Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an on-line coloring heuristic which achieves a competitive ratio of 6 for unit disk graphs. Our heuristics do not need a geometric representation of unit disk graphs. Geometric representations are used only in establishing the performance guarantees of the heuristics. Several of our approximation algorithms can be extended to intersection graphs of circles of arbitrary radii in the plane, intersection graphs of regular polygons, and to intersection graphs of higher dimensional regular objects.

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...

Several polynomial time algorithms finding "good," but not necessarily optimal, tours for the traveling salesman problem are considered. We measure the closeness of a tour by the ratio of the obtained tour length to the minimal tour length. For the nearest neighbor method, we show the ratio is bounded above by a logarithmic function of the number of nodes. We also provide a logarithmic lower bound on the worst case. A class of approximation methods we call insertion methods are studied, and these are also shown to have a logarithmic upper bound. For two specific insertion methods, which we call nearest insertion and cheapest insertion, the ratio is shown to have a constant upper bound of 2, and examples are provided that come arbitrarily close to this upper bound. It is also shown that for any n ≥ 8, there are traveling salesman problems with n nodes having tours which cannot be improved by making n/4 edge changes, but for which the ratio is 2(1 − 1/n).

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