Computational complexity of norm-maximization

Abstract

This paper discusses the problem of maximizing a quasiconvex functionφ over a convex polytopeP inn-space that is presented as the intersection of a finite number of halfspaces. The problem is known to beNP-hard (for variablen) whenφ is thep th power of the classicalp-norm. The present reexamination of the problem establishesNP-hardness for a wider class of functions, and for thep-norm it proves theNP-hardness of maximization overn-dimensionalparallelotopes that are centered at the origin or have a vertex there. This in turn implies theNP-hardness of {−1, 1}-maximization and {0, 1}-maximization of a positive definite quadratic form. On the “good” side, there is an efficient algorithm for maximizing the Euclidean norm over an arbitraryrectangular parallelotope.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    F. Barahona: A solvable case for quadratic 0–1 programming,Discrete Appl. Math.,13 (1986) 23–26.

    Article  Google Scholar 

  2. [2]

    A. Bielecki: Quelques remarques sur la note précédente,Ann. Univ. Mariae Curie-Sklodowska, Sect. A,8 (1954) 101–103.

    Google Scholar 

  3. [3]

    A. Bielecki, andK. Radiszewski: Sur les parallelepipedes inscrits dans les corps convexes,Ann. Univ. Mariae Curie-Sklodowska, Sect. A,8 (1954) 97–100.

    Google Scholar 

  4. [4]

    C. Christensen: Kvadrat inskrevet i konveks figur, Mat. Tidskrift B,1950 (1950) 22–26.

    Google Scholar 

  5. [5]

    S. A. Cook:The complexity of theorem-proving procedures,Proc. Third Ann. ACM Symp. on Theory of Computing, Association for Computing Machinery, New York,1971, 151–158.

    Google Scholar 

  6. [6]

    Y.Crama, P.Hansen, and B.Jaumard: The basic algorithm for pseudo-Boolean programming revisited, preprint, 1988.

  7. [7]

    M. E. Dyer: The complexity of vertex enumeration methods,Math. of Operations Res.,8 (1983) 381–402.

    Google Scholar 

  8. [8]

    R. M. Freund, andJ. B. Orlin: On the complexity of four polyhedral set containment problems,Math. Programming,33 (1985) 139–145.

    Article  Google Scholar 

  9. [9]

    M. R. Garey, andD. S. Johnson: Strong NP-completeness results: motivation, examples, and implications,J. Assoc. Comp. Math.,25 (1978) 499–508.

    Google Scholar 

  10. [10]

    M. R. Garey, andD. S. Johnson:Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, San Francisco,1979.

    Google Scholar 

  11. [11]

    P. Gritzmann, andV. Klee: On the 0–1 maximization of positive definite quadratic forms,Operations Research Proceedings 1988, Springer, Berlin,1989, 222–227.

    Google Scholar 

  12. [12]

    P.Gritzmann, and V.Klee: Computational complexity of inner and outerj-radii of polytopes in finite-dimensional normed spaces,in preparation.

  13. [13]

    B.Grünbaum: Measures of symmetry for convex sets, in Convexity (V. Klee, ed.),Amer. Math. Soc. Proc. Symp. Pure Math.,7 (1963) 233–270.

  14. [14]

    H. Hadwiger, D. G. Larman, andP. Mani: Hyperrhombs inscribed to convex bodies,J. Combinatorial Theory B,24 (1978) 290–293.

    Article  Google Scholar 

  15. [15]

    P. L. Hammer, andB. Simeone: Quasimonotone boolean functions and bistellar graphs,Ann. Discrete Math.,8 (1980) 107–119.

    Google Scholar 

  16. [16]

    P. Hansen, andB. Simeone: Unimodular functions,Discrete Appl. Math.,14 (1986) 269–281.

    Article  Google Scholar 

  17. [17]

    W. M. Hirsch, andA. J. Hoffman: Extreme varietes, concave functions, and the fixed charge problem.Comm. Pure Appl. Math.,14 (1961) 355–369.

    Google Scholar 

  18. [18]

    S. Kapoor, andP. Vaidya: Fast algorithms for convex quadratic programming and multicommodity flows,Proc. Eighteenth Ann. ACM Symp. on Theory of Computing, Association for Computing Machinery, New York,1986, 147–159.

    Google Scholar 

  19. [19]

    N. Karmarkar: A new polynomial-time algorithm for linear programming,Combinatorical,4 (1984) 373–397.

    Google Scholar 

  20. [20]

    R. M. Karp: Reducibility among combinatorial problems, inComplexity of Computer Computations (R. E. Miller and J. W. Thatcher, eds.), Plenum Press, New York,1972, 85–103.

    Google Scholar 

  21. [21]

    L. G. Khachian: Polynomial algorithms in linear programming,USSR Comp. Math. and Math. Phys.,20 (1980) 53–72.

    Article  Google Scholar 

  22. [22]

    M. Kojima, S. Mizuno, andA. Yoshishe: A polynomial-time algorithm for a class of linear complementarity problems,Math. Programming,44 (1989) 18–26.

