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- W3160895500 abstract "In this article, we consider the Euclidean dispersion problems. Let $P={p_{1}, p_{2}, ldots, p_{n}}$ be a set of $n$ points in $mathbb{R}^2$. For each point $p in P$ and $S subseteq P$, we define $cost_{gamma}(p,S)$ as the sum of Euclidean distance from $p$ to the nearest $gamma $ point in $S setminus {p}$. We define $cost_{gamma}(S)=min_{p in S}{cost_{gamma}(p,S)}$ for $S subseteq P$. In the $gamma$-dispersion problem, a set $P$ of $n$ points in $mathbb{R}^2$ and a positive integer $k in [gamma+1,n]$ are given. The objective is to find a subset $Ssubseteq P$ of size $k$ such that $cost_{gamma}(S)$ is maximized. We consider both $2$-dispersion and $1$-dispersion problem in $mathbb{R}^2$. Along with these, we also consider $2$-dispersion problem when points are placed on a line. In this paper, we propose a simple polynomial time $(2sqrt 3 + epsilon )$-factor approximation algorithm for the $2$-dispersion problem, for any $epsilon > 0$, which is an improvement over the best known approximation factor $4sqrt3$ [Amano, K. and Nakano, S. I., An approximation algorithm for the $2$-dispersion problem, IEICE Transactions on Information and Systems, Vol. 103(3), pp. 506-508, 2020]. Next, we develop a common framework for designing an approximation algorithm for the Euclidean dispersion problem. With this common framework, we improve the approximation factor to $2sqrt 3$ for the $2$-dispersion problem in $mathbb{R}^2$. Using the same framework, we propose a polynomial time algorithm, which returns an optimal solution for the $2$-dispersion problem when points are placed on a line. Moreover, to show the effectiveness of the framework, we also propose a $2$-factor approximation algorithm for the $1$-dispersion problem in $mathbb{R}^2$." @default.
- W3160895500 created "2021-05-24" @default.
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- W3160895500 date "2021-05-19" @default.
- W3160895500 modified "2023-09-27" @default.
- W3160895500 title "Approximation Algorithms For The Euclidean Dispersion Problems." @default.
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