On k-Edge-Connectivity Problems with Sharpened Triangle Inequality

The edge-connectivityproblem is to find a minimum-cost k-edge-connected spanning subgraph of an edge-weighted, undirected graph G for any given G and k. Here we consider its APX-hard subproblems with respect to the parameter β, with 1/2 ≤ β < 1, where G = (V, E) is a complete graph with a cost function c satisfying the sharpened triangle inequality c({u, v}) ≤ β ċ (c({u, w}) + c({w, v})) for all u, v, w ∈ V. First, we give a linear-time approximation algorithm for these optimization problems with approximation ratio β/1-β for any 1/2 ≤ β < 1, which does not depend on k. The result above is based on a rough combinatorial argumentation. We sophisticate our combinatorial consideration in order to design a (1 + 5(2β-1)/9(1-β))- approximation algorithm for the 3-edge-connectivitysubgraph problem for graphs satisfying the sharpened triangle inequality for 1/2 ≤ β ≤ 2/3.

[1]  Juraj Hromkovic.  Stability of Approximation Algorithms and the Knapsack Problem , 1999, Jewels are Forever.

[2]  Cristina G. Fernandes.  A better approximation ratio for the minimum k-edge-connected spanning subgraph problem , 1997, SODA '97.

[3]  Michael A. Bender,et al.  Performance guarantees for the TSP with a parameterized triangle inequality , 2000, Inf. Process. Lett..

[4]  Andrzej Lingas,et al.  On approximability of the minimum-cost k-connected spanning subgraph problem , 1999, SODA '99.

[5]  Joseph Cheriyan,et al.  Approximating Minimum-Size k-Connected Spanning Subgraphs via Matching , 1998, Electron. Colloquium Comput. Complex..

[6]  Michael R. Fellows,et al.  Fixed Parameter Tractability and Completeness , 1992, Complexity Theory: Current Research.

[7]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[8]  Hans-Joachim Böckenhauer,et al.  Improved lower bounds on the approximability of the Traveling Salesman Problem , 2000, RAIRO Theor. Informatics Appl..

[9]  Juraj Hromkovic,et al.  An Improved Lower Bound on the Approximability of Metric TSP and Approximation Algorithms for the TSP with Sharpened Triangle Inequality , 2000, STACS.

[10]  Juraj Hromkovic,et al.  Algorithmics for hard problems - introduction to combinatorial optimization, randomization, approximation, and heuristics , 2001 .

[11]  Juraj Hromkovic,et al.  Approximation algorithms for the TSP with sharpened triangle inequality , 2000, Inf. Process. Lett..

[12]  Michal Penn,et al.  Improved Approximation Algorithms for Weighted 2- and 3-Vertex Connectivity Augmentation Problems , 1997, J. Algorithms.

[13]  Hans-Jürgen Bandelt,et al.  Performance Guarantees for Approximation Algorithms Depending on Parametrized Triangle Inequalities , 1995, SIAM J. Discret. Math..

[14]  Guido Proietti,et al.  On the hardness of constructing minimal 2-connected spanning subgraphs in complete graphs with sharpened triangle inequality , 2002, Theor. Comput. Sci..

[15]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[16]  Juraj Hromkovic,et al.  Towards the Notion of Stability of Approximation for Hard Optimization Tasks and the Traveling Salesman Problem , 2000, CIAC.

[17]  Samir Khuller,et al.  Improved Approximation Algorithms for Uniform Connectivity Problems , 1996, J. Algorithms.

[18]  Thomas Andreae,et al.  On the traveling salesman problem restricted to inputs satisfying a relaxed triangle inequality , 2001, Networks.

[19]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .