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Sorting, linear time and the satisfiability problem

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Abstract

Let DLIN denote the class of problems that are computable withinO(n/logn) uniform time on a RAM which can read its input (of lengthn) in blocks and which only uses integersO(n/logn). We prove that the sorting problem belongs to DLIN and formulate the Linear Time Thesis: “Each problem computable by a linear time bounded algorithm (in an intuitive sense) belongs to DLIN”. We also study the subclass DTIMESORT(n) of problems computable within linear time on a Turing machine using in addition a fixed number of sortings, and we show how the reductions of this class, so-called sort-lin reductions, are useful to classify NP problems: e.g. there is a sort-lin reduction from SAT to 3-SAT (using no sorting) and a sort-lin reduction from the NP-complete graph problem KERNEL to SAT (but we do not know of any similar reduction in DTIME(n)). Similarly, problem 2-SAT is linearly equivalent to the problem of Horn renaming (via sort-lin reductions). Finally, SAT is compared with many other combinatorial problems. A problemA is SAT-easy (resp. SAT-hard) if there is a sort-lin reduction fromA to SAT (resp. SAT toA). A SAT-equivalent problem is a problem both SAT-easy and SAT-hard. It is shown that the class of SAT-easy (resp. SAT-equivalent) problems is very large and that its generalization to so-called special clauses or, more generally, regular clauses, does not enlarge it. Moreover, we justify our opinion that problem SAT is, in some sense, a minimal NP-complete problem.

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References

  1. A.V. Aho, J.E. Hopcroft and J.D. Ullman,The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, 1974).

    Google Scholar 

  2. A.V. Aho, J.E. Hopcroft and J.D. Ullman,Data Structures and Algorithms (Addison-Wesley, Reading, 1983).

    Google Scholar 

  3. D. Angluin and L. Valiant, Fast probabilistic algorithms for Hamiltonian circuits and matchings, J. Comput. Syst. Sci. 18(1979)155–193.

    Google Scholar 

  4. B. Apsvall, Recognizing disguised NR(1) instances of the satisfiability problem, J. Algorithms 1(1980)97–103.

    Google Scholar 

  5. A.M. Ben-Amram and Z. Galil, On pointers versus addresses, J. ACM 39(1992)617–648.

    Google Scholar 

  6. C. Berge,Graphs and Hypergraphs (North-Holland, Amsterdam, 1973).

    Google Scholar 

  7. K. Berman, J. Franco and J. Schlipf, Unique satisfiability for Horn sets can be solved in nearly linear time, Technical Report CS-TR-92-5, Comput. Sci. Dept., Univ. Cincinnati, OH (1992).

    Google Scholar 

  8. A. Bertoni, G. Mauri and N. Sabadini, Simulations among classes of random access machines and equivalence among numbers succinctly represented, Ann. Discr. Math. 25(1985)65–90.

    Google Scholar 

  9. K.J. Compton and C.W. Henson, A uniform method for proving lower bounds on the computational complexity of logical theories, Ann. Pure Appl. Logic 48(1990)1–79.

    Google Scholar 

  10. S.A. Cook, The complexity of theorem proving procedures,3rd STOC (1971) pp. 151–158.

  11. A. Cook, Short propositional formulas represent nondeterministic computations, Inf. Proc. Lett. (1987/88) 269–270.

    Google Scholar 

  12. S.A. Cook and R.A. Reckhow, Time-bounded random access machines, J. Comput. Syst. Sci. 7(1973)354–375.

    Google Scholar 

  13. T.H. Cormen, C.E. Leiserson and R.L. Rivest,Introduction to Algorithms (MIT Press/McGraw-Hill, 1991).

  14. N. Creignou, The class of problems that are linearly equivalent to satisfiability or a uniform method for proving NP-completeness,Proc. CSL'92, Lecture Notes in Computer Science 702 (1993) pp. 115–133; Theor. Comput. Sci. 145(1995)111–145.

    Google Scholar 

  15. N. Creignou, Exact complexity of problems of incompletely specified automata, this volume, Ann. of Math. and AI 16(1996).

  16. N. Creignou, Temps linéaire et problèmes NP-complet, Thèse de doctorat, Univ. de Caen, France (1993).

    Google Scholar 

  17. A.K. Dewdney, Linear time transformations between combinatorial problems, Int. J. Comput. Math. 11(1982)91–110.

    Google Scholar 

  18. W.F. Dowling and J.H. Gallier, Linear-time algorithms for testing the satisfiability of propositional Horn formulas, J. Logic Progr. 3(1984)267–284.

