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142
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 38 (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
On The Symmetric Range Assignment Problem In Wireless Ad Hoc Networks
, 2002
"... In this paper we consider a constrained version of the range assignment problem for wireless ad hoc networks, where the value the node transmitting ranges must be assigned in such a way that the resulting communication graph is strongly connected and the energy cost is minimum. We impose the further ..."
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Cited by 37 (1 self)
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In this paper we consider a constrained version of the range assignment problem for wireless ad hoc networks, where the value the node transmitting ranges must be assigned in such a way that the resulting communication graph is strongly connected and the energy cost is minimum. We impose the further requirement of symmetry on the resulting communication graph. We also consider a weaker notion of symmetry, in which only the existence of a set of symmetric edges that renders the communication graph connected is required. Our interest in these problems is motivated by the fact that a (weakly) symmetric range assignment can be more easily integrated with existing higher and lowerlevel protocols for ad hoc networks, which assume that all the nodes have the same transmitting range. We show that imposing symmetry does not change the complexity of the problem, which remains NPhard in two and threedimensional networks. We also show that a weakly symmetric range assignment can reduce the energy cost considerably with respect to the homogeneous case, in which all the nodes have the same transmitting range, and that no further (asymptotic) bene t is expected from the asymmetric range assignment. Hence, the results presented in this paper indicate that weak symmetry is a desirable property of the range assignment.
A ReservationBased Multicast (RBM) Routing Protocol for Mobile Networks: Initial Route Construction Phase
, 1995
"... We propose a combined multicast routing and resource reservation protocol, termed ReservationBased Multicast (RBM), that performs routing in a fashion similar to Protocol Independent Multicast (PIM), but which is intended for mobile operation and routes hierarchicallyencoded data streams based on ..."
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Cited by 34 (0 self)
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We propose a combined multicast routing and resource reservation protocol, termed ReservationBased Multicast (RBM), that performs routing in a fashion similar to Protocol Independent Multicast (PIM), but which is intended for mobile operation and routes hierarchicallyencoded data streams based on userspecified fidelity requirements, realtime delivery thresholds and prevailing network bandwidth constraints. The protocol retains the fully distributed operation, scalability and receiverinitiated orientation of PIM; but, unlike PIM, the protocol is tightly coupled to an underlying, distributed, unicast routing protocol thereby facilitating operation in a dynamic topology. This paper focuses on the initial route construction phase, assumed to occur during a static "snapshot" of the dynamic topology, and is therefore applicable to fixed networks as well, e.g. the Internet. A forthcoming paper will detail the protocol's robustness and adaptivity to arbitrary topological changes during bot...
Computing NearOptimal Solutions to Combinatorial Optimization Problems
 IN COMBINATORIAL OPTIMIZATION, DIMACS SERIES IN DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE
, 1995
"... In the past few years, there has been significant progress in our understanding of the extent to which nearoptimal solutions can be efficiently computed for NPhard combinatorial optimization problems. This paper surveys these recent developments, while concentrating on the advances made in the ..."
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Cited by 31 (0 self)
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In the past few years, there has been significant progress in our understanding of the extent to which nearoptimal solutions can be efficiently computed for NPhard combinatorial optimization problems. This paper surveys these recent developments, while concentrating on the advances made in the design and analysis of approximation algorithms, and in particular, on those results that rely on linear programming and its generalizations.
A distributed algorithm of delaybounded multicast routing for multimedia applications in wide area networks
 IEEE/ACM Transactions on Networking Vol.6 No.6
, 1998
"... Abstract—Multicast routing is to find a tree which is rooted from a source node and contains all multicast destinations. There are two requirements of multicast routing in many multimedia applications: optimal network cost and bounded delay. The network cost of a tree is defined as the sum of the c ..."
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Cited by 30 (1 self)
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Abstract—Multicast routing is to find a tree which is rooted from a source node and contains all multicast destinations. There are two requirements of multicast routing in many multimedia applications: optimal network cost and bounded delay. The network cost of a tree is defined as the sum of the cost of all links in the tree. The bounded delay of a routing tree refers to the feature that the accumulated delay from the source to any destination along the tree shall not exceed a prespecified bound. This paper presents a distributed heuristic algorithm which generates routing trees having a suboptimal network cost under the delay bound constraint. The proposed algorithm is fully distributed, efficient in terms of the number of messages and convergence time, and flexible in dynamic membership changes. A large amount of simulations have been done to show the network cost of the routing trees generated by our algorithm is similar to, or even better than, other existing algorithms. Index Terms — Delaybounded multicast, distributed routing algorithm, multicast routing, multimedia systems, realtime communications. I.
An Improved LPbased Approximation for Steiner Tree
, 2009
"... The Steiner tree problem is one of the most fundamentalhard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum weight tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from to the current best���[Robin ..."
