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An Empirical Study of Algorithms for Point Feature Label Placement
, 1994
"... A major factor affecting the clarity of graphical displays that include text labels is the degree to which labels obscure display features (including other labels) as a result of spatial overlap. Pointfeature label placement (PFLP) is the problem of placing text labels adjacent to point features on ..."
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Cited by 134 (8 self)
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A major factor affecting the clarity of graphical displays that include text labels is the degree to which labels obscure display features (including other labels) as a result of spatial overlap. Pointfeature label placement (PFLP) is the problem of placing text labels adjacent to point features on a map or diagram so as to maximize legibility. This problem occurs frequently in the production of many types of informational graphics, though it arises most often in automated cartography. In this paper we present a comprehensive treatment of the PFLP problem, viewed as a type of combinatorial optimization problem. Complexity analysis reveals that the basic PFLP problem and most interesting variants of it are NPhard. These negative results help inform a survey of previously reported algorithms for PFLP; not surprisingly, all such algorithms either have exponential time complexity or are incomplete. To solve the PFLP problem in practice, then, we must rely on good heuristic methods. We pr...
The Quadratic Assignment Problem: A Survey and Recent Developments
 In Proceedings of the DIMACS Workshop on Quadratic Assignment Problems, volume 16 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1994
"... . Quadratic Assignment Problems model many applications in diverse areas such as operations research, parallel and distributed computing, and combinatorial data analysis. In this paper we survey some of the most important techniques, applications, and methods regarding the quadratic assignment probl ..."
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Cited by 91 (16 self)
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. Quadratic Assignment Problems model many applications in diverse areas such as operations research, parallel and distributed computing, and combinatorial data analysis. In this paper we survey some of the most important techniques, applications, and methods regarding the quadratic assignment problem. We focus our attention on recent developments. 1. Introduction Given a set N = f1; 2; : : : ; ng and n \Theta n matrices F = (f ij ) and D = (d kl ), the quadratic assignment problem (QAP) can be stated as follows: min p2\Pi N n X i=1 n X j=1 f ij d p(i)p(j) + n X i=1 c ip(i) ; where \Pi N is the set of all permutations of N . One of the major applications of the QAP is in location theory where the matrix F = (f ij ) is the flow matrix, i.e. f ij is the flow of materials from facility i to facility j, and D = (d kl ) is the distance matrix, i.e. d kl represents the distance from location k to location l [62, 67, 137]. The cost of simultaneously assigning facility i to locat...
The complexity of approximating a nonlinear program
 IBM Research Report RC 17831
, 1992
"... We consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope. We show that there is no polynomial time approximation algorithm for this problem, even one with a very poor guarantee, unless P = NP. We show that even when the polynomial is quadratic (i.e. quadra ..."
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Cited by 58 (3 self)
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We consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope. We show that there is no polynomial time approximation algorithm for this problem, even one with a very poor guarantee, unless P = NP. We show that even when the polynomial is quadratic (i.e. quadratic programming) there is no polynomial time approximation unless NP is contained in quasipolynomial time. Our results rely on recent advances in the theory of interactive proof systems. They exemplify an interesting interplay of discrete and continuous mathematicsâ€”using a combinatorial argument to get a hardness result for a continuous optimization problem.
A Complexity Theory for Feasible Closure Properties
, 1991
"... The study of the complexity of sets encompasses two complementary aims: (1) establishing  usually via explicit construction of algorithms  that sets are feasible, and (2) studying the relative complexity of sets that plausibly might be feasible but are not currently known to be feasible (such as ..."
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Cited by 47 (3 self)
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The study of the complexity of sets encompasses two complementary aims: (1) establishing  usually via explicit construction of algorithms  that sets are feasible, and (2) studying the relative complexity of sets that plausibly might be feasible but are not currently known to be feasible (such as the NPcomplete sets and the PSPACEcomplete sets). For the study of the complexity of closure properties, a recent urry of results [21, 33, 49, 6, 7, 16] has established an analog of (1); these papers explicitly demonstrate many closure properties possessed by PP and C=P (and the proofs implicitly give closure properties of the function class #P). The present paper presents and develops, for function classes such as #P, SpanP, OptP, and MidP, an analog of (2): a general theory of the complexity of closure properties. In particular, we show that subtraction is hard for the closure properties of each of these classes: each is closed under subtraction if and only if it is closed under every polynom...
Quadratic Optimization
, 1995
"... . Quadratic optimization comprises one of the most important areas of nonlinear programming. Numerous problems in real world applications, including problems in planning and scheduling, economies of scale, and engineering design, and control are naturally expressed as quadratic problems. Moreover, t ..."
