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Hierarchies Based on NP

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Complexity Theory and Cryptology

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5.9 Summary and Bibliographic Remarks

  1. [ACG+03]_G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, M. Marchetti-Spaccamela, and M. Protasi. Complexity and Approximation. Springer-Verlag, second edition, 2003.

    Google Scholar 

  2. E. Allender and L. Hemachandra. Lower bounds for the low hierarchy. Journal of the ACM, 39(1):234–251, 1992.

    Article  Google Scholar 

  3. [AHH+93]_V. Arvind, Y. Han, L. Hemachandra, J. Köbler, A. Lozano, M. Mundhenk, M. Ogiwara, U. Schöning, R. Silvestri, and T. Thierauf. Reductions to sets of low information content. In K. Ambos-Spies, S. Homer, and U. Schöning, editors, Complexity Theory, pages 1–45. Cambridge University Press, 1993.

    Google Scholar 

  4. [ALM+98]_S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and intractability of approximation problems. Journal of the ACM, 45(3):501–555, May 1998.

    Article  Google Scholar 

  5. L. Adleman and K. Manders. Reducibility, randomness, and intractibility. In Proceedings of the 9th ACM Symposium on Theory of Computing, pages 151–153. ACM Press, 1977.

    Google Scholar 

  6. S. Arora. Probabilistic Checking of Proofs and Hardness of Approximation Problems. PhD thesis, University of California at Berkeley, November 1994. Revised version available as Princeton University Technical Report CS-TR-476-94.

    Google Scholar 

  7. K. Arrow. Social Choice and Individual Values. John Wiley and Sons, 1951 (revised editon 1963).

    Google Scholar 

  8. S. Arora and S. Safra. Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM, 45(1):70–122, January 1998. Preliminary version appears as [AS92].

    Article  Google Scholar 

  9. M. Agrawal and T. Thierauf. The formula isomorphism problem. SIAM Journal on Computing, 30(3):990–1009, June 2001.

    Article  Google Scholar 

  10. V. Arvind and J. Torán. Sparse sets, approximable sets, and parallel queries to NP. Information Processing Letters, 69:181–188, February 1999.

    Article  Google Scholar 

  11. J. Balcázar. Simplicity, relativizations and nondeterminism. SIAM Journal on Computing, 14(1):148–157, 1985.

    Article  Google Scholar 

  12. [BBJ+89]_A. Bertoni, D. Bruschi, D. Joseph, M. Sitharam, and P. Young. Generalized boolean hierarchies and boolean hierarchies over RP. In Proceedings of the 7th Conference on Fundamentals of Computation Theory, pages 35–46. Springer-Verlag Lecture Notes in Computer Science #380, August 1989.

    Google Scholar 

  13. J. Balcázar, R. Book, and U. Schöning. Sparse sets, lowness and highness. SIAM Journal on Computing, 15(3):739–746, 1986.

    Article  Google Scholar 

  14. N. Bshouty, R. Cleve, S. Kannan, and C. Tamon. Oracles and queries that are sufficient for exact learning. In Proceedings of the 7th ACM Conference on Computational Learning Theory, pages 130–139. ACM Press, 1994.

    Google Scholar 

  15. R. Beigel, R. Chang, and M. Ogiwara. A relationship between difference hierarchies and relativized polynomial hierarchies. Mathematical Systems Theory, 26(3):293–310, 1993.

    Article  Google Scholar 

  16. D. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104(2):263–283, 1992.

    Article  Google Scholar 

  17. R. Beigel. Bounded queries to SAT and the boolean hierarchy. Theoretical Computer Science, 84(2):199–223, 1991.

    Article  Google Scholar 

  18. H. Buhrman and L. Fortnow. Two queries. In Proceedings of the 13th Annual IEEE Conference on Computational Complexity, pages 13–19. IEEE Computer Society Press, May 1998.

    Google Scholar 

  19. A. Blass and Y. Gurevich. On the unique satisfiability problem. Information and Control, 55(1–3):80–88, 1982.

    Article  Google Scholar 

  20. T. Baker, J. Gill, and R. Solovay. Relativizations of the P=?NP question. SIAM Journal on Computing, 4(4):431–442, 1975.

    Article  Google Scholar 

  21. S. Buss and L. Hay. On truth-table reducibility to SAT and the difference hierarchy over NP. In Proceedings of the 3rd Structure in Complexity Theory Conference, pages 224–233. IEEE Computer Society Press, June 1988.

    Google Scholar 

  22. S. Buss and L. Hay. On truth-table reducibility to SAT. Information and Computation, 91(1):86–102, March 1991.

    Article  Google Scholar 

  23. B. Borchert, L. Hemaspaandra, and J. Rothe. Restrictive acceptance suffices for equivalence problems. London Mathematical Society Journal of Computation and Mathematics, 3:86–95, March 2000.

