Abstract
We study the relationship between the approximation factor for the Set-Cover problem and the parameters Δ : the maximum cardinality of any subset, and k : the maximum number of subsets containing any element of the ground set. We show an LP rounding based approximation of \((k-1)(1-e^{-\frac{\ln \Delta}{k-1}}) +1\), which is substantially better than the classical algorithms in the range k ≈ ln Δ, and also improves on related previous works [19,22]. For the interesting case when k = θ(logΔ) we also exhibit an integrality gap which essentially matches our approximation algorithm. We also prove a hardness of approximation factor of \(\Omega\left(\frac{\log \Delta}{(\log\log \Delta)^2}\right)\) when k = θ(logΔ). This is the first study of the hardness factor specifically for this range of k and Δ, and improves on the only other such result implicitly proved in [18].
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References
Alon, N., Moshkovitz, D., Safra, M.: Algorithmic construction of sets for k-restrictions. The ACM Transactions on Algorithms 2(2), 153–177 (2006)
Austrin, P., Mossel, E.: Approximation Resistant Predicates from Pairwise Independence. Computational Complexity 18(2), 249–271 (2009)
Bansal, N., Khot, S.: Inapproximability of Hypergraph Vertex Cover and Applications to Scheduling Problems. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 250–261. Springer, Heidelberg (2010)
Bar-Yehuda, R., Even, S.: A linear-time approximation algorithm for the weighted vertex cover problem. J. Algorithms 2(2), 198–203 (1981)
Chvatal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4(3), 233–235 (1979)
Dinur, I., Guruswami, V., Khot, S., Regev, O.: A new multilayered PCP and the hardness of hypergraph vertex cover. SIAM J. Comput. 34(5), 1129–1146 (2005)
Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)
Guruswami, V., Raghavendra, P.: Constraint Satisfaction over a Non-Boolean Domain: Approximation Algorithms and Unique-Games Hardness. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds.) APPROX and RANDOM 2008. LNCS, vol. 5171, pp. 77–90. Springer, Heidelberg (2008)
Halperin, E.: Improved Approximation Algorithms for the Vertex Cover Problem in Graphs and Hypergraphs. SIAM J. Comput. 31(5), 1608–1623 (2002)
Hochbaum, D.: Approximation algorithms for the set covering and vertex cover problems. SIAM J. Comput. 11(3), 555–556 (1982)
Holmerin, J.: Improved Inapproximability Results for Vertex Cover on k-Uniform Hypergraphs. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 1005–1016. Springer, Heidelberg (2002)
Johnson, D.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)
Khot, S.: Improved inaproximability results for maxclique, chromatic number and approximate graph coloring. In: Proc. FOCS, pp. 600–609 (2001)
Khot, S.: On the power of unique 2-prover 1-round games. In: Proc. STOC, pp. 767–775 (2002)
Khot, S.: Online lecture notes for Probabilistically Checkable Proofs and Hardness of Approximation, Lecture 3 (scribed by Deeparnab Chakrabarty), www.cs.nyu.edu/~khot/pcp-lecnotes/lec3.ps
Khot, S., Ponnuswami, A.K.: Better Inapproximability Results for MaxClique, Chromatic Number and Min-3Lin-Deletion. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 226–237. Springer, Heidelberg (2006)
Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2 − ε. J. Comput. System Sci. 74(3), 335–349 (2008)
Khot, S., Saket, R.: Hardness of Minimizing and Learning DNF Expressions. In: Proc. FOCS, pp. 231–240 (2008)
Krivelevich, M.: Approximate set covering in uniform hypergraphs. J. Algorithms 25(1), 118–143 (1997)
Lovasz, L.: On the ratio of the optimal integral and fractional covers. Disc. Math. 13, 383–390 (1975)
Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. J. ACM 31(5), 960–981 (1994)
Okun, M.: On the approximation of the vertex cover problem in hypergraphs. Discrete Optimization 2(1), 101–111 (2005)
Raz, R., Safra, M.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proc. STOC, pp. 475–484 (2007)
Samorodnitsky, A., Trevisan, L.: A PCP characterization of NP with optimal amortized query complexity. In: Proc. STOC, pp. 191–199 (2000)
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Saket, R., Sviridenko, M. (2012). New and Improved Bounds for the Minimum Set Cover Problem. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2012 2012. Lecture Notes in Computer Science, vol 7408. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32512-0_25
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