Abstract
The unweighted k-edge-connectivity augmentation problem (kECA for short) is deffned by “Given a σ-edge-connected graph G = (V,E), find an edge set E’ of minimum cardinality such that G’ = (V,E∪ E’) is (σ+δ)-edge-connected and σ+δ = k”, where E’ is called a solution to the problem. Let kECA(S,SA) denote kECA such that both G and G’ are simple.
The subject of the present paper is (σ + 1)ECA(S,SA) (or kECA(S,SA) with k = σ+ 1). Let M be any maximum matching of a certain graph R(G) whose vertex set V R consists of vertices representing all leaves of G. From M we obtain an edge set E’0, with |E’0|= |M|, such that each edge connects vertices in distinct leaves of G. Let L 1 be the set of leaves to be created by adding E’0 to G, and K 1 the set of remaining leaves of G.
The main result is to propose two O(σ 2;V log(V/σ)+E+VR 2) time algorithms for ffnding the following solutions: (1) an optimum solution if G has at least 2σ+6 leaves or if L1 ≤ K1 and G has less than 2σ +6 leaves; (2) a 3/2-approximate solution if L1 > K1 and G has less than 2σ + 6 leaves.
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A. V. Aho, J. E. Hopcroft AND J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, MA, 1974.
J. Bang-Jensen AND T. Jordán, Edge-connectivity augmentation preserving simplicity, SIAM J. Discrete Math., 11 (1998), pp. 603–623.
K. P Eswaran AND R. E. Tarjan, Augmentation problems, SIAM J. Comput., 5 (1976), pp. 653–655.
S. Even, Graph Algorithms, Pitman, London, 1979.
A. Frank, Augmenting graphs to meet edge connectivity requirements, SIAM J. Discrete Mathematics, 5 (1992), pp. 25–53.
H. Frank AND W. Chou, Connectivity considerations in the design of survivable networks, IEEE Trans. Circuit Theory, CT-17 (1970), pp. 486–490.
H. N. Gabow, Applications of a poset representation to edge connectivity and graph rigidity, in Proc. 32nd IEEE Symposium on Foundations of Computer Science, 1991, pp. 812–821.
-, Effcient splitting off algorithms for graphs, in Proc. 26th ACM Symposium on Theory of Computing, 1994, pp. 696–705.
T. C. Hu, Integer Programming and Network Flows, Addison-Wesley, Reading, Mass, 1969.
T. Jordán, Two NP-complete augmentation problems, Tech. Rep. PP-1997–08, Odense University, Denmark, march 1997. http://www.imada.ou.dk/Research/Preprints/j-l.html.
A. V. Karzanov AND E. A. Timofeev, Effcient algorithm for ffnding all minimal edge cuts of a nonoriented graph, Cybernetics, (dy1986), pp. 156–162. Translated from Kibernetika, 2 (1986), 8–12.
H. Nagamochi AND T. Ibaraki, A faster edge splitting algorithm in multigraphs and its application to the edge-connectivity augmentation problem, Tech. Rep. 94017, Kyoto University, 1994.
D. Naor, D. Gusfield, AND C. Martel, A fast algorithm for optimally increasing the edge connectivity, SIAM J. Comput., 26 (1997), pp. 1139–1165.
D. Takafuji, S. Taoka, AND T. Watanabe, Simplicity-preserving augmentation to 4-edge-connect a graph, IPSJ SIG Notes, AL-335 (1993), pp. 33–40.
S. Taoka, D. Takafuji, AND T. Watanabe, Simplicity-preserving augmentation of the edge-connectivity of a graph, Tech. Rep. of IEICE of Japan, COMP9373 (1994), pp. 49–56.
S. Taoka AND T. Watanabe, Effcient algorithms for the edge-connectivity augmentation problem of graphs without increasing edge-multiplicity, IPSJ SIG Notes, AL-42-1 (1994), pp. 1–8.
-, Minimum augmentation to k-edge-connect specified vertices of a graph, in Lecture Notes in Computer Science 834(D-Z du and X-S Zhang(Eds.): Algorithms and Computation, Springer-Verlag, Berlin, 1994, pp. 217–225. (Proc. 5th International Symposium on Algorithms and Computation(ISAAC’94)).
-, Smallest augmentation to k-edge-connect all specified vertices in a graph, IPSJ SIG Notes, AL-383 (1994), pp. 17–24.
R. E. Tarjan, Data Structures and Network Algorithms, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1983.
T. Watanabe, An effcient way for edge-connectivity augmentation, Tec. Rep.ACT-76-UILU-ENG-87-2221, Coordinated Science Lab., University of Illinois at Urbana, Urbana, IL 61801, April 1987. Also presented at Eighteenth Southeastern International Conference on Combinatorics, Graph Theory, Computing, No. 15, Boca Raton, FL, U.S.A., February 1987.
-, A simple improvement on edge-connectivity augmentation, Tech. Rep., IEICE of Japan, CAS87-203 (1987), pp. 43–48.
T. Watanabe AND A. Nakamura, Edge-connectivity augmentation problems, J. Comput. System Sci., 35 (1987), pp. 96–144.
T. Watanabe AND M. Yamakado, A linear time algorithm for smallest augmentation to 3-edge-connect a graph, IEICE Trans. Fundamentals of Japan, E76-A (1993), pp. 518–531.
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Taoka, S., Watanabe, T. (2000). The (σ + 1)-Edge-Connectivity Augmentation Problem without Creating Multiple Edges of a Graph. In: van Leeuwen, J., Watanabe, O., Hagiya, M., Mosses, P.D., Ito, T. (eds) Theoretical Computer Science: Exploring New Frontiers of Theoretical Informatics. TCS 2000. Lecture Notes in Computer Science, vol 1872. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44929-9_14
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