    Google Scholar 

  24. D. Bruschi, D. Joseph, and P. Young. Strong separations for the boolean hierarchy over RP. International Journal of Foundations of Computer Science, 1(3):201–218, 1990.

    Article  Google Scholar 

  25. D. Black. The Theory of Committees and Elections. Cambridge University Press, 1958.

    Google Scholar 

  26. D. Bleichenbacher. Chosen ciphertext attacks against protocols based on the RSA encryption standard PKCS #1. In Advances in Cryptology — CRYPTO’ 98, pages 1–12. Springer-Verlag Lecture Notes in Computer Science #1462, August 1998.

    Google Scholar 

  27. R. Book, P. Orponen, D. Russo, and O. Watanabe. Lowness properties of sets in the exponential-time hierarchy. SIAM Journal on Computing, 17(3):504–516, 1988.

    Article  Google Scholar 

  28. J. Balcázar and D. Russo. Immunity and simplicity in relativizations of probabilistic complexity classes. R.A.I.R.O. Theoretical Informatics and Applications, 22(2):227–244, 1988.

    Google Scholar 

  29. B. Borchert and D. Ranjan. The subfunction relations are Σ p2 -complete. Technical Report MPI-I-93-121, Max-Planck Institut Saarbrücken, Saarbrücken, Germany, 1993.

    Google Scholar 

  30. B. Borchert, D. Ranjan, and F. Stephan. On the computational complexity of some classical equivalence relations on boolean functions. Mathematical Systems Theory, 31(6):679–693, 1998.

    Google Scholar 

  31. D. Bruschi. Strong separations of the polynomial hierarchy with oracles: Constructive separations by immune and simple sets. Theoretical Computer Science, 102(2):215–252, 1992.

    Article  Google Scholar 

  32. H. Buhrman, E. Spaan, and L. Torenvliet. Bounded reductions. In K. Ambos-Spies, S. Homer, and U. Schöning, editors, Complexity Theory, pages 83–99.

    Google Scholar 

  33. H. Buhrmann, E. Spaan, and L. Torenvliet. The relative power of logspace and polynomial time reductions. Computational Complexity, 3(3):231–244, 1993.

    Article  Google Scholar 

  34. J. Bartholdi III, C. Tovey, and M. Trick. The computational difficulty of manipulating an election. Social Choice and Welfare, 6:227–241, 1989.

    Article  Google Scholar 

  35. J. Bartholdi III, C. Tovey, and M. Trick. Voting schemes for which it can be difficult to tell who won the election. Social Choice and Welfare, 6:157–165, 1989.

    Article  Google Scholar 

  36. J. Bartholdi III, C. Tovey, and M. Trick. How hard is it to control an election? Mathematical Comput. Modelling, 16(8/9):27–40, 1992.

    Article  Google Scholar 

  37. J. Cai. S p2 ZPPNP. In Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, pages 620–629. IEEE Computer Society Press, October 2001.

    Google Scholar 

  38. R. Canetti. More on BPP and the polynomial-time hierarchy. Information Processing Letters, 57(5):237–241, 1996.

    Article  Google Scholar 

  39. J. Cai, V. Chakaravarthy, L. Hemaspaandra, and M. Ogihara. Competing provers yield improved Karp-Lipton collapse results. In Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science, pages 535–546.

    Google Scholar 

  40. [CGH+88]_J. Cai, T. Gundermann, J. Hartmanis, L. Hemachandra, V. Sewelson, K. Wagner, and G. Wechsung. The boolean hierarchy I: Structural properties. SIAM Journal on Computing, 17(6):1232–1252, 1988.

    Article  Google Scholar 

  41. [CGH+89]_J. Cai, T. Gundermann, J. Hartmanis, L. Hemachandra, V. Sewelson, K. Wagner, and G. Wechsung. The boolean hierarchy II: Applications. SIAM Journal on Computing, 18(1):95–111, 1989.

    Article  Google Scholar 

  42. J. Cai and L. Hemachandra. The boolean hierarchy: Hardware over NP. In Proceedings of the 1st Structure in Complexity Theory Conference, pages 105–124. Springer-Verlag Lecture Notes in Computer Science #223, June 1986.

    Google Scholar 

  43. R. Chang. On the Structure of NP Computations under Boolean Operators. PhD thesis, Cornell University, Ithaca, NY, 1991.

    Google Scholar 

  44. J. Cai, L. Hemachandra, and J. Vyskoč. Promise problems and access to unambiguous computation. In Proceedings of the 17th Symposium on Mathematical Foundations of Computer Science, pages 162–171. Springer-Verlag Lecture Notes in Computer Science #629, August 1992.

