For two convex bodies, C and D, consider a packing S of n positive homothets of C contained in D. We estimate the total perimeter of the bodies in S, denoted per(S), in terms of n. When all homothets of C touch the boundary of the container D, we show that either per(S) = O(logn) or per(S) = O(1), depending on how C and D “fit together,” and these bounds are the best possible apart from the constant factors. Specifically, we establish an optimal bound per(S) = O(logn) unless D is a convex polygon and every side of D is parallel to a corresponding segment on the boundary of C (for short, D is parallel to C). When D is parallel to C but the homothets of C may lie anywhere in D, we show that per(S) = O((1 + esc(S)) logn/loglogn), where esc(S) denotes the total distance of the bodies in S from the boundary of D. Apart from the constant factor, this bound is also the best possible.


Convex body perimeter maximum independent set homothet traveling salesman approximation algorithm 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Brass, P., Moser, W.O.J., Pach, J.: Research Problems in Discrete Geometry. Springer (2005)Google Scholar
  3. 3.
    Bern, M., Eppstein, D.: Approximation algorithms for geometric problems. In: Approximation Algorithms for NP-hard Problems, pp. 296–345. PWS (1997)Google Scholar
  4. 4.
    de Berg, M., Gudmundsson, J., Katz, M.J., Levcopoulos, C., Overmars, M.H., van der Stappen, A.F.: TSP with neighborhoods of varying size. J. Algorithms 57(1), 22–36 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Dumitrescu, A., Mitchell, J.S.B.: Approximation algorithms for TSP with neighborhoods in the plane. J. Algorithms 48(1), 135–159 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Dumitrescu, A., Tóth, C.D.: Minimum weight convex Steiner partitions. Algorithmica 60(3), 627–652 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Dumitrescu, A., Tóth, C.D.: The traveling salesman problem for lines, balls and planes, in. In: Proc. 24th SODA, pp. 828–843. SIAM (2013)Google Scholar
  8. 8.
    Levcopoulos, C., Lingas, A.: Bounds on the length of convex partitions of polygons. In: Joseph, M., Shyamasundar, R.K. (eds.) FSTTCS 1984. LNCS, vol. 181, pp. 279–295. Springer, Heidelberg (1984)CrossRefGoogle Scholar
  9. 9.
    Mata, C., Mitchell, J.S.B.: Approximation algorithms for geometric tour and network design problems. In: Proc. 11th SOCG, pp. 360–369. ACM (1995)Google Scholar
  10. 10.
    Mitchell, J.S.B.: A constant-factor approximation algorithm for TSP with pairwise-disjoint connected neighborhoods in the plane. In: Proc. 26th SOCG, pp. 183–191. ACM (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • Csaba D. Tóth
    • 2
    • 3
  1. 1.Department of Computer ScienceUniversity of Wisconsin–MilwaukeeUSA
  2. 2.Department of MathematicsCalifornia State UniversityNorthridgeUSA
  3. 3.Department of Mathematics and StatisticsUniversity of CalgaryCanada

Personalised recommendations