On computing the minimum feedback vertex set of a directed graph by contraction operations

Finding the minimum feedback vertex set (MFVS) in a graph is an important problem for a variety of computer-aided design (CAD) applications and graph reduction plays an important role in solving this intractable problem. This paper is largely concerned with three new and powerful reduction operations. Each of these operations defines a new class of graphs, strictly larger than the class of contractible graphs [Levy and Low (1988)] in which the MFVS can be found in polynomial-time complexity. Based on these operations, an exact algorithm run on branch and bound manner is developed. This exact algorithm uses a good heuristic to find out an initial solution and a good bounding strategy to prune the solution space. To demonstrate the efficiency of our algorithms, we have implemented our algorithms and applied them to solving the partial scan problem in ISCASʹ89 benchmarks. The experimental results show that if our three new contraction operations are applied, 27 out of 31 circuits in ISCASʹ89 benchmarks can be fully reduced. Otherwise, only 12 out of 31 can be fully reduced. Furthermore, for all ISCASʹ89 benchmarks our exact algorithm can find the exact cutsets in less than 3 s (CPU time) on a SUN-UltraII workstation. Therefore, the new contraction operations and our algorithms are demonstrated to be very effective in the partial scan application