    Google Scholar 

  45. J. Cai, L. Hemachandra, and J. Vyskoč. Promises and fault-tolerant database access. In K. Ambos-Spies, S. Homer, and U. Schöning, editors, Complexity Theory, pages 101–146. Cambridge University Press, 1993.

    Google Scholar 

  46. R. Chang and J. Kadin. The boolean hierarchy and the polynomial hierarchy: A closer connection. SIAM Journal on Computing, 25(2):340–354, April 1996.

    Article  Google Scholar 

  47. R. Chang, J. Kadin, and P. Rohatgi. On unique satisfiability and the threshold behavior of randomized reductions. Journal of Computer and System Sciences, 50(3):359–373, 1995.

    Article  Google Scholar 

  48. A. Chandra, D. Kozen, and L. Stockmeyer. Alternation. Journal of the ACM, 26(1), 1981.

    Google Scholar 

  49. V. Conitzer, J. Lang, and T. Sandholm. How many candidates are needed to make elections hard to manipulate? In Proceedings of the 9th Conference on Theoretical Aspects of Rationality and Knowledge, pages 201–214. ACM Press, 2003.

    Google Scholar 

  50. J. Cai and G. Meyer. Graph minimal uncolorability is DP-complete. SIAM Journal on Computing, 16(2):259–277, April 1987.

    Article  Google Scholar 

  51. S. Cook. The complexity of theorem-proving procedures. In Proceedings of the 3rd ACM Symposium on Theory of Computing, pages 151–158. ACM Press, 1971.

    Google Scholar 

  52. V. Conitzer and T. Sandholm. Complexity of manipulating elections with few candidates. In Proceedings of the 18th National Conference on Artificial Intelligence, pages 314–319. AAAI Press, 2002.

    Google Scholar 

  53. J. Castro and C. Seara. Characterizations of some complexity classes between Θ p2 and Δ p2 . In Proceedings of the 9th Annual Symposium on Theoretical Aspects of Computer Science, pages 305–317. Springer-Verlag Lecture Notes in Computer Science #577, February 1992.

    Google Scholar 

  54. Z. Chen and S. Toda. The complexity of selecting maximal solutions. Information and Computation, 119:231–239, 1995.

    Article  Google Scholar 

  55. J. Cai and O. Watanabe. Relativized collapsing between BPP and PH under stringent oracle access. Information Processing Letters, 90(3):147–154, 2004.

    Article  Google Scholar 

  56. C. Dwork, R. Kumar, M. Naor, and D. Sivakumar. Rank aggregation methods for the web. In Proceedings of the 10th International World Wide Web Conference, pages 613–622. ACM Press, 2001.

    Google Scholar 

  57. C. Dodgson. A method of taking votes on more than two issues. Pamphlet printed by the Clarendon Press, Oxford, and headed “not yet published” (see the discussions in [MU95, Bla58], both of which reprint this paper), 1876.

    Google Scholar 

  58. T. Eiter and G. Gottlob. Propositional circumscription and extended closed-world reasoning are π p2 -complete. Theoretical Computer Science, 114(2):231–245, 1993. Addendum appears in the same journal, 118(2):315, 1993.

    Article  Google Scholar 

  59. T. Eiter and G. Gottlob. The complexity class Θ p2 : Recent results and applications. In Proceedings of the 11th Conference on Fundamentals of Computation Theory, pages 1–18. Springer-Verlag Lecture Notes in Computer Science #1279, September 1997.

    Google Scholar 

  60. D. Eppstein, L. Hemachandra, J. Tisdall, and B. Yener. Simultaneous strong separations of probabilistic and unambiguous complexity classes. Mathematical Systems Theory, 25(1):23–36, 1992.

    Article  Google Scholar 

  61. W. Espelage. Bewegungsminimierung in der Föorderband-Flow-Shop-Verarbeitung. PhD thesis, Heinrich-Heine-Universität Düsseldorf, Düsseldorf, Germany, 2001. In German.

    Google Scholar 

  62. W. Espelage and E. Wanke. Movement optimization in flow shop processing with buffers. Mathematical Methods of Operations Research, 51(3):495–513, 2000.

    Article  Google Scholar 

  63. W. Espelage and E. Wanke. A 3-approximation algorithmus for movement minimization in conveyor flow shop processing. In Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science, pages 363–374. Springer-Verlag Lecture Notes in Computer Science #2136, 2001.

    Google Scholar 

  64. W. Espelage and E. Wanke. Movement minimization for unit distances in conveyor flow shop processing. Mathematical Methods of Operations Research, 57(2):172–206, 2003.