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Cited by 27 (0 self)
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The Steiner tree problem is one of the most fundamentalhard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum weight tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from to the current best���[Robins,ZelikovskySIDMA’05]. All these algorithms are purely combinatorial. A longstanding open problem is whether there is an LPrelaxation for Steiner tree with integrality gap smaller than [Vazirani,RajagopalanSODA’99]. In this paper we improve the approximation factor for Steiner tree, developing an LPbased approximation a� algorithm. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider a directedcomponent cut relaxation for the�restricted Steiner tree problem. We sample one of these components with probability proportional to the value of the associated variable in the optimal fractional solution and contract it. We iterate this process for a proper number of times and finally output the sampled components together
Highly scalable algorithms for rectilinear and octilinear Steiner trees
 In Proc. Asian and South Pacific Design Automation Conf
, 2003
"... problem, which asks for a minimumlength interconnection of a given set of terminals in the rectilinear plane, is one of the fundamental problems in electronic design automation. Recently there has been renewed interest in this problem due to the need for highly scalable algorithms able to handle ne ..."
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Cited by 27 (3 self)
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problem, which asks for a minimumlength interconnection of a given set of terminals in the rectilinear plane, is one of the fundamental problems in electronic design automation. Recently there has been renewed interest in this problem due to the need for highly scalable algorithms able to handle nets with tens of thousands of terminals. In this paper we give a practical � heuristic for computing nearoptimal rectilinear Steiner trees based on a batched version of the greedy triple contraction algorithm of Zelikovsky [21]. Experiments conducted on both random and industry testcases show that our heuristic matches or exceeds the quality of best known RSMT heuristics, e.g., on random instances with more than 100 terminals our heuristic improves over the rectilinear minimum spanning tree by an average of 11%. Moreover, our heuristic has very well scaling runtime, e.g., it can route a 34kterminals net extracted from a real design in less than 25 seconds compared to over 86 minutes needed by the edgebased heuristic of Borah, Owens, and Irwin [3]. Since our heuristic is graphbased, it can be easily modified to handle practical considerations such as routing obstacles, preferred directions, via costs, and octilinear routing – indeed, experimental results show only a small factor increase in runtime when switching from rectilinear to octilinear routing. I.
An Optimal Bound for the MST Algorithm to Compute Energy Efficient Broadcast Trees in Wireless Networks
 IN ICALP
, 2005
"... Computing energy efficient broadcast trees is one of the most prominent operations in wireless networks. For stations embedded in the Euclidean plane, the best analytic result known to date is a 6.33approximation algorithm based on computing an Euclidean minimum spanning tree. We improve the analy ..."
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Cited by 26 (0 self)
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Computing energy efficient broadcast trees is one of the most prominent operations in wireless networks. For stations embedded in the Euclidean plane, the best analytic result known to date is a 6.33approximation algorithm based on computing an Euclidean minimum spanning tree. We improve the analysis of this algorithm and show that its approximation ratio is 6, which matches a previously known lower bound for this algorithm.
A Comparison of Multicast Trees and Algorithms
, 1994
"... . Multicast trees can be shared across sources or may be sourcespecific. Inspired by recent interests in using shared trees for interdomain multicasting [BFC93] [WLE + 92], this paper investigates the tradeoffs among different algorithms and tree types. Because of the dynamic nature of grap ..."
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Cited by 26 (3 self)
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. Multicast trees can be shared across sources or may be sourcespecific. Inspired by recent interests in using shared trees for interdomain multicasting [BFC93] [WLE + 92], this paper investigates the tradeoffs among different algorithms and tree types. Because of the dynamic nature of graphs, only worst case delay bounds can be calculated using analytical methods. We present simulation results over random graphs that demonstrate the performance of these trees, under different circumstances. We evaluate the performance in terms of path length, link cost, and traffic concentrations. Draft submitted to INFOCOM'94 1 Introduction Pointtomultipoint communications will play a critical role in future computer networks. The problem of computing the optimal multicast path, in the shape of a tree or a group of trees, has many potential solutions; however, to date there have not been systematic comparisons among the different solutions. Today's multicast applications are prima...
An efficient algorithm for minimizing a sum of Euclidean norms with applications
 SIAM Journal on Optimization
, 1997
"... Abstract. In recent years rich theories on polynomialtime interiorpoint algorithms have been developed. These theories and algorithms can be applied to many nonlinear optimization problems to yield better complexity results for various applications. In this paper, the problem of minimizing a sum o ..."
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Cited by 24 (4 self)
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Abstract. In recent years rich theories on polynomialtime interiorpoint algorithms have been developed. These theories and algorithms can be applied to many nonlinear optimization problems to yield better complexity results for various applications. In this paper, the problem of minimizing a sum of Euclidean norms is studied. This problem is convex but not everywhere differentiable. By transforming the problem into a standard convex programming problem in conic form, we show that an ɛoptimal solution can be computed efficiently using interiorpoint algorithms. As applications to this problem, polynomialtime algorithms are derived for the Euclidean single facility location problem, the Euclidean multifacility location problem, and the shortest network under a given tree topology. In particular, by solving the Newton equation in linear time using Gaussian elimination on leaves of a tree, we present an algorithm which computes an ɛoptimal solution to the shortest network under a given full Steiner topology interconnecting N regular points, in O(N √ N(log(¯c/ɛ)+ log N)) arithmetic operations where ¯c is the largest pairwise distance among the given points. The previous bestknown result on this problem is a graphical algorithm which requires O(N 2) arithmetic operations under certain conditions. Key words. polynomial time, interiorpoint algorithm, minimizing a sum of Euclidean norms, Euclidean facilities location, shortest networks, Steiner minimum trees