TY - JOUR
T1 - Biconnectivity approximations and graph carvings
JF - Journal of the ACM (JACM)
Y1 - 1994
A1 - Khuller, Samir
A1 - Vishkin, Uzi
KW - biconnectivity
KW - connectivity
KW - sparse subgraphs
AB - A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2-connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified)? Unfortunately, the problem is known to be NP-hard.We consider the problem of finding a better approximation to the smallest 2-connected subgraph, by an efficient algorithm. For 2-edge connectivity, our algorithm guarantees a solution that is no more than 3/2 times the optimal. For 2-vertex connectivity, our algorithm guarantees a solution that is no more than 5/3 times the optimal. The previous best approximation factor is 2 for each of these problems. The new algorithms (and their analyses) depend upon a structure called a carving of a graph, which is of independent interest. We show that approximating the optimal solution to within an additive constant is NP-hard as well. We also consider the case where the graph has edge weights. For this case, we show that an approximation factor of 2 is possible in polynomial time for finding a k-edge connected spanning subgraph. This improves an approximation factor of 3 for k = 2, due to Frederickson and Ja´Ja´ [1981], and extends it for any k (with an increased running time though).
VL - 41
SN - 0004-5411
UR - http://doi.acm.org/10.1145/174652.174654
CP - 2
M3 - 10.1145/174652.174654
ER -