We consider a problem of dispersing points on disjoint intervals on a line. Given n pairwise disjoint intervals sorted on a line, we want to find a point in each interval such that the minimum pairwise distance of these points is maximized. Based on a greedy strategy, we present a linear time algorithm for the problem. Further, we also solve in linear time the cycle version of the problem where the intervals are given on a cycle.
Wang and Kuo @cite_14 considered the following two problems. Given a set @math of points and a value @math , find a largest subset of @math in which the distance of any two points is at least @math . Given a set @math of points and an integer @math , find a subset of @math points of @math to maximize the minimum distance of all pairs of points in the subset. It was shown in @cite_14 that both problems in 2D are NP-hard but can be solved efficiently in 1D. Refer to @cite_11 @cite_15 @cite_16 @cite_0 @cite_6 for other geometric dispersion problems. Dispersion problems in various non-geometric settings were also considered @cite_12 @cite_13 @cite_7 @cite_3 @cite_2 . These problems are in general NP-hard; approximation and heuristic algorithms were proposed for them.
@cite_14 @cite_7 @cite_6 @cite_3 @cite_0 @cite_2 @cite_15 @cite_16 @cite_13 @cite_12 @cite_11
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Most optimization problems focus on efficiency-based objectives. Given the increasing awareness of system inequity resulting from solely pursuing efficiency, we conceptualize a number of new element-based equity-oriented measures in the dispersion context. We propose the equitable dispersion problem that maximizes the equity among elements based on the introduced measures in a system defined by inter-element distances. Given the proposed optimization framework, we develop corresponding mathematical programming formulations as well as their mixed-integer linear reformulations. We also discuss computational complexity issues, related graph-theoretic interpretations and provide some preliminary computational results. Abstract The problem of finding the maximum diameter of n equal mutually disjoint circles inside a unit square is addressed in this paper. Exact solutions exist for only n = 1, …, 9,10,16,25,36 while for other n only conjectural solutions have been reported. In this work a max-min optimization approach is introduced which matches the best reported solutions in the literature for all n ⩽ 30, yields a better configuration for n = 15, and provides new results for n = 28 and 29. Facility dispersion problem deals with the location of facilities on a network so as to maximize some function of the distances between facilities. We consider the problem under two different optimality criteria, namely maximizing the minimum distance (MAX-MIN) between any pair of facilities and maximizing the average distance (MAX-AVG) between any pair of facilities. Under either criterion, the problem is known to be NP-hard, even when the distances satisfy the triangle inequality. We consider the question of obtaining near-optimal solutions. For the MAX-MIN criterion, 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 do satisfy the triangle inequality, we present an efficient heuristic which provides a performance guarantee of 2, thus improving the performance guarantee of 3 proven in [Wh91]. We also prove that obtaining a performance guarantee of less than 2 is NP-hard. For the MAX-AVG criterion, we present a heuristic which provides a performance guarantee of 4, provided that the distances satisfy the triangle inequality. For the 1-dimensional dispersion problem, we provide polynomial time algorithms for obtaining optimal solutions under both MAX-MIN and MAX-AVG criteria. Using the latter algorithm, we obtain a heuristic which provides a performance guarantee of 4( ( 2 - 1 )) ≈ 1.657 for the 2-dimensional dispersion problem under the MAX-AVG criterion. What is the densest packing of points in an infinite strip of widthw, where any two of the points must be separated by distance at least I? This question was raised by Fejes-Toth a number of years ago. The answer is trivial for MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa83Dai abgsMiJoaakaaabaGaaG4maaWcbeaakiaac+cacaaIYaaaaa!3AF6! @math and, surprisingly, it is not difficult to prove [M2] for MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa83Dai aa-1dacaWGUbWaaOaaaeaacaaIZaaaleqaaOGaai4laiaaikdaaaa!3AF2! @math , wheren is a positive integer, that the regular triangular lattice gives the optimal packing. Kertesz [K] solved the case MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa83DaG qaaiaa+bcacaGF8aWaaOaaaeaacaaIYaaaleqaaaaa!392C! @math . Here we fill the first gap, i.e., the maximal density is determined for MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaie aacaWFZaaaleqaaOGaai4laiaaikdacqGH8aapcaWG3bGaeyizIm6a aOaaaeaacaaIZaaaleqaaaaa!3CCB! @math . 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-ti... Considered is the problem of selecting p out of n given points in some space, such that the minimum distance between pairs of selected points is maximized. This objective may be appropriate if the selected points correspond to facility sites and the objective is to have as ‘dispersed’ a set as possible. This problem is NP-complete. Related graph theoretical problems are discussed, integer programming models are proposed, and an outline is given on a line search procedure to solve this problem optimally. Using a branch-and-bound procedure, problems with n = 40 and p = 16 can be solved on a microcomputer. The heuristic, developed to get an initial lower bound, finds an optimal solution for most of our random test problems. Also described is an extension to the basic problem that allows for preselected points, which may correspond to existing facility locations. This more general version can be solved by slight modifications of the algorithms. The Generalized Maximum Dispersion problem asks for a partition of a given graph into pvertex-disjoint sets, each of them having at most kvertices. The goal is to maximize the total edge-weight of the induced subgraphs. We present the first LP-based approximation algorithm. In the maximum dispersion problem, a given set of objects has to be partitioned into a number of groups. Each object has a non-negative weight and each group has a target weight, which may be different for each group. In addition to meeting the target weight of each group, all objects assigned to the same group should be as dispersed as possible with respect to some distance measure between pairs of objects. Potential applications for this problem come from such diverse fields as the problem of creating study groups or the design of waste collection systems. We develop and compare two different (mixed-) integer linear programming formulations for the problem. We also study a specific relaxation that enables us to derive tight bounds that improve the effectiveness of the formulations. Thereby, we obtain an upper bound by finding in an auxiliary graph subsets of given size with minimal diameter. A lower bound is derived based on the relation of the optimal solution of the relaxation to the chromatic number of a series of auxiliary graphs. Finally, we propose an exact solution scheme for the maximum dispersion problem and present extensive computational experiments to assess its efficiency. We consider the following packing problem. Let α be a fixed real in (0, 1]. We are given a bounding rectangle ρ and a set of n possibly intersecting unit disks whose centers lie in ρ. The task is to pack a set of m disjoint disks of radius α into ρ such that no disk in B intersects a disk in , where m is the maximum number of unit disks that can be packed. In this paper we present a polynomial-time algorithm for α = 2 3. So far only the case of packing squares has been considered. For that case, Baur and Fekete have given a polynomial-time algorithm for α = 2 3 and have shown that the problem cannot be solved in polynomial time for any α > 13 14 unless .