On the Two-Dimensional Knapsack Problem for Convex Polygons

Authors Arturo Merino , Andreas Wiese

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Arturo Merino
  • Technische Universität Berlin, Germany
Andreas Wiese
  • Universidad de Chile, Santiago, Chile

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Arturo Merino and Andreas Wiese. On the Two-Dimensional Knapsack Problem for Convex Polygons. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 84:1-84:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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+δ for some δ > 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Packing and covering problems
  • Approximation algorithms
  • geometric knapsack problem
  • polygons
  • rotation


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