We present a packing-based approximation algorithm for the $k$-Set Cover problem. We introduce a new local search-based $k$-set packing heuristic, and call it Restricted $k$-Set Packing. We analyze its tight approximation ratio via a complicated combinatorial argument. Equipped with the Restricted $k$-Set Packing algorithm, our $k$-Set Cover algorithm is composed of the $k$-Set Packing heuristic cite{schrijver} for $kgeq 7$, Restricted $k$-Set Packing for $k=6,5,4$ and the semi-local $(2,1)$-improvement cite{furer} for 3-Set Cover. We show that our algorithm obtains a tight approximation ratio of $H_k-0.6402+Theta(frac{1}{k})$, where $H_k$ is the $k$-th harmonic number. For small $k$, our results are 1.8667 for $k=6$, 1.7333 for $k=5$ and 1.5208 for $k=4$. Our algorithm improves the currently best approximation ratio for the $k$-Set Cover problem of any $kgeq 4$.
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