    Google Scholar 

  65. P. Fishburn. Condorcet social choice functions. SIAM Journal on Applied Mathematics, 33:469–489, 1977.

    Article  Google Scholar 

  66. L. Fortnow and T. Yamakami. Generic separations. Journal of Computer and System Sciences, 52(1):191–197, 1996.

    Article  Google Scholar 

  67. M. Garey, D. Johnson, and L. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, 1:237–267, 1976.

    Article  Google Scholar 

  68. V. Guruswami and S. Khanna. On the hardness of 4-coloring a 3-colorable graph. In Proceedings of the 15th Annual IEEE Conference on Computational Complexity, pages 188–197. IEEE Computer Society Press, May 2000.

    Google Scholar 

  69. T. Gundermann, N. Nasser, and G. Wechsung. A survey on counting classes. In Proceedings of the 5th Structure in Complexity Theory Conference, pages 140–153. IEEE Computer Society Press, July 1990.

    Google Scholar 

  70. O. Goldreich. A taxonomy of proof systems. In L. Hemaspaandra and A. Selman, editors, Complexity Theory Retrospective II, pages 109–134. Springer-Verlag, 1997.

    Google Scholar 

  71. A. Große, J. Rothe, and G. Wechsung. Relating partial and complete solutions and the complexity of computing smallest solutions. In Proceedings of the Seventh Italian Conference on Theoretical Computer Science, pages 339–356.

    Google Scholar 

  72. T. Gundermann and G. Wechsung. Counting classes with finite acceptance types. Computers and Artificial Intelligence, 6(5):395–409, 1987.

    Google Scholar 

  73. F. Hausdorff. Grundzüge der Mengenlehre. Walter de Gruyter and Co., 1914.

    Google Scholar 

  74. L. Hemachandra. The strong exponential hierarchy collapses. In Proceedings of the 19th ACM Symposium on Theory of Computing, pages 110–122. ACM Press, May 1987.

    Google Scholar 

  75. L. Hemachandra. The strong exponential hierarchy collapses. Journal of Computer and System Sciences, 39(3):299–322, 1989.

    Article  Google Scholar 

  76. L. Hemaspaandra. Lowness: A yardstick for NP-P. SIGACT News, 24 (Spring)(2):10–14, 1993.

    Article  Google Scholar 

  77. H. Hempel. Boolean Hierarchies — On Collapse Properties and Query Order. PhD thesis, Friedrich-Schiller-Universität Jena, Jena, Germany, October 1998.

    Google Scholar 

  78. E. Hemaspaandra and L. Hemaspaandra. Computational politics: Electoral systems. In Proceedings of the 25th International Symposium on Mathematical Foundations of Computer Science, pages 64–83.

    Google Scholar 

  79. E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. Using the no-search easy-hard technique for downward collapse. Technical Report TR-752, University of Rochester, Department of Computer Science, Rochester, NY, June 2001. Earlier versions or parts of this paper appeared in the Proceedings of the Sixth Italian Conference on Theoretical Computer Science (ICTCS'98) and of the 16th Annual Symposium on Theoretical Aspects of Computer Science (STACS'99).

    Google Scholar 

  80. E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. Query order in the polynomial hierarchy. Journal of Universal Computer Science, 4(6):574–588, June 1998.

    Google Scholar 

  81. E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. What's up with downward collapse: Using the easy-hard technique to link boolean and polynomial hierarchy collapses. SIGACT News, 29(3):10–22, 1998.

    Article  Google Scholar 

  82. E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. A downward collapse within the polynomial hierarchy. SIAM Journal on Computing, 28(2):383–393, 1999.

    Article  Google Scholar 

  83. E. Hemaspaandra, L. Hemaspaandra, and J. Rothe. Anyone but him: The complexity of precluding an alternative. In Proceedings of the 20th National Conference on Artificial Intelligence. AAAI Press, 2005. To appear.

    Google Scholar 

  84. E. Hemaspaandra, L. Hemaspaandra, and J. Rothe. Exact analysis of Dodgson elections: Lewis Carroll's 1876 voting system is complete for parallel access to NP. Journal of the ACM, 44(6):806–825, November 1997.

    Article  Google Scholar 

  85. E. Hemaspaandra, L. Hemaspaandra, and J. Rothe. Raising NP lower bounds to parallel NP lower bounds. SIGACT News, 28(2):2–13, June 1997.

    Google Scholar 

  86. L. Hemaspaandra, H. Hempel, and G. Wechsung. Query order. SIAM Journal on Computing, 28(2):637–651, 1999.

    Article  Google Scholar 

  87. L. Hemaspaandra, Z. Jiang, J. Rothe, and O. Watanabe. Polynomial-time multiselectivity. Journal of Universal Computer Science, 3(3):197–229, March 1997.

