Heuristic and Special Case Algorithms for Dispersion Problems

The dispersion problem arises in selecting facilities to maximize some function of the distances between the facilities. The problem also arises in selecting nondominated solutions for multiobjective decision making. It is known to be NP-hard under two objectives: maximizing the minimum distance ( M...

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Vydáno v:	Operations research Ročník 42; číslo 2; s. 299 - 310
Hlavní autoři:	Ravi, S. S, Rosenkrantz, D. J, Tayi, G. K
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Linthicum, MD INFORMS 01.03.1994 Operations Research Society of America Institute for Operations Research and the Management Sciences
Témata:	Algorithms analysis of algorithms: computational complexity Applied sciences Approximation algorithms Dynamic programming Exact sciences and technology facilities/equipment planning: discrete location Flows in networks. Combinatorial problems Functions Heuristic Heuristics Inequality Integers Mathematical theorems Natural satellites Operational research and scientific management Operational research. Management science Operations research Optimal solutions Optimization Polynomials programming: heuristic Studies Triangle inequalities One dimensional model Heuristic method Two dimensional model Facility Algorithm Computational complexity Location problem Dispersion
ISSN:	0030-364X, 1526-5463
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Abstract	The dispersion problem arises in selecting facilities to maximize some function of the distances between the facilities. The problem also arises in selecting nondominated solutions for multiobjective decision making. It is known to be NP-hard under two objectives: maximizing the minimum distance ( MAX-MIN ) between any pair of facilities and maximizing the average distance ( MAX-AVG ). We consider the question of obtaining near-optimal solutions. for MAX-MIN , we show that if the distances do not satisfy the triangle inequality, there is no polynomial-time relative approximation algorithm unless P = NP . When the distances satisfy the triangle inequality, we analyze an efficient heuristic and show that it provides a performance guarantee of two. We also prove that obtaining a performance guarantee of less than two is NP-hard. for MAX-AVG , we analyze an efficient heuristic and show that it provides a performance guarantee of four when the distances satisfy the triangle inequality. We also present a polynomial-time algorithm for the 1-dimensional MAX-AVG dispersion problem. Using that algorithm, we obtain a heuristic which provides an asymptotic performance guarantee of /2 for the 2-dimensional MAX-AVG dispersion problem.
AbstractList	The dispersion problem arises in selecting facilities to maximize some function of the distances between the facilities. The problem also arises in selecting nondominated solutions for multiobjective decision making. It is known to be NP-hard under two objectives: maximizing the minimum distance (MAX-MIN) between any pair of facilities and maximizing the average distance (MAX-AVG). We consider the question of obtaining near-optimal solutions. For MAX-MIN, we show that if the distances do not satisfy the triangle inequality, there is no polynomial-time relative approximation algorithm unless P = NP. When the distances satisfy the triangle inequality, we analyze an efficient heuristic and show that it provides a performance guarantee of two. We also prove that obtaining a performance guarantee of less than two is NP-hard. For MAX-AVG, we analyze an efficient heuristic and show that it provides a performance guarantee of four when the distances satisfy the triangle inequality. We also present a polynomial-time algorithm for the 1-dimensional MAX-AVG dispersion problem. Using that algorithm, we obtain a heuristic which provides an asymptotic performance guarantee of π /2 for the 2-dimensional MAX-AVG dispersion problem. The dispersion problem arises in selecting facilities to maximize some function of the distances between the facilities. The problem also arises in selecting nondominated solutions for multiobjective decision making. It is known to be NP-hard under two objectives: maximizing the minimum distance (MAX-MIN) between any pair of facilities and maximizing the average distance (MAX-AVG). We consider the question of obtaining near-optimal solutions. For MAX-MIN, we show that if the distances do not satisfy the triangle inequality, there is no polynomial-time relative approximation algorithm unless P identical with NP. When the distances satisfy the triangle inequality, we analyze an efficient heuristic and show that it provides a performance guarantee of two. We also prove that obtaining a performance guarantee of less than two is NP-hard. For MAX-AVG, we analyze an efficient heuristic and show that it provides a performance guarantee of four when the distances satisfy the triangle inequality. We also present a polynomial-time algorithm for the 1-dimensional