Vector language: Simple description of hard instances

  • Miroslaw Kowaluk
  • Klaus W. Wagner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 452)


Different versions of vector languages are introduced as input languages for the succinct description of instances to combinatorial problems. For some of these languages we prove: (1) These languages are hard input languages, i.e. all popular non-trivial combinatorial problems have the maximum complexity blow-up if the instances are described by the language and (2) These languages are simpler than (i.e. there are simple compilers to) all other hard input languages investigated so far. To prove (1) we introduce different versions of vector-reducibilities which are restricted AC0 reducibilities. This investigation gives partial answers to the questions: How simple can hard instances to a combinatorial problem be ? How simple can the reductions between the most popular combinatorial problems be


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  1. [Ajt 83]
    Ajtai M., Σ11 formulae on finite structures, Annals of Pure and Applied Logic 24 (1983), 1–48.Google Scholar
  2. [BaImSt 88]
    Barrington D.A.M., Immerman N., Straubing H., On uniformity within NC1, Proc. 3rd Struct. in Complexity Theory Conf., IEEE (1988), 47–59.Google Scholar
  3. [BeOtWi 83]
    Bentley J.L., Ottmann T., Widmayer P., The complexity of manipulating hierarchically defined sets of rectangles, Adv. Comput. Res. 1 (1983), 127–158.Google Scholar
  4. [FuSaSi 81]
    Furst M., Saxe J.B., Sipser M., Parity circuits and the polynomial-time hierarchy, Proc. 22nd IEEE Symp. on Fundations of Computer Science (1981), 260–270; see also MST 17 (1984), 13–27.Google Scholar
  5. [GaJo 79]
    Garey M.R., Johnson D.S., Computers and Intractability. A Guide to the Theory of NP-Completeness, W.H.Freeman, San Francisco (1979).Google Scholar
  6. [GaWi 83]
    Galperin H., Wigderson A., Succinct representations of graphs, Inform. Control 56 (1983), 183–198.Google Scholar
  7. [KaWa 88]
    Karpinski, M., Wagner, K.W., The computational complexity of graph problems with succinct multigraph representation, ZOR 32 (1988), 201–211,.Google Scholar
  8. [Lad 75]
    Ladner R.E., The circuit value problem is logspace complete for P, SIGACT News 7 (1975), 18–20.Google Scholar
  9. [Len 82]
    Lengauer T., The complexity of compacting hierarchically specified layouts of integrated circuits, Proc. 23rd Ann. Symp. on Found. of Comp. Sci. (1982), 358–368.Google Scholar
  10. [LeWa 86]
    Lengauer T., Wagner K.W., The correlation between the complexities of the nonhierarchical and hierarchical versions of graph problems, Bericht Nr.33; Reihe Theoretische Informatik, Universität Paderborn (1986).Google Scholar
  11. [LeWn 88]
    Lengauer T., Wanke E., Efficient analysis of graph properties on context-free graph languages (extended abstract), Proc. of ICALP'88, Springer Lecture Notes in Computer Science (1988), 379–393.Google Scholar
  12. [Lyn 77]
    Lynch N.A., Logspace recognition and translation of parenthesis languages, J. Assoc. Comput. Mach. 24 (1977), 583–590.Google Scholar
  13. [PaYa 86]
    Papadimitriou C.H., Yannakakis M., A note on succinct representations of graphs, Inform. Control 71 (1986), 181–185.Google Scholar
  14. [StMe 73]
    Stockmeyer L.J., Meyer A.R., Word problems requiring exponential time, Proc. 5th ACM Symp. on Theory of Comp. (1973), 1–9.Google Scholar
  15. [Wag 86]
    Wagner K.W., The complexity of combinatorial problems with succinct input representation, Acta Informatica 23 (1986), 325–356.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Miroslaw Kowaluk
    • 1
  • Klaus W. Wagner
    • 1
  1. 1.Institut für InformatikUniversität Würzburg Am HublandWürzburg

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