Previous article Next article Nondeterminism within $P^ * $Jonathan F. Buss and Judy GoldsmithJonathan F. Buss and Judy Goldsmithhttps://doi.org/10.1137/0222038PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractClasses of machines using very limited amounts of nondeterminism are studied. The $P = ?NP$ question is related to questions about classes lying within P. Complete sets for these classes are given.[1] J. A. Abrahamson, , M. R. Ellis, , M. R. Fellows and , M. E. Mata, On the complexity of fixed parameter problems, Proceedings of 30th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Washington, DC, 1989, 210–215 Google Scholar[2] J. A. Abrahamson, , M. R. Ellis, , M. R. Fellows and , M. E. Mata, Completeness for Families of Find Parameter Problems, Report, DCS-141-IR, University of Victoria, Victoria, British Columbia, Canada, 1990 Google Scholar[3] Akeo Adachi, , Shigeki Iwata and , Takumi Kasai, Some combinatorial game problems require $\Omega(n\sp k)$ time, J. Assoc. Comput. Mach., 31 (1984), 361–376 10.1145/62.322433 819 145 0631.68036 CrossrefISIGoogle Scholar[4] Carme Álvarez, , Josep Díaz and , Jacobo Torán, Complexity classes with complete problems between P and NP-CFundamentals of computation theory (Szeged, 1989), Lecture Notes in Comput. Sci., Vol. 380, Springer, New York, 1989, 13–24 91b:68032 0753.68043 CrossrefGoogle Scholar[5] Jonathan F. Buss, Relativized alternation and space-bounded computation, J. Comput. System Sci., 36 (1988), 351–378 10.1016/0022-0000(88)90034-7 90f:68059 0657.68047 CrossrefISIGoogle Scholar[6] S. R. Buss, personal communication Google Scholar[7] Stephen A. Cook, Short propositional formulas represent nondeterministic computations, Inform. Process. Lett., 26 (1988), 269–270 10.1016/0020-0190(88)90152-4 89i:68028 0633.68034 CrossrefISIGoogle Scholar[8] Josep Díaz and , Jacobo Torán, Classes of bounded nondeterminism, Math. Systems Theory, 23 (1990), 21–32 91e:68052 0692.68029 CrossrefGoogle Scholar[9] J. G. Gesice, Ph.D. Thesis, On the Structure of Intractable Sets, Iowa State University, Ames, IA, 1987 Google Scholar[10] Yuri Gurevich and , Saharon Shelah, A. R. Meyer and , M. A. Thitslin, Nearly linear timeLogic at Botik '89, Lecture Notes in Comput. Sci., Vol. 363, Springer, Berlin, 1989, 108–118, New York 91a:68091 0673.03026 CrossrefGoogle Scholar[11] H. B. Hunt, III and , R. E. Stearns, The complexity of very simple Boolean formulas with applications, SIAM J. Comput., 19 (1990), 44–70 10.1137/0219003 91f:68064 0696.68060 LinkISIGoogle Scholar[12] R. Kaye, Masters Thesis, Characterizing some low complexity classes using theories of arithmetic, M. Sc. thesis, University of Manchester, Manchester, U.K., 1985 0746.03032 Google Scholar[13] Chandra M. R. Kintala and , Patrick C. Fischer, Refining nondeterminism in relativized polynomial-time bounded computations, SIAM J. Comput., 9 (1980), 46–53 10.1137/0209003 81b:68049 0447.68044 LinkISIGoogle Scholar[14] N. Pippenger, Fast simulation of combinational logic circuits by machines without random-access storage, Proceedings of the Fifteenth Allerton Conference on Communication, Control, and Computing, 1977, 25–33 Google Scholar[15] Bruno Poizat, ${\scr Q}={\scr N}{\scr Q}$?, J. Symbolic Logic, 51 (1986), 22–32 88g:68040 0592.03027 CrossrefISIGoogle Scholar[16] C. P. Schnorr, Satisfiability is quasilinear complete in NQL, J. Assoc. Comput. Mach., 25 (1978), 136–145 10.1145/322047.322060 58:8495 0364.68056 CrossrefISIGoogle Scholar[17] M. J. Wolf, Ph.D. Thesis, Limited Nondeterminism in Parallel Models of Computation, University of Wisconsin, Madison, WI, 1990 Google ScholarKeywordsnondeterminismquasilinear timecomputational complexity Previous article Next article FiguresRelatedReferencesCited byDetails Dynamic Kernels for Hitting Sets and Set Packing22 June 2022 | Algorithmica, Vol. 84, No. 11 Cross Ref Component Order Connectivity in Directed Graphs23 July 2022 | Algorithmica, Vol. 84, No. 9 Cross Ref Parameterized analysis and crossing minimization problemsComputer Science Review, Vol. 45 Cross Ref Parallel Vertex Cover Algorithms on GPUs Cross Ref Diversity of solutions: An exploration through the lens of fixed-parameter tractability theoryArtificial Intelligence, Vol. 303 Cross Ref Refined notions of parameterized enumeration kernels with applications to matching cut enumerationJournal of Computer and System Sciences, Vol. 123 Cross Ref Sliding window temporal graph coloringJournal of Computer and System Sciences, Vol. 120 Cross Ref Augmented intuition: a bridge between theory and practice31 January 2021 | Journal of Heuristics, Vol. 27, No. 4 Cross Ref Approximation in (Poly-) Logarithmic Space5 April 2021 | Algorithmica, Vol. 83, No. 7 Cross Ref Multistage graph problems on a global budgetTheoretical Computer Science, Vol. 868 Cross Ref Compactors for parameterized counting problemsComputer Science Review, Vol. 39 Cross Ref Clique, Independent Set, and Vertex Cover12 October 2021 Cross Ref Parameterized Complexity of Finding Subgraphs with Hereditary Properties on Hereditary Graph Classes9 September 2021 Cross Ref Parameterized algorithms of fundamental NP-hard problems: a survey2 July 2020 | Human-centric Computing and Information Sciences, Vol. 10, No. 1 Cross Ref Why Is Maximum Clique Often Easy in Practice?Operations Research, Vol. 68, No. 6 Cross Ref Dynamic Parameterized Problems and AlgorithmsACM Transactions on Algorithms, Vol. 16, No. 4 Cross Ref A Brief Survey of Fixed-Parameter Parallelism14 August 2020 | Algorithms, Vol. 13, No. 8 Cross Ref A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms19 June 2020 | Algorithms, Vol. 13, No. 6 Cross Ref Diminishable parameterized problems and strict polynomial kernelizationComputability, Vol. 9, No. 1 Cross Ref Crossing Paths with Hans Bodlaender: A Personal View on Cross-Composition for Sparsification Lower Bounds20 April 2020 Cross Ref On width measures and topological problems on semi-complete digraphsJournal of Combinatorial Theory, Series B, Vol. 138 Cross Ref Cognition and Intractability18 April 2019 | , Vol. 10 Cross Ref On directed covering and domination problemsDiscrete Applied Mathematics, Vol. 259 Cross Ref Rank Vertex Cover as a Natural Problem for Algebraic CompressionSyed M. 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