On the Two-Dimensional Knapsack Problem for Convex Polygons

TL;DR

This paper introduces approximation algorithms for the two-dimensional convex polygon knapsack problem, allowing arbitrary rotations and providing near-optimal solutions with theoretical guarantees.

Contribution

It presents the first approximation algorithms for convex polygons in 2D knapsack, including rotation and resource augmentation techniques, extending beyond rectangles and circles.

Findings

Quasi-polynomial time $O(1)$-approximation for general convex polygons

Polynomial time $O(1)$-approximation for triangles

Resource augmentation algorithm achieving optimal weight

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.