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**1 - 1**of**1**### On the Computational Complexity of Partitioning Weighted Points into a Grid of Quadrilaterals

"... In the paper the computational complexity of the follow-ing partitioning problem is studied: Given a rectangle R in the plane, a set Q of positive-weighted points in R, and two positive integers n1, n2, find a partitioning of R into quadrilaterals whose dual graph is an n1 × n2 grid such that each q ..."

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In the paper the computational complexity of the follow-ing partitioning problem is studied: Given a rectangle R in the plane, a set Q of positive-weighted points in R, and two positive integers n1, n2, find a partitioning of R into quadrilaterals whose dual graph is an n1 × n2 grid such that each quadrilateral contains points of equal total weight. If such a partitioning does not exist, find a solution that minimizes some objective function. This problem is motivated by applications in image process-ing including, among others, image enhancement and similarity retrieval, and it is closely related to the table cartogram problem introduced recently by Evans et al. [ESA 2013]. While there exist fast algorithms that find optimal partitions in 1-dimension, the 2-dimensional case seems to be much harder to solve. Pichon et al. [ICIP 2003] proposed a heuristic yielding admissible solutions, but the computational complexity of the problem has so far remained open. In this paper we prove that a decision version of the problem is NP-hard. 1