The main result of this paper, proved as a theorem, is that a lower bound on the number of test vectors that detect each fault at least N times is N times the minimal test set size for N = 1. Tests with N > 1 have been reported to have a higher defect coverage and hence are of practical interest. We give an integer linear programming (ILP) algorithm for optimally minimizing a given test set for any given N ; in general, the value of N can be separately specified for each fault. Results on benchmark circuits show that optimal N -detection tests are easier to find for circuits that are deep and the input cones of primary outputs have large overlap. However, for small depth circuits, where the primary input overlap between output cones is small or nonexistent, the minimization of the N -detection tests requires further investigation.

-We give a new recursive rounding linear programming (LP) solution to the problem of N -detect test minimzation. This is a polynomialtime solution that closely approximates the exact but NP-hard integer linear programming (ILP) solution. In ILP, a test is represented by a [0, 1] integer variable and the sum of those variables is minimized. Constraints ensure that each fault has at least N tests with non-zero variables. Traditionally, the problem has been transformed to less complex LP by treating the variables as real numbers, regarded as probabilities with which they can be rounded off to 0 or 1. This is known as the randomized rounding method. In the new method, the LP is recursively used, each time rounding the largest variable to 1 and reducing the size of the LP. The method is found to converge to a solution in just a few LP runs and the result is usually better than that of randomized rounding. Experimental results include ISCAS85 benchmarks and a set of multiplier circuits. N -detect tests for N = 1, 5 and 15 are considered. Also, a 10-vector single-detect sequence for c6288 is given.