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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 32 (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.
An Efficient Evolutionary Algorithm for the DegreeConstrained Minimum Spanning Tree Problem
, 2000
"... The representation of candidate solutions and the variation operators are fundamental design choices in an evolutionary algorithm (EA). This paper proposes a novel representation technique and suitable variation operators for the degreeconstrained minimum spanning tree problem. For a weighted, undi ..."
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Cited by 23 (5 self)
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The representation of candidate solutions and the variation operators are fundamental design choices in an evolutionary algorithm (EA). This paper proposes a novel representation technique and suitable variation operators for the degreeconstrained minimum spanning tree problem. For a weighted, undirected graph G(V, E), this problem seeks to identify the shortest spanning tree whose node degrees do not exceed an upper bound d 2. Within the EA, a candidate spanning tree is simply represented by its set of edges. Special initialization, crossover, and mutation operators are used to generate new, always feasible candidate solutions. In contrast to previous spanning tree representations, the proposed approach provides substantially higher locality and is nevertheless computationally efficient; an offspring is always created in O(V time. In addition, it is shown how problemdependent heuristics can be effectively incorporated into the initialization, crossover, and mutation operators without increasing the timecomplexity. Empirical results are presented for hard problem instances with up to 500 vertices. Usually, the new approach identifies solutions superior to those of several other optimization methods within few seconds. The basic ideas of this EA are also applicable to other network optimization tasks.
A Weighted Coding in a Genetic Algorithm for the DegreeConstrained Minimum Spanning Tree Problem
, 2000
"... is a fundamental design choice in a genetic algorithm. This paper describes a novel coding of spanning trees in a genetic algorithm for the degreeconstrained minimum spanning tree problem. For a connected, weighted graph, this problem seeks to identify the shortest spanning tree whose degree does n ..."
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Cited by 19 (4 self)
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is a fundamental design choice in a genetic algorithm. This paper describes a novel coding of spanning trees in a genetic algorithm for the degreeconstrained minimum spanning tree problem. For a connected, weighted graph, this problem seeks to identify the shortest spanning tree whose degree does not exceed an upper bound k 2. In the coding, chromosomes are strings of numerical weights associated with the target graph's vertices. The weights temporarily bias the graph's edge costs, and an extension of Prim's algorithm, applied to the biased costs, identifies the feasible spanning tree a chromosome represents. This decoding algorithm enforces the degree constraint, so that all chromosomes represent valid solutions and there is no need to discard, repair, or penalize invalid chromosomes. On a set of hard graphs whose unconstrained minimum spanning trees are of high degree, a genetic algorithm that uses this coding identifies degreeconstrained minimum spanning trees that are on average shorter than those found by several competing algorithms.
Euclidean BoundedDegree Spanning Tree Ratios
, 2003
"... Let K be the worstcase (supremum) ratio of the weight of the minimum degreeK spanning tree to the weight of the minimum spanning tree, over all finite point sets in the Euclidean plane. It is known that ..."
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Cited by 16 (0 self)
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Let K be the worstcase (supremum) ratio of the weight of the minimum degreeK spanning tree to the weight of the minimum spanning tree, over all finite point sets in the Euclidean plane. It is known that
EdgeSets: An Effective Evolutionary Coding of Spanning Trees
, 2002
"... The fundamental design choices in an evolutionary algorithm are its representation of candidate solutions and the operators that will act on that representation. We propose representing spanning trees in evolutionary algorithms for network design problems directly as sets of their edges, and we d ..."
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Cited by 14 (7 self)
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The fundamental design choices in an evolutionary algorithm are its representation of candidate solutions and the operators that will act on that representation. We propose representing spanning trees in evolutionary algorithms for network design problems directly as sets of their edges, and we describe initialization, recombination, and mutation operators for this representation. The operators offer
A New Evolutionary Approach to the DegreeConstrained Minimum Spanning Tree Problem
 IEEE Transactions on Evolutionary Computation
, 1999
"... Finding the degreeconstrained minimum spanning tree (dMST) of a graph is a wellstudied NPhard problem of importance in communications network design and other networkrelated problems. In this paper we describe some previously proposed algorithms for solving the problem, and then introduce a nove ..."
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Cited by 10 (2 self)
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Finding the degreeconstrained minimum spanning tree (dMST) of a graph is a wellstudied NPhard problem of importance in communications network design and other networkrelated problems. In this paper we describe some previously proposed algorithms for solving the problem, and then introduce a novel tree construction algorithm called the Randomised Primal Method (RPM) which builds degreeconstrained trees of low cost from solution vectors taken as input. RPM is applied in three stochastic iterative search methods: simulated annealing, multistart hillclimbing, and a genetic algorithm. While other researchers have mainly concentrated on finding spanning trees in Euclidean graphs, we consider the more general case of random graph problems. We describe two random graph generators which produce particularly challenging dMST problems. On these and other problems we find that the genetic algorithm employing RPM outperforms simulated annealing and multistart hillclimbing. Our experimental ...
