Approximating survivable networks with β-metric costs

Abstract

The Survivable Network Design ( SND) problem seeks a minimum-cost subgraph that satisfies prescribed node-connectivity requirements. We consider SND on both directed and undirected complete graphs with β- metric costs when c ( x z ) ⩽ β [ c ( x y ) + c ( y z ) ] for all x , y , z ∈ V , which varies from uniform costs ( β = 1 / 2 ) to metric costs ( β = 1 ). For the k- Connected Subgraph ( k- CS) problem our ratios are: 1 + 2 β k ( 1 − β ) − 1 2 k − 1 for undirected graphs, and 1 + 4 β 3 k ( 1 − 3 β 2 ) − 1 2 k − 1 for directed graphs and 1 2 ⩽ β < 1 3 . For undirected graphs this improves the ratios β 1 − β of Böckenhauer et al. (2008) [3] and 2 + β k n of Kortsarz and Nutov (2003) [11] for all k ⩾ 4 and 1 2 + 3 k − 2 2 ( 4 k 2 − 7 k + 2 ) ⩽ β ⩽ k 2 ( k + 1 ) 2 − 2 . We also show that SND admits the ratios 2 β 1 − β for undirected graphs, and 4 β 3 1 − 3 β 2 for directed graphs with 1 / 2 ⩽ β < 1 / 3 . For two important particular cases of SND, so-called Subset k- CS and Rooted SND, our ratios are 2 β 3 1 − 3 β 2 for directed graphs and β 1 − β for subset k- CS on undirected graphs.