Before revision:  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.  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.
After revision:  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.  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.