## A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem (0)

Venue: | Combinatorica |

Citations: | 202 - 5 self |

### BibTeX

@ARTICLE{Jain_afactor,

author = {Kamal Jain},

title = {A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem},

journal = {Combinatorica},

year = {},

volume = {21},

pages = {39--60}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a factor 2 approximation algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, which is also known as the survivable network design problem. Our algorithm first solves the linear relaxation of this problem, and then iteratively rounds off the solution. The key idea in rounding off is that in a basic solution of the LP relaxation, at least one edge gets included at least to the extent of half. We include this edge into our integral solution and solve the residual problem. 1 Introduction We consider the problem of finding a minimum-cost subgraph of a given graph such that the number of edges crossing each cut is at least a specified requirement. Formally, given an undirected multigraph G = (V; E), a non-negative cost function c : E ! Q+ , and a requirement function f : 2 V ! Z , solve the following integer program (IP): min X e2E c e x...

### Citations

384 |
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Citation Context ...mplementation is O(m 2 (T + log m))P (m) +O(m 3 n(T + log m))M(m;n). 9 Implementing the Algorithm in Strongly Polynomial Time Tardos gave a strongly polynomial algorithm to solve combinatorial LPs in =-=[7]-=-. In that algorithm she requires an explicit declaration of all the constraints. If we can write LP (3) compactly with polynomial number of constraints then we can use Tardos' algorithm [7]. Note that... |

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Citation Context ...27308. 1 1. f(V ) = 0 2. For every A; B ` V , at least one of the following holds ffl f(A) + f(B)sf(A \Gamma B) + f(B \Gamma A) ffl f(A) + f(B)sf(A " B) + f(A [ B) The problem was first considere=-=d in [9]-=- with the stronger assumption that f is proper (see Definition 2.1). The authors of [9] give a 2k-approximation algorithm, where k is the maximum requirement of a set. The approximation factor was lat... |

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Citation Context ...mption that f is proper (see Definition 2.1). The authors of [9] give a 2k-approximation algorithm, where k is the maximum requirement of a set. The approximation factor was later improved to 2H k in =-=[2]-=-, where H k = 1 + 1 2 + 1 3 + \Delta \Delta \Delta + 1 k . The algorithm in [2] also works for weakly supermodular functions. No better approximation factor was known even for the generalized Steiner ... |

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Citation Context ...ithm falls into the class of rounding algorithms. Rounding algorithms use an optimal fractional solution to obtain a good integral solution. Some problems, like vertex cover [6] and node multiway cut =-=[1]-=-, have the remarkable property that they admit an optimal fractional solution which is half-integral. When this property holds, rounding up gives an approximation factor of 2. Unfortunately, the half-... |