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Abstract We develop a quasi-polynomial time approximation scheme for the Euclidean version of the Degree-restricted MST problem by adapting techniques used previously by Arora for approximating TSP. Given n points in the plane, d = 3 or 4, and ffl? 0, the scheme finds an approximation with cost within 1 + ffl of the lowest cost spanning tree with the property that all nodes have degree at most d. We also develop a polynomial time approximation scheme for the Euclidean version of the Red-Blue Separation Problem, again extending Arora's techniques. Given ffl? 0, the scheme finds an approximation with cost within 1 + ffl of the cost of the optimum separating polygon of the input nodes, in nearly linear time.

Abstract. Given a graph G and degree bound B on its nodes, the bounded-degree minimum spanning tree (BDMST) problem is to find a minimum cost spanning tree among the spanning trees with maximum degree B. This bi-criteria optimization problem generalizes several combinatorial problems, including the Traveling Salesman Path Problem (TSPP). An (α, f(B))-approximation algorithm for the BDMST problem produces a spanning tree that has maximum degree f(B) andcostwithina factor α of the optimal cost. Könemann and Ravi [13,14] give a polynomialtime (1 + 1 β,bB(1 + β)+logbn)-approximation algorithm for any b>1, β>0. In a recent paper [2], Chaudhuri et al. improved these results with a (1, bB+ √ b logb n)-approximation for any b>1. In this paper, we present a(1+ 1, 2B(1 + β)+o(B(1 + β)))-approximation polynomial-time algo-β rithm. That is, we give the first algorithm that approximates both degree and cost to within a constant factor of the optimal. These results generalize to the case of non-uniform degree bounds. The crux of our solution is an approximation algorithm for the related problem of finding a minimum spanning tree (MST) in which the maximum degree of the nodes is minimized, a problem we call the minimumdegree MST (MDMST) problem. Given a graph G for which the degree of the MDMST solution is Δopt, our algorithm obtains in polynomial time an MST of G of degree at most 2Δopt + o(Δopt). This result improves on a previous result of Fischer [4] that finds an MST of G of degree at most bΔopt +logbnfor any b>1, and on the improved quasipolynomial algorithm of [2]. Our algorithm uses the push-relabel framework developed by Goldberg [7] for the maximum flow problem. To our knowledge, this is the first instance of a push-relabel approximation algorithm for an NP-hard problem, and we believe these techniques may have larger impact. We note that for B = 2, our algorithm gives a tree of cost within a (1 + ɛ)-factor of the optimal solution to TSPP and of maximum degree O ( 1 ɛ)for any ɛ>0, even on graphs not satisfying the triangle inequality. 1

Given a complete undirected graph, a cost function on edges and a degree bound B, the degree bounded network design problem is to find a minimum cost simple subgraph with maximum degree B satisfying given connectivity requirements. Even for simple connectivity requirement such as finding a spanning tree, computing a feasible solution for the degree bounded network design problem is already NP-hard, and thus there is no polynomial factor approximation algorithm for this problem. In this paper, we show that when the cost function satisfies triangle inequalities, there are constant factor approximation algorithms for various degree bounded network design problems. • Global edge-connectivity: There is a (2 + 1 k)approximation algorithm for the minimum bounded degree k-edge-connected subgraph problem. • Local edge-connectivity: There is a 6-approximation algorithm for the minimum bounded degree Steiner network problem. • Global vertex-connectivity: There is a (2 + k−1 n + 1 k)-approximation algorithm for the minimum bounded degree k-vertex-connected subgraph problem. • Spanning tree: There is an (1 + 1 d−1)-approximation algorithm for the minimum bounded degree spanning tree problem. These approximation algorithms return solutions with smallest possible maximum degree, and the cost guarantee is obtained by comparing to the optimal cost when there are no degree constraints. This demonstrates that degree constraints can be incorporated into network design problems with metric costs. Our algorithms can be seen as a generalization of Christofides’ algorithm for metric TSP. The main technical tool is a simplicity-preserving edge splitting-off operation, which is used to “short-cut” vertices with high degree while maintaining connectivity requirements and preserving simplicity of the solutions.

Given n points in the Euclidean plane, the degree-- MST problem asks for a spanning tree of minimum weight in which the degree of each node is at most .