    Article  Google Scholar 

  23. [23]

    H Konno: Maximization of a convex quadratic function over a hypercube,J. Oper. Res. Soc. Japan,23 (1980) 171–189.

    Google Scholar 

  24. [24]

    M. K. Kozlov, S. P. Tarasov, andL. G. Hacijan: Polynomial solvability of convex quadratic programming,Soviet Math. Doklady,20 (1979) 1108–1111.

    Google Scholar 

  25. [25]

    L. Lovász: Coverings and colorings of hypergraphs, inProceedings of the Fourth Southeastern Conference on Combinatorics, Graph Theory, and Computing, Utilitas Mathematica, Winnipeg,1973, 3–12.

    Google Scholar 

  26. [26]

    A. M. Macbeath: A compactness theorem for affine equivalence classes of convex regions,Canad. J. Math.,3 (1951) 54–61.

    Google Scholar 

  27. [27]

    K. Mahler: Ein Übertragungsprinzip für konvexe Körper,Casopis pro Pestovani Mat. a. Fys.,68 (1939) 93–102.

    Google Scholar 

  28. [28]

    O. L. Mangasarian, andT. -H. Shiau: A variable complexity norm maximization problem,SIAM J. Algebraic and Discrete Methods,7 (1986) 455–461.

    Google Scholar 

  29. [29]

    P. McMullen: The maximum number of faces of a convex polytope,Mathematika,17 (1970) 179–184.

    Google Scholar 

  30. [30]

    R. D. C. Monteiro, andI. Adler: Interior path following primal-dual algorithms — Part II: Convex quadratic programming,Math. Programming,44(1989) 43–66.

    Article  Google Scholar 

  31. [31]

    K. G. Murty, andS. N. Kabadi: Some NP-complete problems in quadratic and nonlinear programming,Math. Programming,39 (1987) 117–129.

    Google Scholar 

  32. [32]

    H. Naumann: Beliebige konvexe Polytope als Schnitte und Projektionen höherdimensionaler Würfel, Simplizes und Masspolytope,Math. Z.,65 (1956) 91–103.

    Article  Google Scholar 

  33. [33]

    P. M. Pardalos, andJ. B. Rosen: Constrained Global Optimization,Lecture notes in Comp. Sci. (G. Goos and J. Hartmanis, eds),268, Springer, Berlin, 1987.

    Google Scholar 

  34. [34]

    D. T. Pham: Algorithmes de calcul du maximum de formes quadratiques sur la boule unite de la forme du maximum,Numer Math.,45 (1984) 377–401.

    Article  Google Scholar 

  35. [35]

    J. C. Picard, andM. Queyranne: Selected applications of min cut in networks,INFOR,20 (1982) 395–422.

    Google Scholar 

  36. [36]

    C. Pucci: Sulla inscrivibilita di un ottaedro regolare in un insieme convesso limitato dell spazio ordinaria,Atti. Acad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.,8 (1956) 61–65.

    Google Scholar 

  37. [37]

    S. Sahni: Computationally related problems,SIAM J. Comput.,3 (1974) 262–279.

    Article  Google Scholar 

  38. [38]

    T. J. Schaefer: The complexity of satisfiability problems,Proc. Tenth Ann. ACM. Symp. on Theory of Computing, Association for Computing Machinery, New York,1978, 216–226.

    Google Scholar 

  39. [39]

    R. Seidel: Output-Size Sensitive Algorithms for Constructive Problems in Computational Geometry,Ph.D. Thesis, Department of Computer Science, Cornell University, Ithaca, N.Y.,1987.

    Google Scholar 

  40. [40]

    G. Swart: Finding the convex hull facet by facet,J. of Algorithms,6 (1985) 17–48.

    Article  Google Scholar 

  41. [41]

    Y. Ye, andE. Tse: An extension of Karmarkar's projectíve algorithm for convex quadratic programming,Math. Programming,44(1989) 157–179.

    Article  Google Scholar 

  42. [42]

    K. Zindler: Über konvexe Gebilde,Monatshefte für Math.,31 (1921) 25–57.

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

The authors are indebted to J. O'Rourke, P, Pardalos and R. Freund for useful references. The second and third authors are indebted to the Institute for Mathematics and its Applications in Minneapolis, where much of this paper was written: they acknowledge additional support from the Alexander von Humboldt Stiftung and the National Science Foundation, respectively.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bodlaender, H.L., Gritzmann, P., Klee, V. et al. Computational complexity of norm-maximization. Combinatorica 10, 203–225 (1990). https://doi.org/10.1007/BF02123011

Download citation

AMS subject classification (1980)

  • 90 C 30
  • 68 Q 15
  • 52 A 25
  • 90 C 20
  • 90 C 09
  • 52 A 20