# Improved Approximability and Non-approximability Results for Graph Diameter Decreasing Problems

• Davide Bilò
• Luciano Gualà
• Guido Proietti
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6281)

## 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.

## References

1. 1.
Alon, N., Gyárfás, A., Ruszinkó, M.: Decreasing the diameter of bounded degree graphs. Journal of Graph Theory 35(3), 161–172 (2000)
2. 2.
Brandstädt, A., Le, V.B., Spinrad, J.: Graph classes: a survey. SIAM Monographs on Discrete Mathematics and Applications (1999)Google Scholar
3. 3.
Chandrasekaran, R., Daughety, A.: Location on tree networks: p-centre and n-dispersion problems. Mathematics of Operations Research 6(1), 50–57 (1981)
4. 4.
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)
5. 5.
Chepoi, V., Vaxès, Y.: Augmenting trees to meet biconnectivity and diameter constraints. Algorithmica 33(2), 243–262 (2002)
6. 6.
Chung, F.: Diameters of graph: old problems and new results. Congr. Numer. 60, 295–317 (1987)
7. 7.
Chung, F., Garey, M.: Diameter bounds for altered graphs. Journal of Graph Theory 8(4), 511–534 (1984)
8. 8.
Dodis, Y., Khanna, S.: Designing networks with bounded pairwise distance. In: STOC, pp. 750–759 (1999)Google Scholar
9. 9.
Erdös, P., Gyárfás, A., Ruszinkó, M.: How to decrease the diameter of triangle-free graphs. Combinatorica 18(4), 493–501 (1998)
10. 10.
Erdös, P., Rényi, A.: On a problem in the theory of graphs. Publ. Math. Inst. Hung. Acad. Sci. B(7), 623–639 (1963)Google Scholar
11. 11.
Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theor. Comput. Sci. 38, 293–306 (1985)
12. 12.
Grigorescu, E.: Decreasing the diameter of cycles. J. Graph Theory 43(4), 299–303 (2003)
13. 13.
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)
14. 14.
Kapoor, S., Sarwat, M.: Bounded-diameter minimum-cost graph problems. Theory Comput. Syst. 41(4), 779–794 (2007)
15. 15.
Kariv, O., Hakimi, S.: An algorithmic approach to network location problems. SIAM Journal on Applied Mathematics 37(3), 513–538 (1979)
16. 16.
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)
17. 17.
Plesník, J.: On the computational complexity of centers locating in a graph. Aplikace Mat. 25Google Scholar
18. 18.
Plesník, J.: The complexity of designing a network with minimum diameter. Networks 11(1), 77–85 (1981)
19. 19.
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)Google Scholar
20. 20.
Schoone, A.A., Bodlaender, H.L., van Leeuwen, J.: Diameter increase caused by edge deletion. Journal of Graph Theory 11(3), 409–427 (1987)

## Authors and Affiliations

• Davide Bilò
• 1
• Luciano Gualà
• 2
• Guido Proietti
• 1
• 3
1. 1.Dipartimento di InformaticaUniversità di L’AquilaL’AquilaItaly
2. 2.Dipartimento di MatematicaUniversità di Tor VergataRomaItaly
3. 3.Istituto di Analisi dei Sistemi ed Informatica, CNRRomaItaly