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Additive Approximation for Bounded Degree Survivable Network Design
 STOC'08
, 2008
"... We study a general network design problem with additional degree constraints. Given connectivity requirements ruv for all pairs of vertices, a Steiner network is a graph in which there are at least r_uv edgedisjoint paths between u and v for all pairs of vertices u, v. In the MINIMUM BOUNDEDDEGREE ..."
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Cited by 29 (5 self)
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We study a general network design problem with additional degree constraints. Given connectivity requirements ruv for all pairs of vertices, a Steiner network is a graph in which there are at least r_uv edgedisjoint paths between u and v for all pairs of vertices u, v. In the MINIMUM BOUNDEDDEGREE STEINER NETWORK problem, we are given an undirected graph G with an edge cost for each edge, a connectivity requirement r_uv for each pair of vertices u and v, and a degree upper bound for each vertex v. The task is to find a minimum cost Steiner network which satisfies all the degree upper bounds. The aim of this paper is to design approximation algorithms that minimize the total cost and the degree violation simultaneously. Our main results are the following: • There is a polynomial time algorithm which returns a Steiner forest of cost at most 2OPT and the degree violation at each vertex is at most 3, whereOPT is the cost of an optimal solution which satisfies all the degree bounds. • There is a polynomial time algorithm which returns a Steiner network of cost at most 2OPT and the degree violation at each vertex is at most 6rmax +3,where OPT is the cost of an optimal solution which satisfies all the degree bounds, and rmax: = maxu,v{ruv}. These results achieve the best known guarantees for both the total cost and the degree violation simultaneously. As corollaries, these results provide the first additive approximation algorithms for finding low degree subgraphs including Steiner forests, kedgeconnected subgraphs, and Steiner networks. The algorithms develop on the iterative relaxation method applied to a natural linear programming relaxation as in [10, 16, 22]. The new algorithms avoid paying a multiplicative factor of two on the degree bounds even though the algorithm can only pick edges with fractional value 1/2. This is based on a stronger characterization of the basic so
NetworkDesign with Degree Constraints
"... We study several network design problems with degree constraints. For the degreeconstrained 2connected subgraph problem we obtain a factor 6 violation for the degrees with 4 approximation for the cost. This improves upon the logarithmic degree violation and no cost guarantee obtained by Feder, Mo ..."
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Cited by 7 (3 self)
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We study several network design problems with degree constraints. For the degreeconstrained 2connected subgraph problem we obtain a factor 6 violation for the degrees with 4 approximation for the cost. This improves upon the logarithmic degree violation and no cost guarantee obtained by Feder, Motwani, and Zhu (2006). Then we consider the problem of finding an arborescence with at least k terminals and with minimum maximum“ outdegree. We show that the nat√k” ural LPrelaxation has a gap of Ω or Ω n 1/4 with respect to the multiplicative degree bound violation. We overcome this hurdle by a combinatorial O ( p (k log k)/ ∆ ∗)approximation algorithm, where ∆ ∗ denotes the maximum degree in the optimum solution. We also give an Ω(log n) lower bound on approximating this problem. Then we consider the undirected version of this problem, however, with an extra diameter constraint, and give an Ω(log n) lower bound on the approximability of this version. Finally, we consider a closely related prizecollecting degreeconstrained Steiner Network problem. We obtain several results in this direction by reducing the prizecollecting variant to the regular one.
On some network design problems with degree constraints
, 2013
"... We study several network design problems with degree constraints. For the minimumcost DegreeConstrained 2NodeConnected Subgraph problem, we obtain the first nontrivial bicriteria approximation algorithm, with 5b(v)+3 violation for the degrees and a 4approximation for the cost. This improves up ..."
