Abstract
We study algorithms based on local improvements for the \(k\)-Set Packing problem. The well-known local improvement algorithm by Hurkens and Schrijver [14] has been improved by Sviridenko and Ward [15] from \(\frac{k}{2}+\epsilon \) to \(\frac{k+2}{3}\), and by Cygan [7] to \(\frac{k+1}{3}+\epsilon \) for any \(\epsilon >0\). In this paper, we achieve the approximation ratio \(\frac{k+1}{3}+\epsilon \) for the \(k\)-Set Packing problem using a simple polynomial-time algorithm based on the method by Sviridenko and Ward [15]. With the same approximation guarantee, our algorithm runs in time singly exponential in \(\frac{1}{\epsilon ^2}\), while the running time of Cygan’s algorithm [7] is doubly exponential in \(\frac{1}{\epsilon }\). On the other hand, we construct an instance with locality gap \(\frac{k+1}{3}\) for any algorithm using local improvements of size \(O(n^{1/5})\), where \(n\) is the total number of sets. Thus, our approximation guarantee is optimal with respect to results achievable by algorithms based on local improvements.
Research supported in part by NSF Grant CCF-0964655 and CCF-1320814.
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Fürer, M., Yu, H. (2014). Approximating the \(k\)-Set Packing Problem by Local Improvements. In: Fouilhoux, P., Gouveia, L., Mahjoub, A., Paschos, V. (eds) Combinatorial Optimization. ISCO 2014. Lecture Notes in Computer Science(), vol 8596. Springer, Cham. https://doi.org/10.1007/978-3-319-09174-7_35
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