Approximating the k -Set Packing Problem by Local Improvements

We study algorithms based on local improvements for the k-Set Packing problem. The well-known local improvement algorithm by Hurkens and Schrijver [13] has been improved by Sviridenko and Ward [14] from k 2 + to k+2 3 , and by Cygan [7] to k+1 3 + for any > 0. In this paper, we achieve the approximation ratio k+1 3 + for the k-Set Packing problem using a simple polynomial-time algorithm based on the method by Sviridenko and Ward [14]. With the same approximation guarantee, our algorithm runs in time singly exponential in 1 2 , while the running time of Cygan’s algorithm [7] is doubly exponential in 1 . On the other hand, we construct an instance with locality gap k+1 3 for any algorithm using local improvements of size O(n), here n is the total number of sets. Thus, our approximation guarantee is optimal with respect to results achievable by algorithms based on local improvements.