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The complexity of analog computation
 in Math. and Computers in Simulation 28(1986
"... We ask if analog computers can solve NPcomplete problems efficiently. Regarding this as unlikely, we formulate a strong version of Church’s Thesis: that any analog computer can be simulated efficiently (in polynomial time) by a digital computer. From this assumption and the assumption that P ≠ NP w ..."
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Cited by 36 (0 self)
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We ask if analog computers can solve NPcomplete problems efficiently. Regarding this as unlikely, we formulate a strong version of Church’s Thesis: that any analog computer can be simulated efficiently (in polynomial time) by a digital computer. From this assumption and the assumption that P ≠ NP we can draw conclusions about the operation of physical devices used for computation. An NPcomplete problem, 3SAT, is reduced to the problem of checking whether a feasible point is a local optimum of an optimization problem. A mechanical device is proposed for the solution of this problem. It encodes variables as shaft angles and uses gears and smooth cams. If we grant Strong Church’s Thesis, that P ≠ NP, and a certain ‘‘Downhill Principle’ ’ governing the physical behavior of the machine, we conclude that it cannot operate successfully while using only polynomial resources. We next prove Strong Church’s Thesis for a class of analog computers described by wellbehaved ordinary differential equations, which we can take as representing part of classical mechanics. We conclude with a comment on the recently discovered connection between spin glasses and combinatorial optimization. 1.
Faster Minimization of Linear Wirelength for Global Placement
 IEEE Transactions on ComputerAided Design
, 1997
"... A linear wirelength objective more e#ectively captures timing, congestion, and other global placement considerations than a squared wirelength objective. The GORDIANL cell placement tool #16# minimizes linear wirelength by #rst approximating the linear wirelength objectiveby a modi#ed squared wirel ..."
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Cited by 33 (9 self)
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A linear wirelength objective more e#ectively captures timing, congestion, and other global placement considerations than a squared wirelength objective. The GORDIANL cell placement tool #16# minimizes linear wirelength by #rst approximating the linear wirelength objectiveby a modi#ed squared wirelength objective, then executing the following loop # #1# minimize the current objective to yield some approximate solution, and #2# use the resulting solution to construct a more accurate objective#until the solution converges. In this paper, we #rst show that the GORDIANL loop can be viewed as a special case of a new algorithm that generalizes a 1937 iteration due to Weiszfeld #19#. Speci# cally,we formulate the Weiszfeld iteration using a regularization parameter to control the tradeo# between convergence and solution accuracy; the GORDIANL iteration is equivalent to setting this regularization parameter to zero. Other novel numerical methods described in the paper, the Primal Newton it...
Facility location models for distribution system design
, 2004
"... The design of the distribution system is a strategic issue for almost every company. The problem of locating facilities and allocating customers covers the core topics of distribution system design. Model formulations and solution algorithms which address the issue vary widely in terms of fundamenta ..."
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Cited by 33 (0 self)
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The design of the distribution system is a strategic issue for almost every company. The problem of locating facilities and allocating customers covers the core topics of distribution system design. Model formulations and solution algorithms which address the issue vary widely in terms of fundamental assumptions, mathematical complexity and computational performance. This paper reviews some of the contributions to the current stateoftheart. In particular, continuous location models, network location models, mixedinteger programming models, and applications are summarized.
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 22 (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
The inverse FermatWeber problem
, 2008
"... Given n points in the plane with nonnegative weights, the inverse FermatWeber problem consists in changing the weights at minimum cost such that a prespecified point in the plane becomes the Euclidean 1median. The cost is proportional to the increase or decrease of the corresponding weight. In cas ..."
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Cited by 14 (11 self)
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Given n points in the plane with nonnegative weights, the inverse FermatWeber problem consists in changing the weights at minimum cost such that a prespecified point in the plane becomes the Euclidean 1median. The cost is proportional to the increase or decrease of the corresponding weight. In case that the prespecified point does not coincide with one of the given n points, the inverse FermatWeber problem can be formulated as linear program. We derive a purely combinatorial algorithm which solves the inverse FermatWeber problem with unit cost in O(n log n) time. If the prespecified point coincides with one of the given n points, it is shown that the corresponding inverse problem can be written as convex problem and hence is solvable in polynomial time to any fixed precision. 1 Inverse and reverse location problems In recent years inverse and reverse optimization problems found an increased interest. In a reverse optimization problem, we are given a budget for modifying parameters of the problem. The goal is to modify parameters of the problem such that an objective function attains its best possible value subject to the given budget. The inverse optimization problem consists in changing parameters of the problem at minimum cost such that a prespecified solution becomes optimal. In one of the first papers on this subject, Burton and
An Efficient Algorithm for Minimizing a Sum of PNorms
 SIAM Journal on Optimization
, 1997
"... We study the problem of minimizing a sum of pnorms where p is a fixed real number in the interval [1; 1]. Several practical algorithms have been proposed to solve this problem. However, none of them has a known polynomial time complexity. In this paper, we transform the problem into standard conic ..."
