Abstract
The edge-connectivity problem 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 \( \frac{1} {2} \) ⩽ β < 1, where G = (V, E) is a complete graph with a cost function c satisfying the sharpened triangle inequality
for all u, v, w ∈ V.
First, we give a linear-time approximation algorithm for these optimization problems with approximation ratio \( \frac{\beta } {{1 - \beta }} \) for any \( \frac{1} {2} \) ⩽ 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 \( \left( {1 + \frac{{5\left( {2\beta - 1} \right)}} {{9\left( {1 - \beta } \right)}}} \right) \) approximation algorithm for the 3-edge-connectivity subgraph problem for graphs satisfying the sharpened triangle inequality for \( \frac{1} {2} \) ⩽ β ⩽ \( \frac{2} {3} \) .
This work was partially supported by DFG-grant Hr 14/5-1, the CNR-Agenzia 2000 Program, under Grants No. CNRC00CAB8 and CNRG003EF8, and the Research Project REAL-WINE, partially funded by the Italian Ministry of Education, University and Research.
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Böckenhauer, HJ. et al. (2003). On k-Edge-Connectivity Problems with Sharpened Triangle Inequality. In: Petreschi, R., Persiano, G., Silvestri, R. (eds) Algorithms and Complexity. CIAC 2003. Lecture Notes in Computer Science, vol 2653. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44849-7_24
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DOI: https://doi.org/10.1007/3-540-44849-7_24
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