On k-Edge-Connectivity Problems with Sharpened Triangle Inequality

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

$$ c\left( {\left\{ {u,v} \right\}} \right) \leqslant \beta .\left( {c\left\{ {u,w} \right\}} \right) + c\left( {\left\{ {w,v} \right\}} \right) $$

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.