    Google Scholar 

  88. L. Hemaspaandra, Z. Jiang, J. Rothe, and O. Watanabe. Boolean operations, joins, and the extended low hierarchy. Theoretical Computer Science, 205(1–2):317–327, September 1998.

    Article  Google Scholar 

  89. S. Homer and L. Longpré. On reductions of NP sets to sparse sets. Journal of Computer and System Sciences, 48(2):324–336, April 1994.

    Article  Google Scholar 

  90. S. Homer and W. Maass. Oracle dependent properties of the lattice of NP sets. Theoretical Computer Science, 24(3):279–289, 1983.

    Article  Google Scholar 

  91. L. Hemachandra, M. Ogiwara, and O. Watanabe. How hard are sparse sets? In Proceedings of the 7th Structure in Complexity Theory Conference, pages 222–238. IEEE Computer Society Press, June 1992.

    Google Scholar 

  92. L. Hemaspaandra and J. Rothe. Intersection suffices for boolean hierarchy equivalence. In Proceedings of the First Annual International Computing and Combinatorics Conference, pages 430–435. Springer-Verlag Lecture Notes in Computer Science #959, August 1995.

    Google Scholar 

  93. E. Hemaspaandra and J. Rothe. Recognizing when greed can approximate maximum independent sets is complete for parallel access to NP. Technical Report Math/Inf/97/14, Friedrich-Schiller-Universität Jena, Jena, Germany, May 1997.

    Google Scholar 

  94. L. Hemaspaandra and J. Rothe. Unambiguous computation: Boolean hierarchies and sparse Turing-complete sets. SIAM Journal on Computing, 26(3):634–653, June 1997.

    Article  Google Scholar 

  95. E. Hemaspaandra and J. Rothe. Recognizing when greed can approximate maximum independent sets is complete for parallel access to NP. Information Processing Letters, 65(3):151–156, February 1998.

    Article  Google Scholar 

  96. E. Hemaspaandra, J. Rothe, and H. Spakowski. Recognizing when heuristics can approximate minimum vertex covers is complete for parallel access to NP. R.A.I.R.O. Theoretical Informatics and Applications. To appear.

    Google Scholar 

  97. L. Hemaspaandra, K. Rajasethupathy, P. Sethupathy, and M. Zimand. Power balance and apportionment algorithms for the United States Congress. The ACM Journal of Experimental Algorithmics, 3(1):article 1, 16 pp., 1998.

    Google Scholar 

  98. L. Hemaspaandra, J. Rothe, and G. Wechsung. Easy sets and hard certificate schemes. Acta Informatica, 34(11):859–879, November 1997.

    Article  Google Scholar 

  99. E. Hemaspaandra, H. Spakowski, and J. Vogel. The complexity of Kemeny elections. Theoretical Computer Science. Accepted subject to minor revision.

    Google Scholar 

  100. L. Hemaspaandra and L. Torenvliet. Theory of Semi-Feasible Algorithms. EATCS Monographs in Theoretical Computer Science. Springer-Verlag, 2003.

    Google Scholar 

  101. P. Heggernes and J. Telle. Partitioning graphs into generalized dominating sets. Nordic Journal of Computing, 5(2):128–142, 1998.

    Google Scholar 

  102. E. Hemaspaandra and G. Wechsung. The minimization problem for boolean formulas. SIAM Journal on Computing, 31(6):1948–1958, 2002.

    Article  Google Scholar 

  103. E. Hemaspaandra and G. Wechsung. The minimization problem for boolean formulas. In Proceedings of the 38th IEEE Symposium on Foundations of Computer Science, pages 575–584. IEEE Computer Society Press, October 1997.

    Google Scholar 

  104. L. Hemaspaandra and M. Zimand. Strong self-reducibility precludes strong immunity. Mathematical Systems Theory, 29(5):535–548, 1996.

    Google Scholar 

  105. J. Köbler. Locating P/poly optimally in the extended low hierarchy. Theoretical Computer Science, 134:263–285, 1994.

    Article  Google Scholar 

  106. J. Köbler. On the structure of low sets. In Proceedings of the 10th Structure in Complexity Theory Conference, pages 246–261. IEEE Computer Society Press, 1995.

    Google Scholar 

  107. J. Kadin. The polynomial time hierarchy collapses if the boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263–1282, 1988. Erratum appears in the same journal, 20(2):404, 1991.

    Article  Google Scholar 

  108. J. Kadin. PNP[log n] and sparse Turing-complete sets for NP. Journal of Computer and System Sciences, 39(3):282–298, 1989.

