On approximation algorithms for microcode bit minimization

The bit (or width) minimization problem for microprograms is known to be NP-complete. Motivated by its practical importance, we address the question of obtaining near-optimal solutions. Two main results are presented. First, we establish a tight bound on the quality of solutions produced by algorithms which minimize the number of compatibility classes. Second, we show that the bit minimization problem has a polynomial time relative approximation algorithm only if the vertex coloring problem for graphs with n nodes can be approximated to within a factor of &Ogr;(logn) in polynomial time.