    Google Scholar 

  19. R. Fagin, Generalized first-order spectra and polynomial-time recognizable sets, in:Compl. of Comp., Proc. Symp., New York (1973), SIAM-AMS Proc. 7(1974)43–73.

    Google Scholar 

  20. C. Froidevaux, M.C. Gaudel and M. Soria,Types de Données et Algorithmes (McGraw-Hill).

  21. M.R. Garey and D.S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979).

    Google Scholar 

  22. E. Graedel, on the notion of linear time computability,Proc. 3rd Italian Conf. on Theor. Comput. Sci. (World Scientific, 1989) pp. 323–334, also in Int. J. Found. Comput. Sci. 1(1990)295–307.

  23. E. Grandjean, A natural NP-complete problem with a nontrivial lower bound, SIAM J. Comput. 17(1988)786–809.

    Google Scholar 

  24. E. Grandjean, A nontrivial lower bound for an NP problem on automata, SIAM J. Comput. 19(1990)438–451.

    Google Scholar 

  25. E. Grandjean, Invariance properties of RAMs and linear time, Comput. Complexity 4(1994)62–106.

    Google Scholar 

  26. E. Grandjean, Linear time algorithms and NP-complete problems, SIAM J. Comput. 23(1994)573–597.

    Google Scholar 

  27. E. Grandjean and F. Olive, Monadic logical definability of NP-complete problems, to appear inProc. CSL'94, Lecture Notes Comput. Sci.

  28. E. Grandjean and J.M. Robson, RAM with compact memory: A realistic and robust model of computations,Proc. CSL'90, Lecture Notes Comput. Sci. 533(1991) pp. 195–233.

    Google Scholar 

  29. Y. Gurevich and S. Shelah, Nearly linear time, Lecture Notes Comput. Sci. 363 (Springer, 1989) pp. 108–118.

    Google Scholar 

  30. J. Hartmanis and J. Simon, On the power of multiplication in random access machines,15th IEEE Symp. on Switching and Automata Theory (1974) pp. 13–23.

  31. J.J. Hébrard, A linear algorithm for renaming a set of clauses as a Horn set, Theor. Comput. Sci. 124(1994)343–350.

    Google Scholar 

  32. J.J. Hébrard, Une mise en oeuvre linéaire de la résolution unitaire, Technical Report, Univ. de Caen, Cahiers du LIUC 92-10 (1992).

  33. J.J. Hébrard, Unique Horn renaming and unique 2-satisfiability, Inf. Proc. Lett. 54(1995)235–239.

    Google Scholar 

  34. J.E. Hopcroft, W. Paul and L. Valiant, On time versus space and related problems, J. ACM 24(1977)332–337.

    Google Scholar 

  35. J.E. Hopcroft and J.D. Ullman,Introduction to Automata Theory, Languages and Computation (Addison-Wesley, Reading, 1979).

    Google Scholar 

  36. H.B. Hunt and R.E. Stearns, The complexity of very simple Boolean formulas with applications, SIAM J. Comput. 19(1990)44–70.

    Google Scholar 

  37. N. Immerman, Nondeterministic space is closed under complementation, SIAM J. Comput. 17(1988)935–938.

    Google Scholar 

  38. A. Itai and J.A. Makowsky, On the complexity of Herbrand's theorem. Technical Report 243, Computer Science Dept., Technion, Haifa, Israel (1982).

    Google Scholar 

  39. A. Itai and J.A. Makowsky, Unification as a complexity measure for logic programming, Technical Report 301, Computer Science Dept., Technion, Haifa, Israel (1993), in revised form in J. Logic. Progr. 4(1987)105–117.

    Google Scholar 

  40. N.D. Jones, Space-bounded reducibility among combinatorial problems, J. Comput. Syst. Sci. 11(1975)68–85.

    Google Scholar 

  41. N.D. Jones and W.T. Laaser, Complete problems for deterministic polynomial time, Theor. Comput. Sci. 3(1977)105–117.

    Google Scholar 

  42. N.D. Jones, Y.E. Lien and W.T. Laaser, New problems complete for nondeterministic log space, Math. Syst. Th. 10(1976)1–17.

    Google Scholar 

  43. R.M. Karp, Reducibility among combinatroial problems,IBM Symp. 1972, Complexity of Computer Computations (Plenum Press, New York, 1972).

    Google Scholar 

  44. J. Katajainen, J. van Leuwen and M. Penttonen, Fast simulation of Turing machines by random access machines, SIAM J. Comput. 17(1988)77–88.