    Article  Google Scholar 

  109. R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In Proceedings of the 12th ACM Symposium on Theory of Computing, pages 302–309. ACM Press, April 1980. An extended version has also appeared as: Turing machines that take advice, L'Enseignement Mathématique, 2nd series 28, 1982, pages 191–209.

    Google Scholar 

  110. S. Khanna, N. Linial, and S. Safra. On the hardness of approximating the chromatic number. Combinatorica, 20(3):393–415, 2000.

    Article  Google Scholar 

  111. K. Ko. A note on separating the relativized polynomial time hierarchy by immune sets. R.A.I.R.O. Theoretical Informatics and Applications, 24(3):229–240, 1990.

    Google Scholar 

  112. M. Krentel. The complexity of optimization problems. Journal of Computer and System Sciences, 36:490–509, 1988.

    Article  Google Scholar 

  113. K. Ko and U. Schöning. On circuit-size complexity and the low hierarchy in NP. SIAM Journal on Computing, 14(1):41–51, 1985.

    Article  Google Scholar 

  114. J. Köbler, U. Schöning, and J. Torán. Graph isomorphism is low for PP. Computational Complexity, 2:301–330, 1992.

    Article  Google Scholar 

  115. J. Köbler, U. Schöning, S. Toda, and J. Torán. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44(2):272–286, 1992.

    Article  Google Scholar 

  116. J. Köbler, U. Schöning, and K. Wagner. The difference and truth-table hierarchies for NP. R.A.I.R.O. Informatique théorique et Applications, 21:419–435, 1987.

    Google Scholar 

  117. S. Khuller and V. Vazirani. Planar graph coloring is not self-reducible, assuming P ≠ NP. Theoretical Computer Science, 88(1):183–189, 1991.

    Article  Google Scholar 

  118. J. Köbler and O. Watanabe. New collapse consequences of NP having small circuits. SIAM Journal on Computing, 28(1):311–324, 1998.

    Article  Google Scholar 

  119. G. Lischke. Oracle constructions to prove all possible relationships between relativizations of P, NP, NEL, EP, and NEP. Zeitsch. f. math. Logik und Grundlagen d. Math., 32:257–270, 1986.

    Google Scholar 

  120. G. Lischke. Towards the actual relationship between NP and exponential time. Mathematical Logic Quarterly, 45(1):31–49, 1999. A preliminary version has appeared as: Impossibilities and possibilities of weak separation between NP and exponential time. In Proceedings of the 5th Structure in Complexity Theory Conference, pages 245–253. IEEE Computer Society Press, 1990.

    Google Scholar 

  121. R. Ladner and N. Lynch. Relativization of questions about log space computability. Mathematical Systems Theory, 10(1):19–32, 1976.

    Article  Google Scholar 

  122. R. Ladner, N. Lynch, and A. Selman. A comparison of polynomial time reducibilities. Theoretical Computer Science, 1(2):103–124, 1975.

    Article  Google Scholar 

  123. T. Long. A note on sparse oracles for NP. Journal of Computer and System Sciences, 24:224–232, 1982.

    Article  Google Scholar 

  124. T. Long. Strong nondeterministic polynomial-time reducibilities. Theoretical Computer Science, 21:1–25, 1982.

    Article  Google Scholar 

  125. K.-J. Lange and P. Rossmanith. Unambiguous polynomial hierarchies and exponential size. In Proceedings of the 9th Structure in Complexity Theory Conference, pages 106–115. IEEE Computer Society Press, June/July 1994.

    Google Scholar 

  126. H. Müller. A note on balanced immunity. Mathematical Systems Theory, 26(2):157–167, 1993.

    Article  Google Scholar 

  127. S. Mahaney. Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis. Journal of Computer and System Sciences, 25(2):130–143, 1982.

    Article  Google Scholar 

  128. S. Mahaney. Sparse sets and reducibilities. In R. Book, editor, Studies in Complexity Theory, pages 63–118. John Wiley and Sons, 1986.

    Google Scholar 

  129. S. Mahaney. The isomorphism conjecture and sparse sets. In J. Hartmanis, editor, Computational Complexity Theory, pages 18–46. American Mathematical Society, 1989. Proceedings of Symposia in Applied Mathematics #38.

    Google Scholar 

  130. A. Meyer and L. Stockmeyer. The equivalence problem for regular expressions with squaring requires exponential space. In Proceedings of the 13th IEEE Symposium on Switching and Automata Theory, pages 125–129, 1972.

    Google Scholar 

  131. I. McLean and A. Urken. Classics of Social Choice. University of Michigan Press, Ann Arbor, Michigan, 1995.

    Google Scholar 

  132. R. Niedermeier and P. Rossmanith. Unambiguous computations and locally definable acceptance types. Theoretical Computer Science, 194:137–161, 1998.

