A derandomization using min-wise independent permutations

Min-wise independence is a recently introduced notion of limited independence, similar in spirit to pairwise independence. The latter has proven essential for the derandomization of many algorithms. Here we show that approximate min-wise independence allows similar uses, by presenting a derandomization of the RNC algorithm for approximate set cover due to S. Rajagopalan and V. Vazirani. We also discuss how to derandomize their set multi-cover and multi-set multi-cover algorithms in restricted cases. The multi-cover case leads us to discuss the concept of k-minima-wise independence, a natural counterpart to k-wise independence.

[1]  Andrei Z. Broder,et al.  On the resemblance and containment of documents , 1997, Proceedings. Compression and Complexity of SEQUENCES 1997 (Cat. No.97TB100171).

[2]  Uriel Feige.  A threshold of ln n for approximating set cover (preliminary version) , 1996, STOC '96.

[3]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[4]  Alan M. Frieze,et al.  Min-Wise Independent Permutations , 2000, J. Comput. Syst. Sci..

[5]  Noga Alon,et al.  Simple Construction of Almost k-wise Independent Random Variables , 1992, Random Struct. Algorithms.

[6]  Raymond E. Miller,et al.  Complexity of Computer Computations , 1972 .

[7]  Ketan Mulmuley.  Randomized geometric algorithms and pseudorandom generators , 2005, Algorithmica.

[8]  Larry Carter,et al.  Universal Classes of Hash Functions , 1979, J. Comput. Syst. Sci..

[9]  Vijay V. Vazirani,et al.  Primal-Dual RNC Approximation Algorithms for Set Cover and Covering Integer Programs , 1999, SIAM J. Comput..

[10]  Vijay V. Vazirani,et al.  Primal-dual RNC approximation algorithms for (multi)-set (multi)-cover and covering integer programs , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[11]  Andrei Z. Broder,et al.  Identifying and Filtering Near-Duplicate Documents , 2000, CPM.

[12]  Piotr Indyk,et al.  A small approximately min-wise independent family of hash functions , 1999, SODA '99.

[13]  Geoffrey Zweig,et al.  Syntactic Clustering of the Web , 1997, Comput. Networks.

[14]  Alan M. Frieze,et al.  Min-wise independent permutations (extended abstract) , 1998, STOC '98.

[15]  Avi Wigderson,et al.  Pairwise Independence and Derandomization , 2006, Found. Trends Theor. Comput. Sci..