TY - JOUR

T1 - Approximating k-node connected subgraphs via critical graphs

AU - Kortsarz, Guy

AU - Nutov, Zeev

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2005

Y1 - 2005

N2 - We present two new approximation algorithms for the problem of finding a k-node connected spanning subgraph (directed or undirected) of minimum cost. The best known approximation guarantees for this problem were O(min{k, n/√n-k}) for both directed and undirected graphs, and O(ln k) for undirected graphs with n ≥ 6k 2, where n is the number of nodes in the input graph. Our first algorithm has approximation ratio O(n/n-k ln 2 k), which is O(ln 2 k) except for very large values of k, namely, k = n - o(n). This algorithm is based on a new result on l-connected p-critical graphs, which is of independent interest in the context of graph theory. Our second algorithm uses the primal-dual method and has approximation ratio O(√n ln k) for all values of n, k. Combining these two gives an algorithm with approximation ratio O(ln k·min{√k, n/n-k ln k}), which asymptotically improves the best known approximation guarantee for directed graphs for all values of n, k, and for undirected graphs for k > √n/6. Moreover, this is the first algorithm that has an approximation guarantee better than Θ(k) for all values of n, k. Our approximation ratio also provides an upper bound on the integrality gap of the standard LP-relaxation.

AB - We present two new approximation algorithms for the problem of finding a k-node connected spanning subgraph (directed or undirected) of minimum cost. The best known approximation guarantees for this problem were O(min{k, n/√n-k}) for both directed and undirected graphs, and O(ln k) for undirected graphs with n ≥ 6k 2, where n is the number of nodes in the input graph. Our first algorithm has approximation ratio O(n/n-k ln 2 k), which is O(ln 2 k) except for very large values of k, namely, k = n - o(n). This algorithm is based on a new result on l-connected p-critical graphs, which is of independent interest in the context of graph theory. Our second algorithm uses the primal-dual method and has approximation ratio O(√n ln k) for all values of n, k. Combining these two gives an algorithm with approximation ratio O(ln k·min{√k, n/n-k ln k}), which asymptotically improves the best known approximation guarantee for directed graphs for all values of n, k, and for undirected graphs for k > √n/6. Moreover, this is the first algorithm that has an approximation guarantee better than Θ(k) for all values of n, k. Our approximation ratio also provides an upper bound on the integrality gap of the standard LP-relaxation.

KW - Approximation

KW - Connectivity

KW - Graphs

KW - Network design

UR - http://www.scopus.com/inward/record.url?scp=33644585066&partnerID=8YFLogxK

U2 - 10.1137/S0097539703435753

DO - 10.1137/S0097539703435753

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AN - SCOPUS:33644585066

SN - 0097-5397

VL - 35

SP - 247

EP - 257

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 1

ER -