    Article  Google Scholar 

  133. M. Ogiwara and O. Watanabe. On polynomial-time bounded truth-table reducibility of NP sets to sparse sets. SIAM Journal on Computing, 20(3):471–483, June 1991.

    Article  Google Scholar 

  134. C. Papadimitriou. On the complexity of unique solutions. Journal of the ACM, 31(2):392–400, 1984.

    Article  Google Scholar 

  135. C. Papadimitriou and D. Wolfe. The complexity of facets resolved. Journal of Computer and System Sciences, 37(1):2–13, 1988.

    Article  Google Scholar 

  136. C. Papadimitriou and M. Yannakakis. The complexity of facets (and some facets of complexity). Journal of Computer and System Sciences, 28(2):244–259, 1984.

    Article  Google Scholar 

  137. H. Rogers, Jr. The Theory of Recursive Functions and Effective Computability. McGraw-Hill, 1967.

    Google Scholar 

  138. P. Rohatgi. Saving queries with randomness. Journal of Computer and System Sciences, 50(3):476–492, 1995.

    Article  Google Scholar 

  139. J. Rothe. Exact complexity of Exact-Four-Colorability. Information Processing Letters, 87(1):7–12, July 2003.

    Article  Google Scholar 

  140. J. Rothe. Immunity and simplicity for exact counting and other counting classes. R.A.I.R.O. Theoretical Informatics and Applications, 33(2):159–176, March/April 1999.

    Article  Google Scholar 

  141. T. Riege and J. Rothe. Complexity of the exact domatic number problem and of the exact conveyor flow shop problem. Theory of Computing Systems, December 2004. On-line publication DOI 10.1007/s00224-004-1209-8. Paper publication to appear.

    Google Scholar 

  142. A. Russell and R. Sundaram. Symmetric alternation captures BPP. Computational Complexity, 7(2):152–162, 1998.

    Article  Google Scholar 

  143. J. Rothe, H. Spakowski, and J. Vogel. Exact complexity of Exact-Four-Colorability and of the winner problem for Young elections. In R. Baeza-Yates, U. Montanari, and N. Santoro, editors, Foundations of Information Technology in the Era of Network and Mobile Computing, pages 310–322. Kluwer Academic Publishers, August 2002. Proceedings of the Second IFIP International Conference on Theoretical Computer Science, Stream 1 of the 17th IFIP World Computer Congress.

    Google Scholar 

  144. J. Rothe, H. Spakowski, and J. Vogel. Exact complexity of the winner problem for Young elections. Theory of Computing Systems, 36(4):375–386, June 2003.

    Article  Google Scholar 

  145. S. Reith and K. Wagner. On boolean lowness and boolean highness. Theoretical Computer Science, 261(2):305–321, June 2001.

    Article  Google Scholar 

  146. D. Saari. Chaotic Elections! A Mathematician Looks at Voting. American Mathematical Society, 2001.

    Google Scholar 

  147. D. Saari. Basic Geometry of Voting. Springer-Verlag, Berlin, Germany, 1995.

    Google Scholar 

  148. U. Schöning and R. Book. Immunity, relativization, and nondeterminism. SIAM Journal on Computing, 13(2):329–337, 1984.

    Article  Google Scholar 

  149. C. Schnorr. Optimal algorithms for self-reducible problems. In S. Michaelson and R. Milner, editors, Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming, pages 322–337, University of Edinburgh, July 1976. Edinburgh University Press.

    Google Scholar 

  150. C. Schnorr. On self-transformable combinatorial problems, 1979. Presented at IEEE Symposium on Information Theory, Udine, and Symposium über Mathematische Optimierung, Oberwolfach.

    Google Scholar 

  151. U. Schöning. A low and a high hierarchy within NP. Journal of Computer and System Sciences, 27:14–28, 1983.

    Article  Google Scholar 

  152. U. Schöning. Probabilistic complexity classes and lowness. In Proceedings of the 2nd Structure in Complexity Theory Conference, pages 2–8. IEEE Computer Society Press, June 1987.

    Google Scholar 

  153. U. Schöning. Graph isomorphism is in the low hierarchy. Journal of Computer and System Sciences, 37(3):312–323, 1988.

    Article  Google Scholar 

  154. A. Selman. Polynomial time enumeration reducibility. SIAM Journal on Computing, 7(4):440–457, 1978.

    Article  Google Scholar 

  155. A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13:55–65, 1979.

    Article  Google Scholar 

  156. A. Selman. Analogues of semirecursive sets and effective reducibilities to the study of NP complexity. Information and Control, 52:36–51, 1982.

