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We discuss the compaction of independent test sequences for sequential circuits. Our first contribution is the formulation of this problem as an integer program, which we then solve through a well-known method employing linear programming relaxation and randomized rounding. The key contribution of this approach is that it yields the first polynomial time approximation algorithm for this problem. More specifically, it provides a provably good approximation guarantee while running in time polynomial with respect to the number of vectors in the original test sequences and the number of faults. Another virtue of our approach is that it provides a lower bound for the compacted set of test sequences and, therefore, a quality measure for the test compaction algorithm. Experimental results on benchmark circuits demonstrate that the proposed solution efficiently identifies nearly optimal sets of compacted test sequences.