    Google Scholar 

  45. D.E. Knuth,The Art of Programming, Vols. 1, 2, 3 (Addison-Wesley, Reading, 1968, 1969, 1973).

    Google Scholar 

  46. H.R. Lewis, Renaming a set of clauses as a Horn set, J. ACM 25(1978)134–135.

    Google Scholar 

  47. M.C. Loui and D.R. Luginbuhl, The complexity of on-line simulations between multidimensional Turing machines and random access machines, Math. Syst. Th. 25(1992)293–308.

    Google Scholar 

  48. K. Melhorn,Data Structures and Algorithms, Vols. 1–3 (Springer, 1984).

  49. M. Minoux, LTUR: A simplified linear-time resolution algorithm for Horn formulae and computer implementation, Inform. Proc. Lett. 29(1988)1–12.

    Google Scholar 

  50. B. Monien, About the derivation languages of grammars and machines,4th ICALP, Lecture Notes Comput. Sci 52 (Springer, 1977) pp. 337–351.

    Google Scholar 

  51. W. Paul, N. Pippenger, E. Szemeredi and W.T. Trotter, On determism versus non-determinism and related problems,24th Symp. Found. of Comput. Sci. (1983) pp. 429–438.

  52. S. Ranaivoson, Ph.D. Thesis, Université de Caen, France (1991).

    Google Scholar 

  53. S. Ranaivoson, Nontrivial lower bounds for some NP-complete problems on directed graphs,CSL'90, Lecture Notes Comput. Sci. 533 (1991) pp. 318–339.

    Google Scholar 

  54. J.M. Robson, Random access machines and multi-dimensional memories, Inform. Proc. Lett. 34(1990)265–266.

    Google Scholar 

  55. J.M. Robson, AnO(T logT) reduction from RAM computations to satisfiability, Theor. Comput. Sci. 82(1991)141–149.

    Google Scholar 

  56. J.M. Robson, Subexponential algorithms for some NP-complete problems, unpublished manuscript (1985).

  57. T.J. Schaeffer, The complexity of satisfiability problems,Proc. 10th ACM Symp. on Theory of Computing (1978) pp. 216–226.

  58. C.P. Schnorr, Satisfiability is quasilinear complete in NQL, J. ACM 25(1978)136–145.

    Google Scholar 

  59. A. Schoenhage, Storage modifications machines, SIAM J. Comput. 9(1980)490–508.

    Google Scholar 

  60. A. Schoenhage, A nonlinear lower bound for random access machines under logarithmic cost, J. ACM 35(1988)748–754.

    Google Scholar 

  61. R. Sedgewick,Algorithms in C++ (Addison-Wesley, Reading, 1992).

    Google Scholar 

  62. S.O. Slissenko, Complexity problems of theory of computation, Russ. Math. Surveys 36(1981)23–125.

    Google Scholar 

  63. C. Slot and P. van Emde Boas, The problem of space invariance for sequential machines, Inform. Comput. 77(1988)93–122.

    Google Scholar 

  64. R.E. Stearns and H.B. Hunt III, Power indices and easier hard problems, Math. Syst. Th. 23(1990)209–225.

    Google Scholar 

  65. R. Szelepcsenyi, The method of forced enumeration for nondeterministic automata, Acta Informatica 26(1988)279–284.

    Google Scholar 

  66. R.E. Tarjan, Depth-first search and linear time algorithms, SIAM J. Comput. 1(1972)146–160.

    Google Scholar 

  67. P. van Emde Boas, Machine models and simulations,Handbook of Theoretical Computer Science, Vol. A, ed. J. van Leuwen (Elsevier/MIT Press, 1990) Chap. 1, pp. (1–66).

    Google Scholar 

  68. J. van Leuwen, Graph algorithms,Handbook of Theoretical Computer Science, Vol. A, ed. J. van Leuwen (Elsevier/MIT Press, 1990) Chap. 10, pp. 525–631.

    Google Scholar 

  69. K. Wagner and G. Wechsung,Computational Complexity (Reidel, Berlin, 1986).

    Google Scholar 

  70. J. Wiedermann, Deterministic and nondeterministic simulation of the RAM by the Turing machine,Proc. IFIP Congress 83, ed. R.E.A. Mason (North-Holland, Amsterdam, 1983) pp. 163–168.

    Google Scholar 

  71. J. Wiedermann, Normalizing and accelerating RAM computations and the problem of reasonable space measures, Technical Report OPS-3, Dept. of Programming Systems, Bratislava, Czechoslovakia (1990), andProc. 17th ICALP, Lecture Notes Comput Sci. 443 (Springer, 1990).

    Google Scholar 

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  1. GREYC, Université de Caen, 14032, Caen Cedex, France

    Etienne Grandjean

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Grandjean, E. Sorting, linear time and the satisfiability problem. Ann Math Artif Intell 16, 183–236 (1996). https://doi.org/10.1007/BF02127798

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