How hard are sparse sets?

The frontier of knowledge about the structural properties of sparse sets is explored. A collection of topics that are related to the issue of how hard or easy sparse sets is surveyed. The strongest currently known results, together with the open problems that the results leave, are presented.<<ETX>>

[1]  Joel I. Seiferas,et al.  Limitations on Separating Nondeterministic Complexity Classes , 1981, SIAM J. Comput..

[2]  Juris Hartmanis,et al.  Generalized Kolmogorov complexity and the structure of feasible computations , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[3]  Alan L. Selman,et al.  Qualitative Relativizations of Complexity Classes , 1985, J. Comput. Syst. Sci..

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

[5]  Timothy J. Long,et al.  Quantitative Relativizations of Complexity Classes , 1984, SIAM J. Comput..

[6]  Jacobo Torán,et al.  Turing Machines with Few Accepting Computations and Low Sets for PP , 1992, J. Comput. Syst. Sci..

[7]  Ker-I Ko,et al.  On Sets Truth-Table Reducible to Sparse Sets , 1988, SIAM J. Comput..

[8]  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).

[9]  Timothy J. Long.  On Restricting the Size of Oracles Compared with Restricting Access to Oracles , 1985, SIAM J. Comput..

[10]  Nikolai K. Vereshchagin,et al.  Banishing Robust Turing Completeness , 1992, LFCS.

[11]  Jack H. Lutz,et al.  Category and Measure in Complexity Classes , 1990, SIAM J. Comput..

[12]  P. Vitányi,et al.  Applications of Kolmogorov Complexity in the Theory of Computation , 1990 .

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

[14]  R.E. Ladner,et al.  A Comparison of Polynomial Time Reducibilities , 1975, Theor. Comput. Sci..

[15]  Juris Hartmanis,et al.  On Sparse Oracles Separating Feasible Complexity Classes , 1986, STACS.

[16]  Osamu Watanabe,et al.  Relating Equivalence and Reducibility to Sparse Sets , 1992, SIAM J. Comput..

[17]  Ker-I Ko,et al.  On Circuit-Size Complexity and the Low Hierarchy in NP , 1985, SIAM J. Comput..

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

[19]  Lane A. Hemaspaandra,et al.  Separating Complexity Classes With Tally Oracles , 1992, Theor. Comput. Sci..

[20]  Timothy J. Long.  Erratum: On Restricting the Size of Oracles Compared with Restricting Access to Oracles , 1988, SIAM J. Comput..

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

[22]  Klaus W. Wagner,et al.  Bounded Query Classes , 1990, SIAM J. Comput..

[23]  Klaus-Jörn Lange,et al.  Separation with the Ruzzo, Simon, and Tompa Relativization Implies DSPACE(log n) != NSPACE(log n) , 1987, Inf. Process. Lett..

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

[25]  Pierluigi Crescenzi,et al.  Complexity classes and sparse oracles , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[26]  Osamu Watanabe.  On Polynomial Time One-Truth-Table Reducibility to a Sparse Set , 1992, J. Comput. Syst. Sci..

[27]  José L. Balcázar,et al.  Sparse Oracles and Uniform Complexity Classes , 1984, FOCS.

[28]  C. Papadimitriou,et al.  Two remarks on the power of counting , 1983 .

[29]  Chee-Keng Yap,et al.  Some Consequences of Non-Uniform Conditions on Uniform Classes , 1983, Theor. Comput. Sci..

[30]  Leslie G. Valiant,et al.  Relative Complexity of Checking and Evaluating , 1976, Inf. Process. Lett..

[31]  José L. Balcázar.  Self-Reducibility , 1990, J. Comput. Syst. Sci..

[32]  John Gill,et al.  Computational Complexity of Probabilistic Turing Machines , 1977, SIAM J. Comput..

[33]  Esko Ukkonen.  Two Results on Polynomial Time Truth-Table Reductions to Sparse Sets , 1983, SIAM J. Comput..

[34]  Osamu Watanabe,et al.  Polynomial time reducibility to a set of small density , 1987, SCT.

