An exact algorithm for placement optimization in circuit design

Placement optimization is a crucial phase in chip design, involving the strategic arrangement of cells within a limited region to enhance space utilization and reduce wirelength. Chip design enterprises need to optimize the placement according to design rules to meet customer demands. While mixed-cell-height circuits are widely used in modern chip design, few studies have simultaneously considered the non-overlapping cells, rails alignment, and minimum implantation area constraints in the placement optimization problems. Hence, this study involves preprocessing the non-linear parts and developing a mixed-integer linear programming model to reduce the cost of legalizing chip placements for businesses. Furthermore, this study designs and implements an exact algorithm based on Benders decomposition, utilizing dual theory to obtain an optimal cut and iteratively solve for the coordinates of cells. Numerical experiments across various scales validate the performance of the algorithm. Through a detailed analysis of the shape of the chip region division, the proportion of different types of cells, the total number of cells and bins, and their impact on the placement, we derive some potentially useful design insights that can benefit chip design enterprises.