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Cited by 13 (2 self)
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We study the problem of minimizing a sum of pnorms where p is a fixed real number in the interval [1; 1]. Several practical algorithms have been proposed to solve this problem. However, none of them has a known polynomial time complexity. In this paper, we transform the problem into standard conic form. Unlike those in most convex optimization problems, the cone for the pnorm problem is not selfdual unless p = 2. Nevertheless, we are able to construct two logarithmically homogeneous selfconcordant barrier functions for this problem. The barrier parameter of the first barrier function does not depend on p. The barrier parameter of the second barrier function increases with p. Using both barrier functions, we present a primaldual potential reduction algorithm to compute an ffloptimal solution in polynomial time that is independent of p. Computational experiences of a Matlab implementation are also reported. Key words. Shortest network, Steiner minimum trees, facilities location, po...
The neglected pillar of material computation
 PHYSICA D
, 2008
"... Many novel forms of computational material have been suggested, from using slime moulds to solve graph searching problems, to using packaging foam to solve differential equations. I argue that attempting to force such novel approaches into the conventional Universal Turing computational framework wi ..."
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Cited by 13 (5 self)
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Many novel forms of computational material have been suggested, from using slime moulds to solve graph searching problems, to using packaging foam to solve differential equations. I argue that attempting to force such novel approaches into the conventional Universal Turing computational framework will provide neither insights into theoretical questions of computation, nor more powerful computational machines. Instead, we should be investigating matter from the perspective of its natural computational capabilities. I also argue that we should investigate nonbiological substrates, since these are less complex in that they have not been tuned by evolution to have their particular properties. Only then we will understand both aspects of computation (logical and physical) required to understand the computation occurring in biological systems.
A Newton Acceleration Of The Weiszfeld Algorithm For Minimizing The Sum Of Euclidean Distances
 Comput. Optim. Appl
, 1996
"... . The Weiszfeld algorithm for continuous location problems can be considered as an iteratively reweighted least squares method. It exhibits linear convergence. In this paper, a Newton type algorithm with similar simplicity is proposed to solve a continuous multifacility location problem with Euclide ..."
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Cited by 9 (0 self)
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. The Weiszfeld algorithm for continuous location problems can be considered as an iteratively reweighted least squares method. It exhibits linear convergence. In this paper, a Newton type algorithm with similar simplicity is proposed to solve a continuous multifacility location problem with Euclidean distance measure. Similar to the Weiszfeld algorithm, at each iteration the main computation can be solving a weighted least squares problem. A Cholesky factorization of a symmetric positive definite band matrix, typically with a relatively small band width (e.g., a band width of two for a Euclidean location problem on a plane) is required. This new algorithm can be regarded as a Newton acceleration to the Weiszfeld algorithm with fast global and local convergence. The simplicity and efficiency of the proposed algorithm makes it particularly suitable for largescale Euclidean location problems and parallel implementation. Computational experience also suggests that the proposed algorithm ...
Feedback algorithm for the singlefacility minisum problem
 Annals of the European Academy of Sciences
"... A new accelerated algorithm to solve the singlefacility minisum location problem is developed. The acceleration is achieved using a feedback factor. The proposed algorithm converges faster than the accelerating procedures available in the literature. Being nearly as simple as the classical Weiszfel ..."
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Cited by 1 (0 self)
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A new accelerated algorithm to solve the singlefacility minisum location problem is developed. The acceleration is achieved using a feedback factor. The proposed algorithm converges faster than the accelerating procedures available in the literature. Being nearly as simple as the classical Weiszfeld procedure, the new method can easily be implemented in real applications. Practical subroutines dealing with special cases in the minisum problem are also provided.
Video–on–Demand Network Design And Maintenance Using Fuzzy Optimization
"... Video–on–Demand (VoD) is the entertainment source which, in the future, will likely overtake regular television in many aspects. Even though many companies have deployed working VoD services, some aspects of the VoD should still undergo further improvement, in order for it to reach to the foreseen p ..."
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Video–on–Demand (VoD) is the entertainment source which, in the future, will likely overtake regular television in many aspects. Even though many companies have deployed working VoD services, some aspects of the VoD should still undergo further improvement, in order for it to reach to the foreseen potentials. An important aspect of a VoD system is the underlying network in which it operates. According to the huge number of customers in this network, it should be carefully designed to fulfill certain performance criteria. This process should be capable of finding optimal locations for the nodes of the network as well as determining the content which should be cached in each one. While, this problem is categorized in the general group of network optimization problems, its specific characteristics demand a new solution to be sought for it. In this paper, inspired by the successful use of fuzzy optimization in similar problems in other fields, a fuzzy objective function is derived which is heuristically shown to minimize the communication cost in a VoD network, while also controlling the storage cost. Then, an iterative algorithm is proposed to find a locally optimal solution to the proposed objective function. Capitalizing on the unrepeatable tendency of the proposed algorithm, a heuristic method for picking a good solution, from a bundle of solutions produced by the proposed algorithm, is also suggested. This paper includes formal statement of the problem and its mathematical analysis. Also, different scenarios in which the proposed algorithm can be utilized are discussed