Heuristic and Special Case Algorithms for Dispersion Problems

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. INFORMS is collaborating with JSTOR to digitize, preserve and extend access to Operations Research.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 ir/2 for the 2-dimensional MAX-AVG dispersion problem. M any problems in location theory deal with the placement of facilities on a network to minimize some function of the distances between facilities or between facilities and the nodes of the network (Handler and Mirchandani 1979). Such problems model the placement of "desirable" facilities such as warehouses, hospitals, and fire stations. However, there are situations in which facilities are to be located to maximize some function of the distances between pairs of nodes. Such location problems are referred to as dispersion problems (Chandrasekharan and Daughety 1981, Kuby 1987, Erkut and Neuman 1989, 1990, and Erkut 1990) because they model situations in which proximity of facilities is undesirable. One example of such a situation is the distribution of business franchises in a city (Erkut). Other examples of dispersion problems arise in the context of placing "undesirable" (also called obnoxious) facilities, such as nuclear power plants, oil storage tanks, and ammunition dumps (Kuby 1987, Erkut and Neuman 1989, 1990, and Erkut 1990). Such facilities need to be spread out to the greatest possible extent so that an accident at one of the facilities will not damage any of the others. The concept of dispersion is also useful in the context of multiobjective decision making (Steuer 1986). When the number of nondominated solutions is large, a decision maker may be interested i...

Subset selection from large datasets for Kriging modeling

“…Given n possible locations, locate m < n facilities such that some function of the distances between the facilities is maximized (Ravi et al 1994). Two commonly used functions are the minimum and the sum of the distances (Erkut and Neuman 1989).…”

Comments on the Paper: ‘Heuristic and Special Case Algorithms for Dispersion Problems’ by S. S. Ravi, D. J. Rosenkrantz, and G. K. Tayi

“…These problems are the subject of a paper by Ravi et al (1994). The results presented in that paper include a simple heuristic for MMFD which provides a performance guarantee of 1/2, and a similar heuristic for MAFD with a performance guarantee of 1/4.…”

“…The paper Ravi et al (1994) also discusses the onedimensional versions of MMFD and MAFD, where the vertex set V consists of a set of n points on the real line and w (v …”

“…Section 2 of Ravi et al (1994) is devoted to the heuristic for MMFD. It should be noted that this heuristic was already analyzed in Tamir (1991).…”

“…Section 3 of Ravi et al (1994) contains an algorithm for obtaining approximations for MAFD. The algorithm can be executed in O(n 2 ) time.…”