Maximum k-coverings of weighted transitive graphs with applications

A weighted transitive graph is considered where each vertex is assigned a positive weight. Given a positive integer k, the maximum k-covering problem is to find k disjoint cliques covering a set of vertices with maximum total weight. An O (kn²) time algorithm to solve the problem in a transitive graph is proposed, where n is the number of vertices. Based on this algorithm the weighted version of a number of VLSI problems can be solved-for example, k-layer topological via minimization, assignment of intervals into k tracks (with applications to channel routing), k-layer subset of multiterminal nets, and k-layer subset of nets in a leveled routing region (e.g. standard cells). The proposed technique solves several fundamental geometric and graph-theoretic problems