Discrete & Computational Geometry

, Volume 40, Issue 3, pp 469–479 | Cite as

On the Hardness of Computing Intersection, Union and Minkowski Sum of Polytopes



For polytopes P 1,P 2⊂ℝ d , we consider the intersection P 1P 2, the convex hull of the union CH(P 1P 2), and the Minkowski sum P 1+P 2. For the Minkowski sum, we prove that enumerating the facets of P 1+P 2 is NP-hard if P 1 and P 2 are specified by facets, or if P 1 is specified by vertices and P 2 is a polyhedral cone specified by facets. For the intersection, we prove that computing the facets or the vertices of the intersection of two polytopes is NP-hard if one of them is given by vertices and the other by facets. Also, computing the vertices of the intersection of two polytopes given by vertices is shown to be NP-hard. Analogous results for computing the convex hull of the union of two polytopes follow from polar duality. All of the hardness results are established by showing that the appropriate decision version, for each of these problems, is NP-complete.


Minkowski addition Extended convex hull Polytope intersection Polytopes Turing reduction 


  1. 1.
    Avis, D., Bremner, D., Seidel, R.: How good are convex hull algorithms? Comput. Geom. 7, 265–301 (1997) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Balas, E.: On the convex hull of the union of certain polyhedra. Oper. Res. Lett. 7, 279–283 (1988) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Freund, R.M., Orlin, J.B.: On the complexity of four polyhedral set containment problems. Math. Program. 33(2), 139–145 (1985) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fukuda, K.: From the zonotope construction to the Minkowski addition of convex polytopes. J. Symb. Comput. 38(4), 1261–1272 (2004) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Fukuda, K., Liebling, T.M., Lutolf, C.: Extended convex hull. Comput. Geom. 20(1–2), 13–23 (2001) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Fukuda, K., Weibel, C.: Computing all faces of the Minkowski sum of \(\mathcal{V}\) -polytopes. In: Proceedings of the 17th Canadian Conference on Computational Geometry (2005) Google Scholar
  7. 7.
    Fukuda, K., Weibel, C.: f-vectors of Minkowski additions of convex polytopes. Discrete Comput. Geom. 37, 503–516 (2007) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gritzmann, P., Sturmfels, B.: Minkowski addition of polytopes: Computational complexity and applications to Grobner bases. SIAM J. Discrete Math. 6, 246–269 (1993) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Algorithms and Combinatorics, vol. 2. Springer, Berlin (1993). MATHGoogle Scholar
  10. 10.
    Grünbaum, B.: Convex Polytopes, 2nd edn. Graduate Texts in Mathematics, vol. 221. Springer, Berlin (2003). Prepared by V. Kaibel, V.L. Klee, G.M. Ziegler Google Scholar
  11. 11.
    Huber, B., Rambau, J., Santos, F.: The Cayley trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings. J. Eur. Math. Soc. 2(2), 179–198 (2000) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Khachiyan, L., Boros, E., Borys, K., Elbassioni, K.M., Gurvich, V.: Generating all vertices of a polyhedron is hard. Discrete Comput. Geom. 39(1–3), 174–190 (2008) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Khachiyan, L.G.: A polynomial algorithm in linear programming. Sov. Math. Dokl. 20, 191–194 (1979) MATHGoogle Scholar
  14. 14.
    Strumfels, B.: On the Newton polytope of the resultant. J. Algebr. Comb. 3(2), 207–236 (1994) CrossRefGoogle Scholar
  15. 15.
    Yap, C.K.: Fundamental Problems in Algorithmic Algebra. Oxford University Press, Oxford (2000) Google Scholar
  16. 16.
    Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, Berlin (1995) MATHGoogle Scholar

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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.FR InformatikUniversität des SaarlandesSaarbrückenGermany

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