Linear Time Algorithms and NP-Complete Problems

This paper defines and studies a computational model (a random access machine with powerful input/output instructions), and shows that the classes ${\text{DLINEAR}}$ and ${\text{NLINEAR}}$ of problems computable in deterministic (respectively, nondeterministic) linear time in this model of computation are robust and powerful. In particular, ${\text{DLINEAR}}$ includes most of the concrete problems commonly regarded as computable in linear time (such as graph problems, topological sorting, strong connectivity, etc.). Most combinatorial NP-complete problems are in ${\text{NLINEAR}}$. The interest of ${\text{NLINEAR}}$ class is enhanced by the fact that some natural NP-complete problems, for example, “reduction of incompletely specified automata” $({\text{RISA}})$, are ${\text{NLINEAR}}$-complete (consequently, ${\text{NLINEAR}} \ne {\text{ DLINEAR}}$ if and only if ${\text{RISA}} \notin {\text{DLINEAR}}$). This notion strengthens NP-completeness, as this paper argues that propositional satisfiability is not...

[1]  Arnold Schönhage.  Storage Modification Machines , 1980, SIAM J. Comput..

[2]  Jirí Wiedermann.  Deterministic and Nondeterministic Simulation of the RAM by the Turing Machine , 1983, IFIP Congress.

[3]  Endre Szemerédi,et al.  On determinism versus non-determinism and related problems , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[4]  Michel Minoux,et al.  LTUR: A Simplified Linear-Time Unit Resolution Algorithm for Horn Formulae and Computer Implementation , 1988, Inf. Process. Lett..

[5]  Prof. Dr. Kurt Mehlhorn,et al.  Data Structures and Algorithms 3 , 2012, EATCS Monographs on Theoretical Computer Science.

[6]  Etienne Grandjean,et al.  A Nontrivial Lower Bound for an NP Problem on Automata , 1990, SIAM J. Comput..

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

[8]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[9]  Claus-Peter Schnorr.  Satisfiability Is Quasilinear Complete in NQL , 1978, JACM.

[10]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[11]  C. Ward Henson,et al.  A Uniform Method for Proving Lower Bounds on the Computational Complexity of Logical Theories , 1990, Ann. Pure Appl. Log..

[12]  Leslie G. Valiant,et al.  Fast probabilistic algorithms for hamiltonian circuits and matchings , 1977, STOC '77.

[13]  Jean H. Gallier,et al.  Linear-Time Algorithms for Testing the Satisfiability of Propositional Horn Formulae , 1984, J. Log. Program..

[14]  Charles P. Pfleeger,et al.  State Reduction in Incompletely Specified Finite-State Machines , 1973, IEEE Transactions on Computers.

[15]  A. K. Dewdney.  Linear time transformations between combinatorial problems , 1982 .

[16]  H. Scholz.  Review: Y. Bar-Hillel, Bolzano's Definition of Analytic Propositions , 1952 .

[17]  Solomampionona Ranaivoson.  Nontrivial Lower Bounds for some NP-Problems on Directed Graphs , 1990, CSL.

[18]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[19]  John Michael Robson.  Random Access Machines with Multi-Dimensional Memories , 1990, Inf. Process. Lett..

[20]  Arnold Schönhage.  A nonlinear lower bound for random-access machines under logarithmic cost , 1988, JACM.

[21]  Etienne Grandjean.  A Natural NP-Complete Problem with a Nontrivial Lower Nound , 1988, SIAM J. Comput..

[22]  Juris Hartmanis,et al.  On the Power of Multiplication in Random Access Machines , 1974, SWAT.

[23]  Leslie G. Valiant,et al.  On time versus space and related problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[24]  Saharon Shelah,et al.  Nearly Linear Time , 1989, Logic at Botik.

[25]  Peter van Emde Boas,et al.  The Problem of Space Invariance for Sequential Machines , 1988, Inf. Comput..

[26]  Leslie G. Valiant,et al.  On Time Versus Space , 1977, JACM.

[27]  Andrew Chi-Chih Yao,et al.  Near-optimal time-space tradeoff for element distinctness , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[28]  Burkhard Monien.  About the Derivation Languages of Grammars and Machines , 1977, ICALP.

[29]  Stephen A. Cook,et al.  Short Propositional Formulas Represent Nondeterministic Computations , 1988, Inf. Process. Lett..

[30]  Erich Grädel.  On the Notion of Linear Time Computability , 1990, Int. J. Found. Comput. Sci..

[31]  Judy Goldsmith,et al.  Nondterminism Within P , 1991, STACS.

[32]  Harry B. Hunt,et al.  The Complexity of Very Simple Boolean Formulas with Applications , 1990, SIAM J. Comput..

[33]  Stephen A. Cook,et al.  Time Bounded Random Access Machines , 1973, J. Comput. Syst. Sci..

[34]  Jirí Wiedermann.  Normalizing and Accelerating RAM Computations and the Problem of Reasonable Space Measures , 1990, ICALP.

[35]  Jan van Leeuwen,et al.  Fast Simulation of Turing Machines by Random Access Machines , 1988, SIAM J. Comput..

[36]  John Michael Robson,et al.  RAM with Compact Memory: A Realistic and Robust Model of Computation , 1990, CSL.

[37]  John Michael Robson,et al.  An O (T log T) Reduction from RAM Computations to Satisfiability , 1991, Theor. Comput. Sci..

[38]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[39]  Giancarlo Mauri,et al.  Simulations Among Classes of Random Access Machines and Equivalence Among Numbers Succinctly Represented11A preliminary version of the first part of this paper has been published in the Proceedings of the ACM Symp. on theory of Computation, Milwaukee, 1991. , 1985 .