Approximation Schemes for Degree-Restricted MST and Red-Blue Separation Problem

Arora, Sanjeev; Chang, Kevin L.

doi:10.1007/3-540-45061-0_16

“…Notice that u 1 ∈ W as u i is the only u-neighbor in W but that u 1 and u i are adjacent after the addition of u 1 u i , so u 1 must be in S; otherwise u 1 is a new neighbor of W . Also, since S ∪ {u} is a minimal cutset before the operations, there exists some u 1 We remark that the following fact similar to Lemma 4.3 is known. If G is a simple k-vertex-connected graph, u is a vertex of degree at least k + 1, and uv is an edge incident to u, then u has a neighbor w such that G − uv + vw is k-vertex-connected.…”

Section: Claim 422 G Is K-vertex-connected If There Are K Internallmentioning

confidence: 83%

“…On the Euclidean plane, there is a minimum spanning tree of maximum degree 5 [37], and Khuller, Raghavachari, and Young [24] showed how to convert such a spanning tree to a spanning tree with maximum degrees 3 and 4 with cost no more than 1.5 and 1.25 times the minimum spanning tree, respectively. Further improvements are made in [7,23], and there is a quasi-polynomial time approximation scheme in [1]. For higher dimensional Euclidean space, Khuller, Raghavachari, and Young [24] showed that the problem of finding a spanning tree with maximum degree 3 is approximable within a factor of 5/3.…”

Section: Spanning Treesmentioning

confidence: 99%

“…Formally, we are given a complete undirected graph G = (V, E), a connectivity requirement function r : V ×V → Z on pairs of vertices, a cost function c : V ×V → Q 1 We remark that if parallel edges are allowed in the solutions, then a statement similar to Theorem 1.1(1) is proved by Bienstock, Brickell, and Monma [6]. However, for degree bounded network design problems, there are capacity constraints on edges, and so their result cannot be applied directly.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Degree Bounded Network Design with Metric Costs

Chan

¹

,

Fung

²

,

Lau

³

et al. 2008

2008 49th Annual IEEE Symposium on Foundations of Computer Science

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Abstract. 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 a 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 the triangle inequality, there are constant factor approximation algorithms for various degree bounded network design problems. In global edge-connectivity, there is a (2 + 1 k )-approximation algorithm for the minimum bounded degree k-edge-connected subgraph problem. In local edge-connectivity, there is a 4-approximation algorithm for the minimum bounded degree Steiner network problem when rmax is even, and a 5.5-approximation algorithm when rmax is odd, where rmax is the maximum connectivity requirement. In global vertex-connectivity, there is a (2 + k−1 n + 1 k )-approximation algorithm for the minimum bounded degree k-vertex-connected subgraph problem when n ≥ 2k, where n is the number of vertices. For spanning tree, there is a (1 + 1 B−1 )-approximation algorithm for the minimum bounded degree spanning tree problem. These approximation algorithms return solutions with the smallest possible maximum degree, and in most cases 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 the metric traveling salesman problem. 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. 1. Introduction. Consider finding a minimum cost k-edge-subgraph with maximum degree at most B in a weighted undirected graph. This problem is a generalization of the traveling salesman problem (TSP) when k = B = 2 and the minimum bounded degree spanning tree problem when k = 1. In general this problem does not admit any polynomial time approximation algorithm, since the feasibility problem is already NP-hard. Recent research has thus focused on obtaining bicriteria approximation algorithms for degree bounded network design problems [19,31,39,32].In some network design problems the cost function satisfies the triangle inequality, and stronger algorithmic results are known [27,9,11]. For the TSP, although there is no polynomial factor approximation algorithm in general, it is well known that there is a 1.5-approximation algorithm assuming the triangle inequality [10]. This motivates us to study the more general degree bounded network design problems with metric costs.Formally, we are given a complete ...

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“…This result has been generalized to k-vertex-connectivity in [6], which combined with [27] implies a bicriteria (2 + k−1 n , +1)-approximation algorithm for the minimum bounded degree k-vertex-connected subgraph problem. For bounded degree spanning trees, there is a simple 2-approximation algorithm, and improvement over this 2-approximation algorithm was known for Euclidean space [37,12,24,1,7,23].…”

Section: Related Workmentioning

confidence: 99%

Degree Bounded Network Design with Metric Costs

Chan¹,

Fung²,

Lau³

et al. 2011

View full text Add to dashboard Cite

Abstract. 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 a 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 the triangle inequality, there are constant factor approximation algorithms for various degree bounded network design problems. In global edge-connectivity, there is a (2 + 1 k )-approximation algorithm for the minimum bounded degree k-edge-connected subgraph problem. In local edge-connectivity, there is a 4-approximation algorithm for the minimum bounded degree Steiner network problem when rmax is even, and a 5.5-approximation algorithm when rmax is odd, where rmax is the maximum connectivity requirement. In global vertex-connectivity, there is a (2 + k−1 n + 1 k )-approximation algorithm for the minimum bounded degree k-vertex-connected subgraph problem when n ≥ 2k, where n is the number of vertices. For spanning tree, there is a (1 + 1 B−1 )-approximation algorithm for the minimum bounded degree spanning tree problem. These approximation algorithms return solutions with the smallest possible maximum degree, and in most cases 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 the metric traveling salesman problem. 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. 1. Introduction. Consider finding a minimum cost k-edge-subgraph with maximum degree at most B in a weighted undirected graph. This problem is a generalization of the traveling salesman problem (TSP) when k = B = 2 and the minimum bounded degree spanning tree problem when k = 1. In general this problem does not admit any polynomial time approximation algorithm, since the feasibility problem is already NP-hard. Recent research has thus focused on obtaining bicriteria approximation algorithms for degree bounded network design problems [19,31,39,32].In some network design problems the cost function satisfies the triangle inequality, and stronger algorithmic results are known [27,9,11]. For the TSP, although there is no polynomial factor approximation algorithm in general, it is well known that there is a 1.5-approximation algorithm assuming the triangle inequality [10]. This motivates us to study the more general degree bounded network design problems with metric costs.Formally, we are given a complete ...

show abstract

“…For red-blue separation by a polygon, a natural problem is the minimum-perimeter polygon that separates the bichromatic point set. This problem is NP-hard by reduction from Euclidean traveling salesperson [4,11]; polynomial-time approximation schemes follow from the m-guillotine method of Mitchell [19] and from Arora's method [7]. Minimum-link separation has also received attention [6,1,18].…”

Section: Introductionmentioning

confidence: 99%

Delineating Boundaries for Imprecise Regions

Reinbacher

¹

,

Benkert

²

,

Kreveld

³

et al. 2005

Algorithms – ESA 2005

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In geographic information retrieval, queries often name geographic regions that do not have a well-defined boundary, such as "Southern France." We provide two algorithmic approaches to the problem of computing reasonable boundaries of such regions based on data points that have evidence indicating that they lie either inside or outside the region. Our problem formulation leads to a number of subproblems related to red-blue point separation and minimum-perimeter polygons, many of which we solve algorithmically. We give experimental results from our implementation and a comparison of the two approaches.

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Approximation Schemes for Degree-Restricted MST and Red-Blue Separation Problem

Cited by 5 publications

References 11 publications

Degree Bounded Network Design with Metric Costs

Degree Bounded Network Design with Metric Costs

Degree Bounded Network Design with Metric Costs

Delineating Boundaries for Imprecise Regions

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