On VLSI interconnect optimization and linear ordering problem

Some well-known VLSI interconnect optimizations problems for timing, power and cross-coupling noise immunity share a property that enables mapping them into a specialized Linear Ordering Problem (LOP). Unlike the general LOP problem which is NP-complete, this paper proves that the specialized one has a closed-form solution. Let f(x,y):ℝ²→ℝ be symmetric, non-negative, defined for x≥0 and y≥0, and let f(x,y) be twice differentiable, satisfying ∂²f(x,y)/∂x∂y<0. Let π be a permutation of {1,...,n}. The specialized LOP comprises n objects, each associated with a real value parameter r_i, 1≤i≤n, and a cost f(r_i,r_j) associated to any two objects if {pipe}π(i)-π(j){pipe}=1,1≤i,j≤n, and f(r_i,r_j)=0 otherwise. We show that the permutation π which minimizes, called "symmetric hill", is determined upfront by the relations between the parameter values r_i.