We study several multi-criteria undirected network design problems with node costs and lengths with all problems related to the node costs Multicommodity Buy at Bulk (mbb) problem in which we are given a graph G = (V,E), demands {d st : s,t ∈ V}, and a family {c v : v ∈ V} of subadditive cost functions. For every s,t ∈ V we seek to send d st flow units from s to t on a single path, so that ∑  v c v (f v ) is minimized, where f v the total amount of flow through v. In the Multicommodity Cost-Distance (mcd) problem we are also given lengths {ℓ(v):v ∈ V}, and seek a subgraph H of G that minimizes c(H) + ∑  s,t ∈ V d st ·ℓ H (s,t), where ℓ H (s,t) is the minimum ℓ-length of an st-path in H. The approximation for these two problems is equivalent up to a factor arbitrarily close to 2. We give an O(log3 n)-approximation algorithm for both problems for the case of demands polynomial in n. The previously best known approximation ratio for these problems was O(log4 n) [Chekuri et al., FOCS 2006] and [Chekuri et al., SODA 2007]. This technique seems quite robust and was already used in order to improve the ratio of Buy-at-bulk with protection (Antonakopoulos et al FOCS 2007) from log3 h to log2 h. See ?.

We also consider the Maximum Covering Tree (maxct) problem which is closely related to mbb: given a graph G = (V,E), costs {c(v):v ∈ V}, profits {p(v):v ∈ V}, and a bound C, find a subtree T of G with c(T) ≤ C and p(T) maximum. The best known approximation algorithm for maxct [Moss and Rabani, STOC 2001] computes a tree T with c(T) ≤ 2C and p(T) = Ω(opt/logn). We provide the first nontrivial lower bound and in fact provide a bicriteria lower bound on approximating this problem (which is stronger than the usual lower bound) by showing that the problem admits no better than Ω(1/(loglogn)) approximation assuming \(\mbox{NP}\not\subseteq \mbox{Quasi(P)}\) even if the algorithm is allowed to violate the budget by any universal constant ρ. This disproves a conjecture of [Moss and Rabani, STOC 2001].

Another related to mbb problem is the Shallow Light Steiner Tree (slst) problem, in which we are given a graph G = (V,E), costs {c(v):v ∈ V}, lengths {ℓ(v):v ∈ V}, a set U ⊆ V of terminals, and a bound L. The goal is to find a subtree T of G containing U with diam (T) ≤ L and c(T) minimum. We give an algorithm that computes a tree T with c(T) = O(log2 n) ·opt and diam (T) = O(logn) ·L. Previously, a polylogarithmic bicriteria approximation was known only for the case of edge costs and edge lengths.


Network design Node costs Multicommodity Buy at Bulk Covering tree Approximation algorithm Hardness of approximation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andrews, M.: Hardness of buy-at-bulk network design. In: Proc. FOCS, pp. 115–124 (2004)Google Scholar
  2. 2.
    Andrews, M., Zhang, L.: Approximation algorithms for access network design. Algorithmica 34(2), 197–215 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Antonakopoulos, S., Chekuri, C., Shepherd, F.B., Zhang, L.: Buy-at-bulk network design with protection. In: FOCS, pp. 634–644 (2007)Google Scholar
  4. 4.
    Charikar, M., Karagiozova, A.: On non-uniform multicommodity buy-at-bulk network design. In: Proc. STOC, pp. 176–182 (2005)Google Scholar
  5. 5.
    Chekuri, C., Hajiaghayi, M.T., Kortsarz, G., Salavatipour, M.R.: Approximation algorithms for non-uniform buy-at-bulk network design. In: Proc. FOCS, pp. 677–686 (2006)Google Scholar
  6. 6.
    Chekuri, C., Hajiaghayi, M.T., Kortsarz, G., Salavatipour, M.R.: Approximation algorithms for node-weighted buy-at-bulk network design. In: SODA, pp. 1265–1274 (2007)Google Scholar
  7. 7.
    Feige, U.: A threshold of for approximating set cover. J. ACM 45, 634–652 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Guha, S., Moss, A., Naor, J.S., Schieber, B.: Efficient recovery from power outage. In: Proc. STOC, pp. 574–582 (1999)Google Scholar
  9. 9.
    Hajiaghayi, M.T., Kortsarz, G., Salavatipour, M.R.: Approximating buy-at-bulk and shallow-light k-steiner trees. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds.) APPROX 2006. LNCS, vol. 4110, pp. 152–163. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Klein, C., Ravi, R.: A nearly best-possible approximation algorithm for node-weighted steiner trees. Journal of Algorithms 19(1), 104–115 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Marathe, M.V., Ravi, R., Sundaram, R., Ravi, S.S., Rosenkrantz, D.J., Hunt III., H.B.: Bicriteria network design problems. J. Algorithms 28(1), 142–171 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Moss, A., Rabani, Y.: Approximation algorithms for constrained node weighted Steiner tree problems. In: Proc. STOC, pp. 373–382 (2001)Google Scholar
  13. 13.
    Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proc. STOC, pp. 475–484 (1997)Google Scholar
  14. 14.
    Salman, F.S., Cheriyan, J., Ravi, R., Subramanian, S.: Approximating the single-sink link-installation problem in network design. SIAM J. on Optimization 11(3), 595–610 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Guy Kortsarz
    • 1
  • Zeev Nutov
    • 2
  1. 1.Rutgers UniversityCamdenUSA
  2. 2.The Open University of IsraelIsrael

Personalised recommendations