On Two Dimensional Orthogonal Knapsack Problem

TL;DR

This paper introduces a PTAS for the two-dimensional orthogonal knapsack problem involving packing squares with profits into a rectangular bin, achieving a 5+\u03b5 approximation ratio, advancing efficient packing algorithms.

Contribution

The paper develops a PTAS for packing weighted squares into rectangular bins and applies it to maximize profits, providing a new approximation algorithm with a 5+ ratio.

Findings

Existence of a PTAS for packing weighted squares into large rectangular bins.

Application of PTAS to profit maximization in square packing.

Achieved a 5+ approximation ratio for the problem.

Abstract

In this paper, we study the following knapsack problem: Given a list of squares with profits, we are requested to pack a sublist of them into a rectangular bin (not a unit square bin) to make profits in the bin as large as possible. We first observe there is a Polynomial Time Approximation Scheme (PTAS) for the problem of packing weighted squares into rectangular bins with large resources, then apply the PTAS to the problem of packing squares with profits into a rectangular bin and get a $\frac65+\epsilon$ approximation algorithm.