    Article  Google Scholar 

  157. A. Selman. Reductions on NP and P-selective sets. Theoretical Computer Science, 19:287–304, 1982.

    Article  Google Scholar 

  158. A. Selman. Natural self-reducible sets. SIAM Journal on Computing, 17(5):989–996, 1988.

    Article  Google Scholar 

  159. M. Sheu and T. Long. The extended low hierarchy is an infinite hierarchy. SIAM Journal on Computing, 23(3):488–509, 1994.

    Article  Google Scholar 

  160. R. Soare. Computational complexity, speedability, and levelable sets. Journal of Symbolic Logic, 42:545–563, 1977.

    Google Scholar 

  161. R. Soare. Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets. Perspectives in Mathematical Logic. Springer-Verlag, 1987.

    Google Scholar 

  162. L. Stockmeyer. Planar 3-colorability is NP-complete. SIGACT News, 5(3):19–25, 1973.

    Article  Google Scholar 

  163. L. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science, 3(1):1–22, 1977.

    Article  Google Scholar 

  164. M. Schaefer and C. Umans. Completeness in the polynomial-time hierarchy: Part I: A compendium. SIGACT News, 33(3):32–49, September 2002.

    Google Scholar 

  165. M. Schaefer and C. Umans. Completeness in the polynomial-time hierarchy: Part II. SIGACT News, 33(4):22–36, December 2002.

    Article  Google Scholar 

  166. M. Sudan. Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems. Springer-Verlag Lecture Notes in Computer Science #1001, 1995. ACM Distinguished Thesis. Based on the author's Ph.D. thesis, UC Berkeley, 1992.

    Google Scholar 

  167. H. Spakowski and J. Vogel. Θ p2 -completeness: A classical approach for new results. In Proceedings of the 20th Conference on Foundations of Software Technology and Theoretical Computer Science, pages 348–360. Springer-Verlag Lecture Notes in Computer Science #1974, December 2000.

    Google Scholar 

  168. L. Torenvliet and P. van Emde Boas. Simplicity, immunity, relativizations and nondeterminism. Information and Computation, 80(1):1–17, 1989.

    Article  Google Scholar 

  169. C. Umans. The Minimum Equivalent DNF problem and shortest implicants. In Proceedings of the 39th IEEE Symposium on Foundations of Computer Science, pages 556–563. IEEE Computer Society Press, November 1998.

    Google Scholar 

  170. V. Vazirani. Approximation Algorithms. Springer-Verlag, second edition, 2003.

    Google Scholar 

  171. L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85–93, 1986.

    Article  Google Scholar 

  172. K. Wagner. More complicated questions about maxima and minima, and some closures of NP. Theoretical Computer Science, 51:53–80, 1987.

    Article  Google Scholar 

  173. K. Wagner. More complicated questions about maxima and minima, and some closures of NP. Theoretical Computer Science, 51:53–80, 1987. p. 70

    Article  Google Scholar 

  174. K. Wagner. Number-of-query hierarchies. Institut für Mathematik 158, Universit ät Augsburg, Augsurg, Germany, October 1987.

    Google Scholar 

  175. K. Wagner. Number-of-query hierarchies. Institut für Informatik 4, Universität Würzburg, Würzburg, Germany, February 1989.

    Google Scholar 

  176. K. Wagner. Bounded query classes. SIAM Journal on Computing, 19(5):833–846, 1990.

    Article  Google Scholar 

  177. G. Wechsung. On the boolean closure of NP. In Proceedings of the 5th Conference on Fundamentals of Computation Theory, pages 485–493. Springer-Verlag Lecture Notes in Computer Science #199, 1985. (An unpublished precursor of this paper was coauthored by K. Wagner).

    Google Scholar 

  178. C. Wrathall. Complete sets and the polynomial-time hierarchy. Theoretical Computer Science, 3:23–33, 1977.

    Article  Google Scholar 

  179. H. Young. Extending Condorcet's rule. Journal of Economic Theory, 16:335–353, 1977.

    Article  Google Scholar 

  180. P. Young. How reductions to sparse sets collapse the polynomial-time hierarchy: A primer. SIGACT News, 1992. Part I (#3, pages 107–117), Part II (#4, pages 83–94), and Corrigendum to Part I (#4, page 94).

    Google Scholar 

  181. M. Zimand. Computational Complexity: A Quantitative Perspective, volume 196 of North-Holland Mathematics Studies. Elsevier, 2004.

    Google Scholar 

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(2005). Hierarchies Based on NP. In: Complexity Theory and Cryptology. Texts in Theoretical Computer Science An EATCS Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-28520-2_5

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