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Cited by 46 (3 self)
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. Quadratic optimization comprises one of the most important areas of nonlinear programming. Numerous problems in real world applications, including problems in planning and scheduling, economies of scale, and engineering design, and control are naturally expressed as quadratic problems. Moreover, the quadratic problem is known to be NPhard, which makes this one of the most interesting and challenging class of optimization problems. In this chapter, we review various properties of the quadratic problem, and discuss different techniques for solving various classes of quadratic problems. Some of the more successful algorithms for solving the special cases of bound constrained and large scale quadratic problems are considered. Examples of various applications of quadratic programming are presented. A summary of the available computational results for the algorithms to solve the various classes of problems is presented. Key words: Quadratic optimization, bilinear programming, concave pro...
Computational experience with an interior point algorithm on the satisfiability problem
 Annals of Operations Research
, 1990
"... We apply the zeroone integer programming algorithm described in Karmarkar [12] and Karmarkar, Resende and Ramakrishnan [13] to solve randomly generated instances of the satisfiability problem (SAT). The interior point algorithm is briefly reviewed and shown to be easily adapted to solve large insta ..."
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Cited by 43 (4 self)
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We apply the zeroone integer programming algorithm described in Karmarkar [12] and Karmarkar, Resende and Ramakrishnan [13] to solve randomly generated instances of the satisfiability problem (SAT). The interior point algorithm is briefly reviewed and shown to be easily adapted to solve large instances of SAT. Hundreds of instances of SAT (having from 100 to 1000 variables and 100 to 32,000 clauses) are randomly generated and solved. For comparison, we attempt to solve the problems via linear programming relaxation with MINOS.
Approximating The Minimum Equivalent Digraph
, 1995
"... . The MEG (minimum equivalent graph) problem is the following: "Given a directed graph, find a smallest subset of the edges that maintains all reachability relations between nodes." This problem is NPhard; this paper gives an approximation algorithm achieving a performance guarantee of about 1:64 i ..."
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Cited by 35 (2 self)
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. The MEG (minimum equivalent graph) problem is the following: "Given a directed graph, find a smallest subset of the edges that maintains all reachability relations between nodes." This problem is NPhard; this paper gives an approximation algorithm achieving a performance guarantee of about 1:64 in polynomial time. The algorithm achieves a performance guarantee of 1:75 in the time required for transitive closure. The heart of the MEG problem is the minimum SCSS (strongly connected spanning subgraph) problem  the MEG problem restricted to strongly connected digraphs. For the minimum SCSS problem, the paper gives a practical, nearly lineartime implementation achieving a performance guarantee of 1:75. The algorithm and its analysis are based on the simple idea of contracting long cycles. The analysis applies directly to 2Exchange, a general "local improvement" algorithm, showing that its performance guarantee is 1:75. AMS subject classifications. 68R10, 90C27, 90C35, 05C85, 68Q20....
An Interior Point Algorithm to Solve Computationally Difficult Set Covering Problems
, 1990
"... ..."
Approximation Algorithms for Quadratic Programming
, 1998
"... We consider the problem of approximating the global minimum of a general quadratic program (QP) with n variables subject to m ellipsoidal constraints. For m = 1, we rigorously show that an fflminimizer, where error ffl 2 (0; 1), can be obtained in polynomial time, meaning that the number of arithme ..."
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Cited by 23 (5 self)
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We consider the problem of approximating the global minimum of a general quadratic program (QP) with n variables subject to m ellipsoidal constraints. For m = 1, we rigorously show that an fflminimizer, where error ffl 2 (0; 1), can be obtained in polynomial time, meaning that the number of arithmetic operations is a polynomial in n, m, and log(1=ffl). For m 2, we present a polynomialtime (1 \Gamma 1 m 2 )approximation algorithm as well as a semidefinite programming relaxation for this problem. In addition, we present approximation algorithms for solving QP under the box constraints and the assignment polytope constraints. Key words. Quadratic programming, global minimizer, polynomialtime approximation algorithm The work of the first author was supported by the Australian Research Council; the second author was supported in part by the Department of Management Sciences of the University of Iowa where he performed this research during a research leave, and by the Natural Scien...
Efficient Algorithms for Function Approximation with Piecewise Linear Sigmoidal Networks
, 1998
"... This paper presents a computationally efficient algorithm for function approximation with piecewise linear sigmoidal nodes. A one hidden layer network is constructed one node at a time using the wellknown method of fitting the residual. The task of fitting an individual node is accomplished using ..."
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Cited by 19 (1 self)
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This paper presents a computationally efficient algorithm for function approximation with piecewise linear sigmoidal nodes. A one hidden layer network is constructed one node at a time using the wellknown method of fitting the residual. The task of fitting an individual node is accomplished using a new algorithm that searches for the best fit by solving a sequence of Quadratic Programming problems. This approach offers significant advantages over derivativebased search algorithms (e.g. backpropagation and its extensions). Unique characteristics of this algorithm include: finite step convergence, a simple stopping criterion, solutions that are independent of initial conditions, good scaling properties and a robust numerical implementation. Empirical results are included to illustrate these characteristics.