On the Existence of Hard Sparse Sets under Weak Reductions

Recently a 1978 conjecture by Hartmanis was resolved by Cai and Sivakumar, following progress made by Ogihara. It was shown that there is no sparse set that is hard for P under logspace many-one reductions, unless P=LOGSPACE. We extend these results to the case of sparse sets that are hard under more general reducibilities. Furthermore, the proof technique can be applied to resolve open questions about hard sparse sets for NP as well. Using algebraic and probabilistic techniques, we show the following results.

[1]  Osamu Watanabe,et al.  On polynomial time bounded truth-table reducibility of NP sets to sparse sets , 1990, STOC '90.

[2]  Osamu Watanabe,et al.  On Polynomial-Time Bounded Truth-Table Reducibility of NP Sets to Sparse Sets , 1991, SIAM J. Comput..

[3]  Richard J. Lipton,et al.  Some connections between nonuniform and uniform complexity classes , 1980, STOC '80.

[4]  Moni Naor,et al.  Small-bias probability spaces: efficient constructions and applications , 1990, STOC '90.

[5]  Leonid A. Levin,et al.  A hard-core predicate for all one-way functions , 1989, STOC '89.

[6]  Mitsunori Ogihara.  Sparse Hard Sets for P Yield Space-Efficient Algorithms , 1996, Chic. J. Theor. Comput. Sci..

[7]  R. Ladner.  The circuit value problem is log space complete for P , 1975, SIGA.

[8]  Samuel R. Buss,et al.  An Optimal Parallel Algorithm for Formula Evaluation , 1992, SIAM J. Comput..

[9]  Noga Alon,et al.  Simple construction of almost k-wise independent random variables , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[10]  Juris Hartmanis,et al.  On Isomorphisms and Density of NP and Other Complete Sets , 1977, SIAM J. Comput..

[11]  Marek Karpinski,et al.  On the Power of Two-Way Random Generators and the Impossibility of Deterministic Poly-Space Simulation , 1986, Inf. Control..

[12]  Stephen R. Mahaney.  Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[13]  Stephen R. Mahaney.  Sparse Complete Sets of NP: Solution of a Conjecture of Berman and Hartmanis , 1982, J. Comput. Syst. Sci..

[14]  J. Hopcroft,et al.  Fast parallel matrix and GCD computations , 1982, FOCS 1982.

[15]  Mitsunori Ogihara,et al.  On one query, self-reducible sets , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[16]  Leslie G. Valiant,et al.  NP is as easy as detecting unique solutions , 1985, STOC '85.

[17]  Stephen A. Cook,et al.  A Taxonomy of Problems with Fast Parallel Algorithms , 1985, Inf. Control..

[18]  Juris Hartmanis,et al.  On isomorphisms and density of NP and other complete sets , 1976, STOC '76.

[19]  Desh Ranjan,et al.  On randomized reductions to sparse sets , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[20]  L. Berman.  Polynomial reducibilities and complete sets. , 1977 .

[21]  J. H. van Lint,et al.  Introduction to Coding Theory , 1982 .

[22]  S. Homer,et al.  On reductions of NP sets to sparse sets , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[23]  Juris Hartmanis,et al.  On Log-Tape Isomorphisms of Complete Sets , 1978, Theor. Comput. Sci..

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

[25]  Jin-Yi Cai,et al.  The resolution of a Hartmanis conjecture , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[26]  Paul Young.  How reductions to sparse sets collapse the polynomial-time hierarchy: a primer: Part II restricted polynomial-time reductions , 1992, SIGA.

[27]  Mitsunori Ogihara.  Sparse P-hard sets yield space-efficient algorithms , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[28]  Jin-Yi Cai,et al.  Resolution of Hartmanis' Conjecture for NL-Hard Sparse Sets , 1997, COCOON.

[29]  Vikraman Arvind,et al.  On Bounded Truth-Table, Conjunctive, and Randomized Reductions to Sparse Sets , 1992, FSTTCS.

[30]  Osamu Watanabe,et al.  How hard are sparse sets? , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[31]  Alexander L. Chistov,et al.  Fast parallel calculation of the rank of matrices over a field of arbitrary characteristic , 1985, FCT.

[32]  Moni Naor,et al.  Small-Bias Probability Spaces: Efficient Constructions and Applications , 1993, SIAM J. Comput..

[33]  Paul Young.  How reductions to sparse sets collapse the polynomial-time hierarchy: a primer; part I: polynomial-time Turing reductions , 1992, SIGA.

[34]  Samuel R. Buss,et al.  The Boolean formula value problem is in ALOGTIME , 1987, STOC.

[35]  Steven Fortune,et al.  A Note on Sparse Complete Sets , 1978, SIAM J. Comput..

[36]  K. Mulmuley.  A fast parallel algorithm to compute the rank of a matrix over an arbitrary field , 1987, Comb..

[37]  Nancy A. Lynch,et al.  Comparison of polynomial-time reducibilities , 1974, STOC '74.

[38]  Allan Borodin,et al.  Fast parallel matrix and GCD computations , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).