Cluster-Cover: A Theoretical Framework for a Class of VLSI-CAD Optimization Problems

This article introduces a mathematical framework called cluster-cover. We show that this framework captures the combinatorial structure of a class of VLSI design optimization problems, including two-level logic minimization, constrained encoding, multilayer topological planar routing, application timing assignment for delay-fault testing, and minimization of monitoring logic for BIST enhancement. These apparently unrelated problems can all be cast into two metaproblems in our framework: finding a maximum cluster and finding a minimum cover. We describe paradigms for developing algorithms for these problems. First, a simple heuristic called greedy peeling is presented and characterized. We derive sufficient conditions that guarantee optimum solutions by greedy peeling. We generalize the performance analysis of a multilayer topological planar routing heuristic to greedy peeling for the general cluster-cover problems. We propose a performance bound of greedy set covering that can be computed efficiently for a given problem instance; this bound is much tighter than the previously known bounds. Second, prime covering—originally developed for logic minimization—is generalized to finding exact solutions for cluster-cover problems. Previously, only the connection betwee