TY - JOUR

T1 - Small ℓ-edge-covers in k-connected graphs

AU - Nutov, Zeev

PY - 2013/9

Y1 - 2013/9

N2 - Let G=(V,E) be a k-edge-connected graph with edge-costs {c(e):e ε E} and minimum degree d. We show by a simple and short proof, that for any integer ℓ with dk≤ℓ≤d(1-1k), G contains an ℓ-edge cover I such that: c(I)≤ℓdc(E) if G is bipartite, or if ℓ|V| is even, or if |E|≥d|V|2+d2ℓ; otherwise, c(I)≤(ℓd+1d|V|)c(E). The particular case d=k=ℓ+1 and unit costs already includes a result of Cheriyan and Thurimella (2000) [1], that G contains a (k-1)-edge-cover of size |E|-|V|/2. Using our result, we slightly improve the approximation ratios for the k-Connected Subgraph problem (the node-connectivity version) with uniform and β-metric costs. We then consider the dual problem of finding a spanning subgraph of maximum connectivity k* with a prescribed number of edges. We give an algorithm that computes a (k*-1)-connected subgraph, which is tight, since the problem is NP-hard.

AB - Let G=(V,E) be a k-edge-connected graph with edge-costs {c(e):e ε E} and minimum degree d. We show by a simple and short proof, that for any integer ℓ with dk≤ℓ≤d(1-1k), G contains an ℓ-edge cover I such that: c(I)≤ℓdc(E) if G is bipartite, or if ℓ|V| is even, or if |E|≥d|V|2+d2ℓ; otherwise, c(I)≤(ℓd+1d|V|)c(E). The particular case d=k=ℓ+1 and unit costs already includes a result of Cheriyan and Thurimella (2000) [1], that G contains a (k-1)-edge-cover of size |E|-|V|/2. Using our result, we slightly improve the approximation ratios for the k-Connected Subgraph problem (the node-connectivity version) with uniform and β-metric costs. We then consider the dual problem of finding a spanning subgraph of maximum connectivity k* with a prescribed number of edges. We give an algorithm that computes a (k*-1)-connected subgraph, which is tight, since the problem is NP-hard.

KW - Approximation algorithms

KW - k-connectivity

KW - k-edge-cover

UR - http://www.scopus.com/inward/record.url?scp=84878324374&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2013.02.021

DO - 10.1016/j.dam.2013.02.021

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AN - SCOPUS:84878324374

SN - 0166-218X

VL - 161

SP - 2101

EP - 2106

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

IS - 13-14

ER -