[35]  José L. Balcázar,et al.  Sparse Sets, Lowness and Highness , 1986, SIAM J. Comput..

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

[37]  Pierluigi Crescenzi,et al.  A Uniform Approach to Define Complexity Classes , 1992, Theor. Comput. Sci..

[38]  Neil Immerman,et al.  Sparse Sets in NP-P: EXPTIME versus NEXPTIME , 1985, Inf. Control..

[39]  Christopher B. Wilson.  A Measure of Relativized Space Which Is Faithful With Respect to Depth , 1988, J. Comput. Syst. Sci..

[40]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

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

[42]  Lane A. Hemaspaandra,et al.  On sets with efficient implicit membership tests , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.

[43]  Uwe Schöning,et al.  A Low and a High Hierarchy within NP , 1983, J. Comput. Syst. Sci..

[44]  Edith Hemaspaandra,et al.  SPARSE reduces conjunctively to TALLY , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[45]  Seinosuke Toda.  Restricted Relativizations of Probablistic Polynomial Time , 1992, Theor. Comput. Sci..

[46]  Yaacov Yesha,et al.  On Certain Polynomial-Time Truth-Table Reducibilities of Complete Sets to Sparse Sets , 1983, SIAM J. Comput..

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

[48]  Jim Kadin,et al.  P^(NP[O(log n)]) and Sparse Turing-Complete Sets for NP , 1989, J. Comput. Syst. Sci..

[49]  Shouwen Tang,et al.  Reducibilities on tally and sparse sets , 1991, RAIRO Theor. Informatics Appl..

[50]  Lane A. Hemaspaandra,et al.  Reductions to Sets of Low Information Content , 1992, Complexity Theory: Current Research.

[51]  Timothy J. Long,et al.  Relativizing complexity classes with sparse oracles , 1986, JACM.

[52]  Albert R. Meyer,et al.  On time-space classes and their relation to the theory of real addition , 1978, STOC '78.

[53]  Jacobo Torán,et al.  Turning machines with few accepting computations and low sets for PP , 1989, [1989] Proceedings. Structure in Complexity Theory Fourth Annual Conference.

[54]  Hans Heller,et al.  On Relativized Exponential and Probabilistic Complexity Classes , 1986, Inf. Control..

[55]  Ker-I Ko.  Distinguishing Conjunctive and Disjunctive Reducibilities by Sparse Sets , 1989, Inf. Comput..

[56]  Eric Allender,et al.  P-Printable Sets , 1988, SIAM J. Comput..

[57]  Ronald V. Book.  Towards a Theory of Relativizations: Positive Relativizations , 1987, STACS.

[58]  Ian Parberry,et al.  On the Construction of Parallel Computers from Various Bases of Boolean Functions , 1986, Theor. Comput. Sci..

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

[60]  Juris Hartmanis.  Solvable problems with conflicting relativizations , 1985, Bull. EATCS.

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

[62]  José L. Balcázar,et al.  The polynomial-time hierarchy and sparse oracles , 1986, JACM.

[63]  Lane A. Hemachandra,et al.  The strong exponential hierarchy collapses , 1989 .

[64]  Adi Shamir,et al.  IP = PSPACE , 1992, JACM.

[65]  Eric Allender,et al.  Lower bounds for the low hierarchy , 1992, JACM.

[66]  R. Solovay,et al.  Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question , 1975 .

[67]  Jack H. Lutz,et al.  Measure, Stochasticity, and the Density of Hard Languages , 1993, STACS.

[68]  Albert R. Meyer,et al.  The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space , 1972, SWAT.

[69]  Timothy J. Long.  Strong Nondeterministic Polynomial-Time Reducibilities , 1982, Theor. Comput. Sci..

[70]  David S. Johnson,et al.  A Catalog of Complexity Classes , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[71]  Osamu Watanabe,et al.  On the Computational Complexity of Small Descriptions , 1993, SIAM J. Comput..

[72]  Seinosuke Toda.  Positive Relativizations for Log Space Computability , 1990, Theor. Comput. Sci..