MAX-AVG dispersion problem. Using that algorithm, we obtain a heuristic which provides an asymptotic performance guarantee of pi /2 for the 2-dimensional MAX-AVG dispersion problem. The dispersion problem arises in selecting facilities to maximize some function of the distances between the facilities. The problem also arises in selecting nondominated solutions for multiobjective decision making. It is known to be NP-hard under 2 objectives: 1. maximizing the minimum distance (MAX-MIN) between any pair of facilities and 2. maximizing the average distance (MAX-AVG). The question of obtaining near-optimal solutions is considered. For MAX-MIN, it is shown that if the distances do not satisfy the triangle inequality, there is no polynomial-time relative approximation algorithm unless P=NP. When the distances satisfy the triangle inequality, an efficient heuristic is analyzed and it is shown that it provides a performance guarantee of 2. It is also proven that obtaining a performance gurantee of less than 2 is NP-hard. For MAX-AVG, an efficient heuristic is analyzed and it is shown that it provides a performance guarantee of 4 when the distances satisfy the triangle inequality. The dispersion problem arises in selecting facilities to maximize some function of the distances between the facilities. The problem also arises in selecting nondominated solutions for multiobjective decision making. It is known to be NP-hard under two objectives: maximizing the minimum distance ( MAX-MIN ) between any pair of facilities and maximizing the average distance ( MAX-AVG ). We consider the question of obtaining near-optimal solutions. for MAX-MIN , we show that if the distances do not satisfy the triangle inequality, there is no polynomial-time relative approximation algorithm unless P = NP . When the distances satisfy the triangle inequality, we analyze an efficient heuristic and show that it provides a performance guarantee of two. We also prove that obtaining a performance guarantee of less than two is NP-hard. for MAX-AVG , we analyze an efficient heuristic and show that it provides a performance guarantee of four when the distances satisfy the triangle inequality. We also present a polynomial-time algorithm for the 1-dimensional MAX-AVG dispersion problem. Using that algorithm, we obtain a heuristic which provides an asymptotic performance guarantee of π/2 for the 2-dimensional MAX-AVG dispersion problem. The dispersion problem arises in selecting facilities to maximize some function of the distances between the facilities. The problem also arises in selecting nondominated solutions for multiobjective decision making. It is known to be NP-hard under two objectives: maximizing the minimum distance ( MAX-MIN ) between any pair of facilities and maximizing the average distance ( MAX-AVG ). We consider the question of obtaining near-optimal solutions. for MAX-MIN , we show that if the distances do not satisfy the triangle inequality, there is no polynomial-time relative approximation algorithm unless P = NP . When the distances satisfy the triangle inequality, we analyze an efficient heuristic and show that it provides a performance guarantee of two. We also prove that obtaining a performance guarantee of less than two is NP-hard. for MAX-AVG , we analyze an efficient heuristic and show that it provides a performance guarantee of four when the distances satisfy the triangle inequality. We also present a polynomial-time algorithm for the 1-dimensional MAX-AVG dispersion problem. Using that algorithm, we obtain a heuristic which provides an asymptotic performance guarantee of /2 for the 2-dimensional MAX-AVG dispersion problem.
Author	Rosenkrantz, D. J Ravi, S. S Tayi, G. K
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Copyright	Copyright 1994 The Operations Research Society of America 1994 INIST-CNRS Copyright Institute for Operations Research and the Management Sciences Mar/Apr 1994
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Snippet	The dispersion problem arises in selecting facilities to maximize some function of the distances between the facilities. The problem also arises in selecting... The dispersion problem arises in selecting facilities to maximize some function of the distances between the facilities. The problem also arises in selecting...
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SubjectTerms	Algorithms analysis of algorithms: computational complexity Applied sciences Approximation algorithms Dynamic programming Exact sciences and technology facilities/equipment planning: discrete location Flows in networks. Combinatorial problems Functions Heuristic Heuristics Inequality Integers Mathematical theorems Natural satellites Operational research and scientific management Operational research. Management science Operations research Optimal solutions Optimization Polynomials programming: heuristic Studies Triangle inequalities
Title	Heuristic and Special Case Algorithms for Dispersion Problems
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