{%DIFAUXCMD bezakova }\hspace{0pt}%DIFAUXCMD we get our 4-approximate solution in polynomial time}%DIFDELCMD < }%%%
In this paper, we consider the restricted case of the problem and improve the current best approximation ratio by presenting a polynomial time 12-approximation algorithm using linear programming and semi-definite programming . Our algorithm starts by solving the configuration LP and uses the optimum value to get a 12-gap instance. This is then followed by the well-known clustering technique of Bansal and Sviridenkobansal. We then apply the analysis of Asadpour et al. AFS,AFS2 to show that the clustered instance has an integer solution which is at least 1{6} times the best possible value, which was computed by solving the configuration LP. To find this solution, we formulate a problem called the Extended Assignment Problem, and formulate it as an LP. We then, show that the associated polytope is integral and gives us an fractional solution of value at least \frac{1 From this solution we find a solution to a new quadratic program that we introduce to select one machine from each cluster, and then we show that the resulting instance has an Assignment LP fractional solution of value at least 1{6} times the optimum. We then use the well known rounding technique due to Bezakova and Dani bezakova on the 12-gap instance to get our 12-approximate solution. %DIFDELCMD < {\em %%% 0pt%DIFAUXCMD . We show that the soltuion can be used to select a single machine from each cluster so that the resulting assignment LP instance has a fractional solution of value at least the 1{2} times the optimum. Using the rounding technique in Bezakova and Dani \mbox{%DIFAUXCMD bezakova }\hspace{0pt}%DIFAUXCMD we get our 4-approximate solution in polynomial time}%DIFDELCMD < }%%%
R
We
In this paper, we
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Santa Claus
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novel quadratic program that we consider
semi-definite programming
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To find this solution, we formulate a problem called the Extended Assignment Problem, and formulate it as an LP. We then, show that the associated polytope is integral and gives us an fractional solution of value at least \frac{1
D
More recently, we show that our technique gives a 4-approximation algorithm. We consider the 4-gap instance and the solution to the clustered instance obtained by Bansal and Sviridenko \mbox{%DIFAUXCMD bansal