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).

No abstract

No abstract

We study the problem of nding small trees. Classical network design problems are considered with the additional constraint that only a speci ed number k of nodes are required to be connected in the solution. A prototypical example is the kMST problem in which we require a tree of minimum weight spanning at least k nodes in an edge-weighted graph. We show that the kMST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio 2 p k for the general edge-weighted case and O(k 1=4 ) for the case of points in the plane.Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane.We also investigate the problem of nding short trees, and more generally, that of nding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for nding k-trees of minimum diameter. We identify easy and hard problems arising in nding short networks using a framework due to T. C. Hu.This paper appeared in a preliminary form as 28].

We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a ¡subgraph from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria -the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented.First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same we present a "black box" parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms -the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.

Left corner parsing refers to a class of parsing procedures in which the productions are recognized in a particular order which is different than both bottom up and top down. Each production is recognized after its left descendant but before its other descendants. Procedures in this class have occurred frequently in the compiler literature.