Linear Time Algorithms and NP-Complete Problems

Abstract

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 ${\text{NLINEAR}}$ complete.