Logical definability of NP-optimisation problems with monadic auxiliary predicates

  • Clemens Lautemann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 702)


Given a first-order formula ϕ with predicate symbols e1...el, so,...,sr, an NP-optimisation problem on <e1,...,el>-structures can be defined as follows: for every <e1,...,el>-structure G, a sequence <S0,...,Sr> of relations on G is a feasible solution iff <G, S0,....Sr> satisfies ϕ, and the value of such a solution is defined to be ¦S0¦. In a strong sense, every polynomially bounded NP-optimisation problem has such a representation, however, it is shown here that this is no longer true if the predicates s1, ...,sr are restricted to be monadic. The result is proved by an Ehrenfeucht-Fraïssé game and remains true in several more general situations.


Chromatic Number Predicate Symbol Winning Strategy Clique Number Chromatic Index 
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  1. [AF90]
    Miklos Ajtai, Ronald Fagin, Reachability is harder for directed than for undirected finite graphs. JSL 55, pp. 113–150.Google Scholar
  2. [ALMSS]
    Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, Mario Szegedy, Proof verification and intractability of approximation problems (preliminary version). Preprint, Computer Science Division, University of California, Berkeley, April 1992.Google Scholar
  3. [ALS91]
    Stefan Arnborg, Jens Lagergren, Detlev Seese, Easy problems for tree decomposable graphs. J. of Algorithms, 12, pp.308–340.Google Scholar
  4. [BJY90]
    Danilo Bruschi, Deborah Joseph, Paul Young, A structural overview of NP optimization problems. Rapporto Interno 75/90, Dipartimento di Scienze dell'Informazione, Université degli Studi di Milano.Google Scholar
  5. [Co90]
    Bruno Courcelle, On the expression of monadic second-order graph properties without quantifications over sets of edges. Proc. 5th Ann. IEEE Symposium on Logic in Computer Science, pp. 190–196.Google Scholar
  6. [dR87]
    Michel de Rougemont, Second-order and inductive definability on finite structures. Zeitschrift f. math. Logik und Grundlagen d. Math. 33, pp. 47–63.Google Scholar
  7. [Fa74]
    Ronald Fagin, Generalized first-order spectra and polynomial-time recognizable sets. In Richard Karp, (Ed.):”Complexity of Computation”, SIAM-AMS Proc. 7, pp. 43–73.Google Scholar
  8. [Fa75]
    Ronald Fagin, Monadic generalized spectra. Zeitschr. f. math. Logik und Grundlagen d. Math. 21, pp. 89–96.Google Scholar
  9. [GJ79]
    Michael R. Garey, David S. Johnson, Computers and Intractability. Freeman, N.Y.Google Scholar
  10. [Ih90]
    Edmund Ihler, Approximation and existential second-order logic. Bericht 26, Institut für Informatik, Universität Freiburg.Google Scholar
  11. [Im89]
    Neil Immerman, Descriptive and computational complexity. In: Juris Hartmanis (Ed.):”Computational Complexity Theory”, Proc. AMS Symp. Appl. Math. 38, pp. 75–91.Google Scholar
  12. [Kn92]
    Dieter Knobloch, Zur Komplexität kombinatorischer Optimierungsprobleme. Diplomarbeit, FB Mathematik, Johannes Gutenberg-Universität Mainz.Google Scholar
  13. [KT90]
    Phokion G. Kolaitis, Madhukar N. Thakur, Logical definability of NP optimization problems. Technical Report UCSC-CRL-90-48, Computer Research Laboratory, University of California, Santa Cruz.Google Scholar
  14. [KT91]
    Phokion G. Kolaitis, Madhukar N. Thakur, Approximation properties of NP minimization classes. Proc. 7th Structure in Complexity Theory Conference, pp. 353–366.Google Scholar
  15. [Ly82]
    James F. Lynch, Complexity Classes and Theories of Finite Models. Math. Syst. Theory 15, pp. 127–144.Google Scholar
  16. [PR90]
    Alessandro Panconesi, Desh Ranjan, Quantifiers and approximation (extended abstract). Proc. 22nd ACM STOC, pp. 446–456.Google Scholar
  17. [PY91]
    Christos H. Papadimitriou, Mihalis Yannakakis, Optimization, approximation, and complexity classes. JCSS 43, pp. 425–440.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Clemens Lautemann
    • 1
  1. 1.Institut für InformatikJohannes Gutenberg-Universität MainzDeutschland

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