Abstract
In this paper we study two variants of the problem of adding edges to a graph so as to reduce the resulting diameter. More precisely, given a graph G = (V,E), and two positive integers D and B, the Minimum-Cardinality Bounded-Diameter Edge Addition (MCBD) problem is to find a minimum cardinality set F of edges to be added to G in such a way that the diameter of G + F is less than or equal to D, while the Bounded-Cardinality Minimum-Diameter Edge Addition (BCMD) problem is to find a set F of B edges to be added to G in such a way that the diameter of G + F is minimized. Both problems are well known to be NP-hard, as well as approximable within O(logn logD) and 4 (up to an additive term of 2), respectively. In this paper, we improve these long-standing approximation ratios to O(logn) and to 2 (up to an additive term of 2), respectively. As a consequence, we close, in an asymptotic sense, the gap on the approximability of the MCBD problem, which was known to be not approximable within c logn, for some constant c > 0, unless P=NP. Remarkably, as we further show in the paper, our approximation ratio remains asymptotically tight even if we allow for a solution whose diameter is optimal up to a multiplicative factor approaching \(\frac{5}{3}\). On the other hand, on the positive side, we show that at most twice of the minimal number of additional edges suffices to get at most twice of the required diameter.
This work was partially supported by the PRIN 2008 research project COGENT (COmputational and GamE-theoretic aspects of uncoordinated NeTworks), funded by the Italian Ministry of Education, University, and Research.
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
Alon, N., Gyárfás, A., Ruszinkó, M.: Decreasing the diameter of bounded degree graphs. Journal of Graph Theory 35(3), 161–172 (2000)
Brandstädt, A., Le, V.B., Spinrad, J.: Graph classes: a survey. SIAM Monographs on Discrete Mathematics and Applications (1999)
Chandrasekaran, R., Daughety, A.: Location on tree networks: p-centre and n-dispersion problems. Mathematics of Operations Research 6(1), 50–57 (1981)
Chepoi, V., Estellon, B., Nouioua, K., Vaxès, Y.: Mixed covering of trees and the augmentation problem with odd diameter constraints. Algorithmica 45(2), 209–226 (2006)
Chepoi, V., Vaxès, Y.: Augmenting trees to meet biconnectivity and diameter constraints. Algorithmica 33(2), 243–262 (2002)
Chung, F.: Diameters of graph: old problems and new results. Congr. Numer. 60, 295–317 (1987)
Chung, F., Garey, M.: Diameter bounds for altered graphs. Journal of Graph Theory 8(4), 511–534 (1984)
Dodis, Y., Khanna, S.: Designing networks with bounded pairwise distance. In: STOC, pp. 750–759 (1999)
Erdös, P., Gyárfás, A., Ruszinkó, M.: How to decrease the diameter of triangle-free graphs. Combinatorica 18(4), 493–501 (1998)
Erdös, P., Rényi, A.: On a problem in the theory of graphs. Publ. Math. Inst. Hung. Acad. Sci. B(7), 623–639 (1963)
Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theor. Comput. Sci. 38, 293–306 (1985)
Grigorescu, E.: Decreasing the diameter of cycles. J. Graph Theory 43(4), 299–303 (2003)
Ishii, T., Yamamoto, S., Nagamochi, H.: Augmenting forests to meet odd diameter requirements. In: Ibaraki, T., Katoh, N., Ono, H. (eds.) ISAAC 2003. LNCS, vol. 2906, pp. 434–443. Springer, Heidelberg (2003)
Kapoor, S., Sarwat, M.: Bounded-diameter minimum-cost graph problems. Theory Comput. Syst. 41(4), 779–794 (2007)
Kariv, O., Hakimi, S.: An algorithmic approach to network location problems. SIAM Journal on Applied Mathematics 37(3), 513–538 (1979)
Li, C.-L., McCormick, S.T., Simchi-Levi, D.: On the minimum-cardinality-bounded-diameter and the bounded-cardinality-minimum-diameter edge addition problem. Operations Research Letters 11, 303–308 (1992)
Plesník, J.: On the computational complexity of centers locating in a graph. Aplikace Mat. 25
Plesník, J.: The complexity of designing a network with minimum diameter. Networks 11(1), 77–85 (1981)
Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability pcp characterization of np. In: STOC, pp. 475–484 (1997)
Schoone, A.A., Bodlaender, H.L., van Leeuwen, J.: Diameter increase caused by edge deletion. Journal of Graph Theory 11(3), 409–427 (1987)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bilò, D., Gualà, L., Proietti, G. (2010). Improved Approximability and Non-approximability Results for Graph Diameter Decreasing Problems. In: Hliněný, P., Kučera, A. (eds) Mathematical Foundations of Computer Science 2010. MFCS 2010. Lecture Notes in Computer Science, vol 6281. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15155-2_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-15155-2_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15154-5
Online ISBN: 978-3-642-15155-2
eBook Packages: Computer ScienceComputer Science (R0)