Abstract. In this paper we introduce and study cooperative variants of the Traveling Salesperson Problem. In these problems a salesperson has to make deliveries to customers who are willing to help in the process. Customer cooperativeness may be manifested in several modes: they may assist by approaching the salesperson, by reselling the goods they purchased to other customers, or by doing both. Several objectives are of interest: minimizing the total distance traveled by all the participants, minimizing the maximal distance traveled by a participant and minimizing the total time until all the deliveries are made. All the combinations of cooperation-modes and objective functions are considered, both in weighted undirected graphs and in Euclidean space. We show that most of the problems have a constant approximation algorithm, many of the others admit a PTAS, and a few are solvable in polynomial time. On the intractability side we provide NP-hardness proofs and inapproximability factors, some of which are tight. 1

Abstract Connectivity in wireless sensor networks may be established using either omnidirectional or directional antennae. The former radiate power uniformly in all directions while the latter emit greater power in a specified direction thus achieving increased transmission range and encountering reduced interference from unwanted sources. Regardless of the type of antenna being used the transmission cost of each antenna is proportional to the coverage area of the antenna. It is of interest to design efficient algorithms that minimize the overall transmission cost while at the same time maintaining network connectivity. Consider a set S of n points in the plane modeling sensors of an ad hoc network. Each sensor is equipped with a fixed number of directional antennae modeled as a circular sector with a given spread (or angle) and range (or radius). Construct a network with the sensors as the nodes and with directed edges (u,v) connecting sensors u and v if v lies within u’s sector. We survey recent algorithms and study trade-offs on the maximum angle, sum of angles, maximum range and the number of antennae per sensor for the problem of establishing strongly connected networks of sensors.

In the minimum-degree minimum spanning tree (MDMST) problem, we are given a graph G, and the goal is to find a minimum spanning tree (MST) T, such that the maximum degree of T is as small as possible. This problem is NP-hard and generalizes the Hamiltonian path problem. We give an algorithm that outputs an MST of degree at most 2∆opt(G)+o(∆opt(G)), where ∆opt(G) denotes the degree of the optimal tree. This result improves on a previous result of Fischer [5] that finds an MST of degree at most b∆opt(G) + log b n, for any b> 1. The MDMST problem is a special case of the following problem: given a k-ary hypergraph G = (V,E) and weighted matroid M with E as its ground set, find a minimum-cost basis (MCB) T of M such that the degree of T in G is as small as possible. Our algorithm immediately generalizes to this problem, finding an MCB of degree at most k 2 ∆opt(G,M)+O(k � k∆opt(G,M)). We use the push-relabel framework developed by Goldberg [8] for the maximum-flow problem. To our knowledge, this is the first use of the push-relabel technique in an approximation algorithm for an NP-hard problem. The MDMST problem is closely connected to the bounded-degree minimum spanning tree (BDMST) problem. Given a graph G and degree bound B on its nodes, the BDMST problem is to find a minimum cost spanning tree among the spanning trees with maximum degree B. Previous algorithms for this problem by Könemann and Ravi [13, 14] and by Chaudhuri et al. [2] incur a near-logarithmic additive error in the degree. We give the first BDMST algorithm that approximates both the degree and the cost to within a constant factor of the optimal. These results generalize to the case of non-uniform degree bounds. 1

We show that for every set S of n points in the plane and a designated point rt ∈ S, there exists a tree T that has small maximum degree, depth and weight. Moreover, for every point v ∈ S, the distance between rt and v in T is within a factor of (1+ɛ) close to their Euclidean distance ‖rt, v‖. We call these trees narrow-shallow-low-light (NSLLTs). We demonstrate that our construction achieves optimal (up to constant factors) tradeoffs between all parameters of NSLLTs. Our construction extends to point sets in R d, for an arbitrarily large constant d. The running time of our construction is O(n · log n). We also study this problem in general metric spaces, and show that NSLLTs with small maximum degree, depth and weight can always be constructed if one is willing to compromise the root-distortion. On the other hand, we show that the increased root-distortion is inevitable, even if the point set S resides in a Euclidean space of dimension Θ(log n). On the bright side, we show that if one is allowed to use Steiner points then it is possible to achieve root-distortion (1+ɛ) together with small maximum degree, depth and weight for general metric spaces. Finally, we establish some lower bounds on the power of Steiner points in the context of Euclidean spanning trees and spanners.