Abstract
Let G = (V,E) be a connected multigraph, whose edges are associated with labels specified by an integer-valued function \({\mathcal{L}} : E \rightarrow \mathbb{N}\). In addition, each label ℓ ∈ ℕ to which at least one edge is mapped has a non-negative cost c( ℓ). The minimum label spanning tree problem (MinLST) asks to find a spanning tree in G that minimizes the overall cost of the labels used by its edges. Equivalently, we aim at finding a minimum cost subset of labels I ⊆ ℕ such that the edge set \(\{ e \in E : {\mathcal {L}}( e ) \in I \}\) forms a connected subgraph spanning all vertices. Similarly, in the minimum label s -t path problem (MinLP) the goal is to identify an s-t path minimizing the combined cost of its labels, where s and t are provided as part of the input.
The main contributions of this paper are improved approximation algorithms and hardness results for MinLST and MinLP. As a secondary objective, we make a concentrated effort to relate the algorithmic methods utilized in approximating these problems to a number of well-known techniques, originally studied in the context of integer covering.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ageev, A.A., Sviridenko, M.: Pipage rounding: A new method of constructing algorithms with proven performance guarantee. Journal of Combinatorial Optimization 8(3), 307–328 (2004)
Arora, S.: Personal communication (November 2005)
Arora, S., Sudan, M.: Improved low-degree testing and its applications. Combinatorica 23(3), 365–426 (2003)
Avidor, A., Zwick, U.: Approximating MIN 2-SAT and MIN 3-SAT. Theory of Computing Systems 38(3), 329–345 (2005)
Bellare, M., Goldwasser, S., Lund, C., Russell, A.: Efficient probabilistically checkable proofs and applications to approximations. In: Proceedings of the 25th Annual ACM Symposium on Theory of Computing, pp. 294–304 (1993)
Broersma, H., Li, X., Woeginger, G., Zhang, S.: Paths and cycles in colored graphs. Australasian Journal on Combinatorics 31, 299–311 (2005)
Brüggemann, T., Monnot, J., Woeginger, G.J.: Local search for the minimum label spanning tree problem with bounded color classes. Operations Research Letters 31(3), 195–201 (2003)
Carr, R.D., Doddi, S., Konjevod, G., Marathe, M.V.: On the red-blue set cover problem. In: Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 345–353 (2000)
Chang, R.-S., Leu, S.-J.: The minimum labeling spanning trees. Information Processing Letters 63(5), 277–282 (1997)
Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Annals of Mathematics 162(1), 439–486 (2005)
Goldschmidt, O., Hochbaum, D.S., Yu, G.: A modified greedy heuristic for the set covering problem with improved worst case bound. Information Processing Letters 48(6), 305–310 (1993)
Hassin, R.: Approximation schemes for the restricted shortest path problem. Mathematics of Operations Research 17(1), 36–42 (1992)
Hassin, R., Monnot, J., Segev, D.: Approximation algorithms and hardness results for labeled connectivity problems (2006), available at http://www.math.tau.ac.il/~segevd
Karger, D.R., Motwani, R., Ramkumar, G.D.S.: On approximating the longest path in a graph. Algorithmica 18(1), 82–98 (1997)
Khuller, S., Moss, A., Naor, J.: The budgeted maximum coverage problem. Information Processing Letters 70(1), 39–45 (1999)
Krumke, S.O., Wirth, H.-C.: On the minimum label spanning tree problem. Information Processing Letters 66(2), 81–85 (1998)
Lorenz, D.H., Raz, D.: A simple efficient approximation scheme for the restricted shortest path problem. Operations Research Letters 28(5), 213–219 (2001)
Marathe, M.V., Ravi, S.S.: On approximation algorithms for the minimum satisfiability problem. Information Processing Letters 58(1), 23–29 (1996)
Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pp. 475–484 (1997)
Srinivasan, A.: Distributions on level-sets with applications to approximation algorithms. In: Proceedings of the 42nd Annual Symposium on Foundations of Computer Science, pp. 588–597 (2001)
Wan, Y., Chen, G., Xu, Y.: A note on the minimum label spanning tree. Information Processing Letters 84(2), 99–101 (2002)
Wirth, H.-C.: Multicriteria Approximation of Network Design and Network Upgrade Problems. PhD thesis, Department of Computer Science, Würzburg University (2001)
Xiong, Y., Golden, B., Wasil, E.: Worst-case behavior of the MVCA heuristic for the minimum labeling spanning tree problem. Operations Research Letters 33(1), 77–80 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hassin, R., Monnot, J., Segev, D. (2006). Approximation Algorithms and Hardness Results for Labeled Connectivity Problems. In: Královič, R., Urzyczyn, P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11821069_42
Download citation
DOI: https://doi.org/10.1007/11821069_42
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37791-7
Online ISBN: 978-3-540-37793-1
eBook Packages: Computer ScienceComputer Science (R0)