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Survivable network design with degree or order constraints
 SIAM J. on Computing
, 2009
"... Abstract. We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivit ..."
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Cited by 55 (8 self)
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Abstract. We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds Bv on the vertices. This includes the wellstudied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a (2, 2Bv +3)approximation algorithm for the edgeconnectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex v is at most 2Bv + 3. This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly kEdgeConnected Subgraph problem. In contrast, we show that the vertexconnectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic
A Matter of Degree: Improved Approximation Algorithms for DegreeBounded Minimum Spanning Trees
 SIAM Journal on Computing
, 2000
"... A7 80 05 2B C D ; 84 6 E> 7 , 6 F,G < =3H 3D I: 7J ,F,G K L MON4P I: K L MN4P , 6 E : O Q 9 6 E7 , ,<= 2 = , ?6 ES8 6 Q9 , O Q 2OT , 3 O 6 US8 ; 6 Q 7 6 9 ,7 3 O , 6 79 ..."
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Cited by 49 (6 self)
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A7 80 05 2B C D ; 84 6 E> 7 , 6 F,G < =3H 3D I: 7J ,F,G K L MON4P I: K L MN4P , 6 E : O Q 9 6 E7 , ,<= 2 = , ?6 ES8 6 Q9 , O Q 2OT , 3 O 6 US8 ; 6 Q 7 6 9 ,7 3 O , 6 79 82 1.
Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal
 In Proc. of ACM Symposium on Theory of computing (STOC
, 2007
"... ABSTRACT In the MINIMUM BOUNDED DEGREE SPANNING TREE problem,we are given an undirected graph with a degree upper bound Bv oneach vertex v, and the task is to find a spanning tree of minimumcost which satisfies all the degree bounds. Let OPT be the costof an optimal solution to this problem. In this ..."
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Cited by 48 (6 self)
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ABSTRACT In the MINIMUM BOUNDED DEGREE SPANNING TREE problem,we are given an undirected graph with a degree upper bound Bv oneach vertex v, and the task is to find a spanning tree of minimumcost which satisfies all the degree bounds. Let OPT be the costof an optimal solution to this problem. In this paper, we present a polynomial time algorithm which returns a spanning tree T ofcost at most OPT and dT (v) ^ Bv + 1 for all v, where dT (v)denotes the degree of v in T. This generalizes a result of Furerand Raghavachari [8] to weighted graphs, and settles a 15yearold conjecture of Goemans [10] affirmatively. The algorithm generalizes when each vertex v has a degree lower bound Av and a degreeupper bound Bv, and returns a spanning tree with cost at most OPTand Av \Gamma 1 ^ dT (v) ^ Bv + 1 for all v. This is essentially thebest possible. The main technique used is an extension of the iterative rounding method introduced by Jain [12] for the design ofapproximation algorithms.
Primaldual meets local search: Approximating MST's with nonuniform degree bounds
 STOC'03
, 2003
"... ... Our previous algorithm [9] with similar guarantees worked only in the case of uniform degree bounds (i.e. Bv = B for all vertices v). While the new algorithm is based on ideas from Lagrangean relaxation as is our previous work, it does not rely on computing a solution to a linear program. Instea ..."
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Cited by 34 (4 self)
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... Our previous algorithm [9] with similar guarantees worked only in the case of uniform degree bounds (i.e. Bv = B for all vertices v). While the new algorithm is based on ideas from Lagrangean relaxation as is our previous work, it does not rely on computing a solution to a linear program. Instead it uses a repeated application of Kruskal's MST algorithm interleaved with a combinatorial update of approximate Lagrangean nodemultipliers maintained by the algorithm. These updates cause subsequent repetitions of the spanning tree algorithm to run for longer and longer times, leading to overall progress and a proof of the performance guarantee.
Approximation Algorithms for DegreeConstrained MinimumCost NetworkDesign Problems
, 2001
"... We study networkdesign problems with two different design objectives: the total cost of the edges and nodes in the network and the maximum degree of any node in the network. A prototypical example is the degreeconstrained nodeweighted Steiner tree problem: We are given an undirected graph ..."
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Cited by 31 (2 self)
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We study networkdesign problems with two different design objectives: the total cost of the edges and nodes in the network and the maximum degree of any node in the network. A prototypical example is the degreeconstrained nodeweighted Steiner tree problem: We are given an undirected graph , with a nonnegative integral function that specifies an upper bound on the degree of each vertex in the Steiner tree to be constructed, nonnegative costs on the nodes, and a subset of nodes called terminals. The goal is to construct a Steiner containing all the terminals such that the degree of any node is at most the specified upper bound and the total cost of the nodes in is minimum. Our main result is a bicriteria approximation algorithm whose output is approximate in terms of both the degree and cost criteria  the degree of any node in the output Steiner tree is and the cost of the tree is times that of a minimumcost Steiner tree that obeys the degree bound for each node . Our result extends to the more general problem of constructing oneconnected networks such as generalized Steiner forests. We also consider the special case in which the edge costs obey the triangle inequality and present simple approximation algorithms with better performance guarantees.
