In very deep-submicron VLSI, manufacturing steps involving chemical-mechanical polishing CMP have v arying e ects on device and interconnect features, depending on local characteristics of the layout. To reduce manufacturing variation due to CMP and to improve performance predictability and yield, layout must be made uniform with respect to certain density criteria, by inserting ll" geometries into the layout. To date, only foundries and special mask data processing tools perform layout post-processing for density control. In the future, better convergence of performance veri cation ows will depend on such l a yout manipulations being embedded within the layout synthesis place-and-route ow. In this paper, we give the rst realistic formulation of the lling problem that arises in layout optimization for manufacturability. Our formulation seeks to add features to a given process layer, such that i feature area densities satisfy prescribed upper and lower bounds in all windows of given size, and ii the maximum variation of such densities over all possible window positions in the layout is minimized. We present e cient algorithms for density analysis, notably a multilevel approach that a ords user-tunable accuracy. W e also develop exact solutions to the problem of ll synthesis, based on a linear programming approach. These include an LP formulation for the xed-dissection regime where density bounds are imposed on a predetermined set of in the layout and an LP formulation that is automatically generated by our multilevel density analysis. We brie y review criteria for ll pattern synthesis, and the paper then concludes with computational results and directions for future research.