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. Given n points in the plane, the degree-K 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 low-weight degree-K 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...

We present algorithmic and hardness results for network design problems with degree or order constraints. We first consider the SURVIVABLE NETWORK DESIGN problem with degree constraints on vertices: the objective is to find a minimum cost subgraph satisfying certain connectivity requirements as well as degree upper bounds on the vertices. A well known special case is the MIN-IMUM BOUNDED DEGREE SPANNING TREE problem which has attracted much attention recently. Denote by Bv the degree constraint of vertex v. We present a (2, 2Bv + 3)-approximation algorithm for the element-connectivity SURVIVABLE NETWORK DE-SIGN problem with degree constraints on terminals, i.e., the cost of the solution is at most twice the optimum solution (satisfying the degree bounds), and the degree of each terminal vertex v is at most 2Bv + 3. This extends the most general network design model which admits a 2-approximation algorithm (with no degree constraints), and implies the first constant factor (bicriteria) approximation algorithms for many network design problems with degree constraints, including the MINIMUM BOUNDED DEGREE STEINER TREE problem. In the edge connectivity SURVIVABLE ∗ This work was partly done during a visit to Egerváry Research

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 15-year-old conjecture of Goemans [10] affirmatively. The algorithm general-izes 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.

We study network-design 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 degree-constrained node-weighted Steiner tree problem: We are given an undirected graph , with a non-negative 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 minimum-cost Steiner tree that obeys the degree bound for each node . Our result extends to the more general problem of constructing one-connected 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.

... 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 node-multipliers 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.

Given an undirected graph G, finding a spanning tree of G with maximum number of leaves is NP-complete. 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 local-improvement steps than the former and hence has higher running time. This may indicate an interesting trade-off between the performance ratios and the running times of the series of algorithms we describe. Keywords: Approximation algorithms, NP-complete 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 ...

Given a graph with edge weights satisfying the triangle inequality, and a degree bound for each vertex, the problem of computing a low weight spanning tree such that the degree of each vertex is at most its specified bound is considered. In particular, modifying a given spanning tree T using adoptions to meet the degree constraints is considered. A novel network-flow based algorithm for finding a good sequence of adoptions is introduced. The method yields a better performance guarantee than any previously obtained. Equally importantly, it takes this approach to the limit in the following sense: if any performance guarantee that is solely a function of the topology and edge weights of a given tree holds for any algorithm at all, then it also holds for our algorithm. The performance guarantee is the following. If the degree constraint d(v) for each v is at least 2, the algorithm is guaranteed to find a tree whose weight is at most the weight of the given tree times 2 \Gamma min n d(v)\Gamma2 deg T (v)\Gamma2 : deg T (v) ? 2 o ; where deg T (v) is the initial degree of v. Examples are provided in which no lighter tree meeting the degree constraint exists. Linear-time algorithms are provided with the same worst-case performance guarantee. Choosing T to be a minimum spanning tree yields approximation algorithms for the general problem on geometric graphs with distances induced by various Lp norms. Finally, examples of Euclidean graphs are provided in which the ratio of the lengths of an optimal Traveling Salesman path and a minimum spanning tree can be arbitrarily close to 2.

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 l-edge-connected subgraphs is briefly discussed.

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