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arXiv · 2007.16144

On the Two-Dimensional Knapsack Problem for Convex Polygons

Arturo Merino·Andreas Wiese
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Abstract

We study the two-dimensional geometric knapsack problem for convex polygons. Given a set of weighted convex polygons and a square knapsack, the goal is to select the most profitable subset of the given polygons that fits non-overlappingly into the knapsack. We allow to rotate the polygons by arbitrary angles. We present a quasi-polynomial time $O(1)$-approximation algorithm for the general case and a polynomial time $O(1)$-approximation algorithm if all input polygons are triangles, both assuming polynomially bounded integral input data. Also, we give a quasi-polynomial time algorithm that computes a solution of optimal weight under resource augmentation, i.e., we allow to increase the size of the knapsack by a factor of $1+\delta$ for some $\delta>0$ but compare ourselves with the optimal solution for the original knapsack. To the best of our knowledge, these are the first results for two-dimensional geometric knapsack in which the input objects are more general than axis-parallel rectangles or circles and in which the input polygons can be rotated by arbitrary angles.

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Arturo Merino, Andreas Wiese. 2020-07-31. On the Two-Dimensional Knapsack Problem for Convex Polygons. https://doi.org/10.4230/lipics.icalp.2020.84

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