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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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... We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger. par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 . 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.