Relationship between density and deterministic complexity of MP-complete languages

  • Piotr Berman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 62)


Let Σ be an arbitrary alphabet.
denotes ɛ∪Σ∪...∪Σn. We say that a function t, t :
is f-sparse iff card
for every natural n. The main theorem of this paper establishes that if CLIQUE has some f-sparse translation into another set, which is calculable by a deterministic Turing machine in time bounded by f, then all the sets belonging to NP are calculable in time bounded by a function polynomially related to f. The proof is constructive and shows the way of constructing a proper algorithm. The simplest and most significant corollary says that if there is an NP-complete language over a single letter alphabet, then P=NP.


Polynomial Time Single Letter Partial Correctness Marked Vertex Proper Subgraph 
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    J.Hartmanis, L.Berman: On isomorphisms and density of NP and other complete sets. Proc. Eight ACM Sym. Theory of Computing, 30–40 (1976).Google Scholar
  2. [2]
    R.Book, C.Wrathhall, A.Selman and D.Dobkin: Inclusion Complete Tally Languages and Hartmanis-Berman Conjecture (1977).Google Scholar
  3. [3]
    A.V.Aho, J.E.Hopcroft and J.D.Ullman: The Design and Analysis of Computer Algorithms, Addison-Wesley Publishing Company, Readin, Mass., 1074.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1978

Authors and Affiliations

  • Piotr Berman
    • 1
  1. 1.Uniwersytet WarszawskiWarszawaPoland

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