Solving constrained via minimization by compact linear programming

Via minimization is an important problem in integrated circuit layout and printed circuit board design. In this paper, a linear (non-integral) programming approach to twolayer constrained via minimization (CVM) is presented. The approach nds optimum solutions for routings containing no more than three way splits, and guarantees provably good results for the general case. Most importantly, the size of linear programming formulation is polynomial in terms of the size of the CVM problem. The signi cance of our work lies in three aspects. First, since linear programming can be solved in polynomial time, our work thus provides, for the rst time, a mathematical programming solution with computational e ciency comparable to known combinatorial CVM algorithms. Second, our compact linear programming approach is provably good and natural for general CVM, while previous restricted CVM algorithms are di cult to be extended to the general case. Third, our approach can handle additional constraints in a uni ed manner, and thus provides an e cient method for performance-driven layer assignment. Our approach is based on some new graphtheoretic and polyhedron-combinatorial results presented in this paper on the structure of the CVM problem.