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.
On degree constrained shortest paths
 European Symposium on Algorithms (ESA
, 2005
"... Abstract. Traditional shortest path problems play a central role in both the design and use of communication networks and have been studied extensively. In this work, we consider a variant of the shortest path problem. The network has two kinds of edges, “actual ” edges and “potential ” edges. In ad ..."
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Cited by 3 (1 self)
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Abstract. Traditional shortest path problems play a central role in both the design and use of communication networks and have been studied extensively. In this work, we consider a variant of the shortest path problem. The network has two kinds of edges, “actual ” edges and “potential ” edges. In addition, each vertex has a degree/interface constraint. We wish to compute a shortest path in the graph that maintains feasibility when we convert the potential edges on the shortest path to actual edges. The central difficulty is when a node has only one free interface, and the unconstrained shortest path chooses two potential edges incident on this node. We first show that this problem can be solved in polynomial time by reducing it to the minimum weighted perfect matching problem. The number of steps taken by this algorithm isÇ���ÐÓ����for the singlesource singledestination case. In other words, for eachÚwe compute the shortest pathÈÚsuch that converting the potential edges onÈÚto actual edges, does not violate any degree constraint. We then develop more efficient algorithms by extending Dijkstra’s shortest path algorithm. The number of steps taken by the latter algorithm isÇ����Î�, even for the singlesource all destination case. 1
Approximation Algorithms for Minimumcost Lowdegree Subgraphs
, 2003
"... In this thesis we address problems in minimumcost lowdegree network design. In the design of communication networks we often face the problem of building a network connecting a large number of endhosts. Available hardware or software often imposes additional restrictions on the topology of the ne ..."
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Cited by 3 (1 self)
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In this thesis we address problems in minimumcost lowdegree network design. In the design of communication networks we often face the problem of building a network connecting a large number of endhosts. Available hardware or software often imposes additional restrictions on the topology of the network. One example for such a requirement arises in the area of computer networks: We are given a number of client hosts in a network that communicate with each other using a certain communication protocol that limits the number of simultaneously open connections for each host. In graph theoretic terms, where hosts correspond to nodes and network connections are represented by edges, this restriction naturally translates to a bound on the maximum nodedegree of the constructed network. In this thesis we first address the problem of finding a spanning tree T for a given undirected graph G = (V, E) with maximum nodedegree at most a given parameter B> 1. Among all such trees, we aim to find one that minimizes the total edgecost for a given cost function c on the edges of G. We develop an algorithm based on Lagrangean relaxation. We show how to compute a spanning tree with maximum nodedegree O(B + log(n)) and total cost at most a constant factor worse than the cost of
Approximating minimum cost multigraphs of specified edgeconnectivity under degree bounds
 in Proceedings of the 9th JapanKorea Joint Workshop on Algorithms and Computation
, 2006
"... In this paper, we consider the problem of constructing a minimum cost graph with a specified edgeconnectivity under a degree constraint. For a set V of vertices, let r: � � V → Z+ 2 be a connectivity demand, a: V → Z+ be a lower capacity, b: V → Z+ be an upper capacity and c: � � V 2 → Q+ be a ..."
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Cited by 3 (2 self)
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In this paper, we consider the problem of constructing a minimum cost graph with a specified edgeconnectivity under a degree constraint. For a set V of vertices, let r: � � V → Z+ 2 be a connectivity demand, a: V → Z+ be a lower capacity, b: V → Z+ be an upper capacity and c: � � V 2 → Q+ be a metric edge cost. The problem (V, r, a, b, c) asks to find a minimum cost multigraph G = (V, E) with no selfloops such that λ(u, v) ≥ r(u, v) for each pair u, v ∈ V and a(v) ≤ d(v) ≤ b(v) for each v ∈ V, where λ(u, v) (resp., d(v)) denotes the localedgeconnectivity between u and v (resp., the degree of v) in G. We show several conditions on functions r, a, b and c for which the above problem admits an approximation algorithm. For example, we give a (2 + 1 / ⌊k/2⌋)approximation algorithm to (V, r, a, b, c) with r(u, v) ≥ 2, u, v ∈ V and a uniform b(v), v ∈ V, where k = minu,v∈V r(u, v). To design the algorithms in this paper, we use our new results on edgesplitting and detachment, which are graph transformations to split vertices while preserving edgeconnectivity.