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Cited by 2 (0 self)
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We study several network design problems with degree constraints. For the minimumcost DegreeConstrained 2NodeConnected Subgraph problem, we obtain the first nontrivial bicriteria approximation algorithm, with 5b(v)+3 violation for the degrees and a 4approximation for the cost. This improves upon the logarithmic degree violation and no cost guarantee obtained by Feder, Motwani, and Zhu (2006). Then we consider the problem of finding an arborescence with at least k terminals and with minimum maximum outdegree. We show that the natural LPrelaxation has a gap of Ω ( √ k) or Ω(n 1/4) with respect to the multiplicative degree bound violation. We overcome this hurdle by a combinatorial O ( √ (klogk)/ ∆ ∗)approximation algorithm, where ∆ ∗ denotes the maximum degree in the optimum solution. We also give an Ω(logn) lower bound on approximating this problem. Then we consider the undirected version of this problem, however, with an extra diameter constraint, and give an Ω(log n) lower bound on the approximability of this version. We also consider a closely related PrizeCollecting DegreeConstrained EdgeConnectivity Survivable Network problem, and obtain several
Improved Approximation Algorithms for Degreebounded Network Design Problems with Node Connectivity Requirements
"... We consider degree bounded network design problems with element and vertex connectivity requirements. In the degree bounded Survivable Network Design (SNDP) problem, the input is an undirected graph G = (V,E) with weights w(e) on the edges and degree bounds b(v) on the vertices, and connectivity r ..."
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We consider degree bounded network design problems with element and vertex connectivity requirements. In the degree bounded Survivable Network Design (SNDP) problem, the input is an undirected graph G = (V,E) with weights w(e) on the edges and degree bounds b(v) on the vertices, and connectivity requirements r(uv) for each pair uv of vertices. The goal is to select a minimumweight subgraph H of G that meets the connectivity requirements and it satisfies the degree bounds on the vertices: for each pair uv of vertices, H has r(uv) disjoint paths between u and v; additionally, each vertex v is incident to at most b(v) edges in H. We give the first (O(1), O(1) · b(v)) bicriteria approximation algorithms for the degreebounded SNDP problem with element connectivity requirements and for several degreebounded SNDP problems with vertex connectivity requirements. Our algorithms construct a subgraph H whose weight is at most O(1) times the optimal such that each vertex v is incident to at most O(1) · b(v) edges in H. We can also extend our approach to network design problems in directed graphs with outdegree constraints to obtain (O(1), O(1) · b+(v)) bicriteria approximation. 1.
Approximating some NetworkDesign problems with Degree Constraints
"... We study several network design problems with degree constraints. For the degreeconstrained 2 vertexconnected subgraph problem we obtain a factor 6 violation for the degrees with 4 approximation for the cost. This improves upon the logarithmic degree violation and no cost guarantee obtained by Fed ..."
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We study several network design problems with degree constraints. For the degreeconstrained 2 vertexconnected subgraph problem we obtain a factor 6 violation for the degrees with 4 approximation for the cost. This improves upon the logarithmic degree violation and no cost guarantee obtained by Feder, Motwani, and Zhu (2006). Then we consider the problem of finding an arborescence with at least k terminals and ( with minimum maximum outdegree. We show that the √k) natural LPrelaxation has a gap of Ω or Ω ( n1/4) with respect to the multiplicative degree bound violation. We overcome this hurdle by a combinatorial O ( √ (k log k)/ ∆ ∗)approximation algorithm, where ∆ ∗ denotes the maximum degree in the optimum solution. We also give an Ω(log n) lower bound on approximating this problem. Then we consider the undirected version of this problem, however, with an extra diameter constraint, and give an Ω(log n) lower bound on the approximability of this version. We consider a closely related prizecollecting degreeconstrained Steiner Network problem. We obtain several results in this direction by reducing the prizecollecting variant to the regular one. Finally, we consider the a degree constraint problem in which the degree bounds have hard capacity and a violation of this capacity is not allowed. We study the Steiner tree problem in which some of the terminals are required to be leaves. We show that this seemingly simple problem is equivalent to the Group Steiner problem in trees.
Degree Constrained NodeConnectivity Problems
 ALGORITHMICA
"... We consider Degree Constrained Survivable Network problems. For ... slightly improve the best known approximation ratio, by a simple proof. Our main contribution is giving a framework to handle nodeconnectivity degree constrained problems with the iterative rounding method. In particular, for the d ..."
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We consider Degree Constrained Survivable Network problems. For ... slightly improve the best known approximation ratio, by a simple proof. Our main contribution is giving a framework to handle nodeconnectivity degree constrained problems with the iterative rounding method. In particular, for the degree constrained versions of the ElementConnectivity Survivable Network problem on undirected graphs, and of the kOutconnected Subgraph problem on both directed and undirected graphs, our algorithm computes a solution J of cost O(logk) times the optimal, with degrees O(2 k)·b(v). Similar result are obtained for the kConnected Subgraph problem. The latter improves on the only degree approximation O(klogn) · b(v) in O(n k) time on undirected graphs by Feder, Motwani, and Zhu.