Low Degree Spanning Trees Of Small Weight
, 1996
"... . Given n points in the plane, the degreeK spanning tree problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most K. This paper addresses the problem of computing lowweight degreeK spanning trees for K ? 2. It is shown that for an arbitrary collection of n ..."
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Cited by 31 (2 self)
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. Given n points in the plane, the degreeK spanning tree problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most K. This paper addresses the problem of computing lowweight degreeK spanning trees for K ? 2. It is shown that for an arbitrary collection of n points in the plane, there exists a spanning tree of degree three whose weight is at most 1.5 times the weight of a minimum spanning tree. It is shown that there exists a spanning tree of degree four whose weight is at most 1.25 times the weight of a minimum spanning tree. These results solve open problems posed by Papadimitriou and Vazirani. Moreover, if a minimum spanning tree is given as part of the input, the trees can be computed in O(n) time. The results are generalized to points in higher dimensions. It is shown that for any d 3, an arbitrary collection of points in ! d contains a spanning tree of degree three, whose weight is at most 5/3 times the weight of a minimum spanning tre...
Minimum Bounded Degree Spanning Trees
, 2006
"... We consider the minimum cost spanning tree problem under the restriction that all degrees must be at most a given value k. We show that we can efficiently find a spanning tree of maximum degree at most k + 2 whose cost is at most the cost of the optimum spanning tree of maximum degree at most k. Thi ..."
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Cited by 30 (0 self)
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We consider the minimum cost spanning tree problem under the restriction that all degrees must be at most a given value k. We show that we can efficiently find a spanning tree of maximum degree at most k + 2 whose cost is at most the cost of the optimum spanning tree of maximum degree at most k. This is almost best possible. The approach uses a sequence of simple algebraic, polyhedral and combinatorial arguments. It illustrates many techniques and ideas in combinatorial optimization as it involves polyhedral characterizations, uncrossing, matroid intersection, and graph orientations (or packing of spanning trees). The result generalizes to the setting where every vertex has both upper and lower bounds and gives then a spanning tree which violates the bounds by at most two units and whose cost is at most the cost of the optimum tree. It also gives a better understanding of the subtour relaxation for both the symmetric and asymmetric traveling salesman problems. The generalization to ledgeconnected subgraphs is briefly discussed.
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 27 (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
Additive Guarantees for Degree Bounded Directed Network Design
 STOC'08
, 2008
"... We present polynomialtime approximation algorithms for some degreebounded directed network design problems. Our main result is for intersecting supermodular connectivity with degree bounds: given a directed graph G = (V, E) with nonnegative edgecosts, a connectivity requirement specified by an in ..."
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Cited by 22 (2 self)
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We present polynomialtime approximation algorithms for some degreebounded directed network design problems. Our main result is for intersecting supermodular connectivity with degree bounds: given a directed graph G = (V, E) with nonnegative edgecosts, a connectivity requirement specified by an intersecting supermodular function f, and upper bounds {av, bv}v∈V on indegrees and outdegrees of vertices, find a minimumcost fconnected subgraph of G that satisfies the degree bounds. We give a bicriteria approximation algorithm that for any 0 ≤ ɛ ≤ 1, computes an fconnected sub2 graph with indegrees at most ⌈ av ⌉+4, outdegrees at most
The Power of Local Optimization: Approximation Algorithms for Maximumleaf Spanning Tree
 In Proceedings, Thirtieth Annual Allerton Conference on Communication, Control and Computing
, 1996
"... Given an undirected graph G, finding a spanning tree of G with maximum number of leaves is NPcomplete. We use the simple technique of local optimization to provide the first approximation algorithms for this problem. Our algorithms run in polynomial time to produce locally optimal solutions. We pro ..."
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Cited by 21 (3 self)
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Given an undirected graph G, finding a spanning tree of G with maximum number of leaves is NPcomplete. We use the simple technique of local optimization to provide the first approximation algorithms for this problem. Our algorithms run in polynomial time to produce locally optimal solutions. We prove that locally optimal solutions to this problem are globally nearoptimal. In particular, we prove that two such algorithms have performance ratios of 5 and 3. The latter algorithm employs more powerful localimprovement steps than the former and hence has higher running time. This may indicate an interesting tradeoff between the performance ratios and the running times of the series of algorithms we describe. Keywords: Approximation algorithms, NPcomplete problems, Performance ratio, Local optimization, Communication network design, Combinatorial algorithms. 1 Introduction Given an undirected graph G = (V; E), the Maximum Leaf Spanning Tree problem is to find a spanning tree of G with ...