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15
Approximation schemes for NPhard geometric optimization problems: A survey
 Mathematical Programming
, 2003
"... NPhard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (di ..."
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Cited by 39 (2 self)
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NPhard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (distance between nodes (x1, y1) and (x2, y2) is ((x1−x2) 2 +(y1−y2) 2) 1/2) then the problem is called Euclidean TSP. More generally the distance could be defined using other norms, such as ℓp norms for any p> 1. All these are subcases of the more general notion of a geometric norm or Minkowski norm. We will refer to the version of the problem with a general geometric norm as geometric TSP. Some other NPhard geometric optimization problems are Minimum Steiner Tree (“Given n points, find the smallest network connecting them,”), kTSP(“Given n points and a number k, find the shortest salesman tour that visits k points”), kMST (“Given n points and a number k, find the shortest tree that contains k points”), vehicle routing, degree restricted minimum
What Would Edmonds Do? Augmenting Paths and Witnesses for DegreeBounded MSTs
 IN PROCEEDINGS OF APPROX/RANDOM
, 2005
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Approximation schemes for degreerestricted MST and redblue separation problem
 Algorithmica
, 2003
"... Abstract We develop a quasipolynomial time approximation scheme for the Euclidean version of the Degreerestricted 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 with ..."
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Cited by 13 (1 self)
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Abstract We develop a quasipolynomial time approximation scheme for the Euclidean version of the Degreerestricted 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 RedBlue 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.
A pushrelabel algorithm for approximating degreebounded spanning trees
 In Proc. of International Colloquium on Automata, Languages and Programming (ICALP
, 2006
"... Abstract. Given a graph G and degree bound B on its nodes, the boundeddegree minimum spanning tree (BDMST) problem is to find a minimum cost spanning tree among the spanning trees with maximum degree B. This bicriteria optimization problem generalizes several combinatorial problems, including the ..."
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Cited by 7 (2 self)
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Abstract. Given a graph G and degree bound B on its nodes, the boundeddegree minimum spanning tree (BDMST) problem is to find a minimum cost spanning tree among the spanning trees with maximum degree B. This bicriteria 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 polynomialtime algoβ 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 nonuniform 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 pushrelabel framework developed by Goldberg [7] for the maximum flow problem. To our knowledge, this is the first instance of a pushrelabel approximation algorithm for an NPhard 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
Degree Bounded Network Design with Metric Costs
"... 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 ..."
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Cited by 7 (3 self)
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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 NPhard, 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 edgeconnectivity: There is a (2 + 1 k)approximation algorithm for the minimum bounded degree kedgeconnected subgraph problem. • Local edgeconnectivity: There is a 6approximation algorithm for the minimum bounded degree Steiner network problem. • Global vertexconnectivity: There is a (2 + k−1 n + 1 k)approximation algorithm for the minimum bounded degree kvertexconnected 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 simplicitypreserving edge splittingoff operation, which is used to “shortcut” vertices with high degree while maintaining connectivity requirements and preserving simplicity of the solutions.
DegreeBounded Minimum Spanning Trees
, 2004
"... 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 . ..."
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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 .
A PushRelabel Approximation Algorithm for Approximating the MinimumDegree MST Problem and its Generalization to Matroids
, 2007
"... In the minimumdegree 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 NPhard and generalizes the Hamiltonian path problem. We give an algorithm that out ..."
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In the minimumdegree 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 NPhard 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 kary hypergraph G = (V,E) and weighted matroid M with E as its ground set, find a minimumcost 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 pushrelabel framework developed by Goldberg [8] for the maximumflow problem. To our knowledge, this is the first use of the pushrelabel technique in an approximation algorithm for an NPhard problem. The MDMST problem is closely connected to the boundeddegree 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 nearlogarithmic 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 nonuniform degree bounds. 1
NarrowShallowLowLight Trees with and without Steiner Points
"... 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 ca ..."
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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 narrowshallowlowlight (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 rootdistortion. On the other hand, we show that the increased rootdistortion 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 rootdistortion (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.
Maintaining Connectivity in Sensor Networks Using Directional Antennae
"... 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 i ..."
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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 tradeoffs 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.