[Paper Review] Approximation Algorithms for Generalized Multidimensional Knapsack

Arindam Khan, Eklavya Sharma+1 more|arXiv (Cornell University)|Feb 11, 2021

Optimization and Packing Problems20 references4 citations

TL;DR

This paper presents a $(2+\varepsilon)$-approximation algorithm for the generalized $(2,d)$ Knapsack problem, where items have both 2D geometric dimensions and $d$-dimensional vector weights, and must be packed non-overlappingly into a unit square knapsack without exceeding capacity in any dimension. The approach uses corridor decomposition and introduces a new problem, Vector-Max-GAP, for which a PTAS is developed to enable efficient search over structured packings.

ABSTRACT

We study a generalization of the knapsack problem with geometric and vector constraints. The input is a set of rectangular items, each with an associated profit and $d$ nonnegative weights ($d$-dimensional vector), and a square knapsack. The goal is to find a non-overlapping axis-parallel packing of a subset of items into the given knapsack such that the vector constraints are not violated, i.e., the sum of weights of all the packed items in any of the $d$ dimensions does not exceed one. We consider two variants of the problem: $(i)$ the items are not allowed to be rotated, $(ii)$ items can be rotated by 90 degrees. We give a $(2+ε)$-approximation algorithm for this problem (both versions). In the process, we also study a variant of the maximum generalized assignment problem (Max-GAP), called Vector-Max-GAP, and design a PTAS for it.

Motivation & Objective

• To design an approximation algorithm for the $(2,d)$ Knapsack problem, a generalization of the 2D geometric knapsack with $d$-dimensional vector constraints.
• To address the challenge of combining geometric packing constraints with multidimensional vector capacity limits in a single optimization framework.
• To develop a new problem variant, Vector-Max-GAP, which extends the Maximum Generalized Assignment Problem to include global vector constraints.
• To provide a $(2+\varepsilon)$-approximation algorithm for both the rotation and non-rotation variants of the $(2,d)$ Knapsack problem.
• To lay the foundation for future approximation algorithms in related problems, such as $(2,d)$ Bin Packing, by using the proposed knapsack algorithm as a subroutine.

Proposed method

• Apply corridor decomposition to restructure an optimal packing by removing either vertical or horizontal items, resulting in a 'nice' packing structure that is amenable to algorithmic search.
• Introduce the Vector-Max-GAP problem, where items have machine-dependent sizes and values, and a global $d$-dimensional weight constraint limits the total weight across all machines.
• Design a PTAS for Vector-Max-GAP using dynamic programming and rounding techniques to handle the multidimensional constraints.
• Use the PTAS for Vector-Max-GAP as a subroutine to efficiently search for high-value packings in the decomposed corridor structure.
• Guess the set of containers (large, wide, tall, area) used in the optimal packing, leveraging the fact that only a constant number of containers are needed.
• Combine the structural decomposition with the PTAS for Vector-Max-GAP to achieve a $(2+\varepsilon)$-approximation for the $(2,d)$ Knapsack problem.

Experimental results

Research questions

• RQ1 Can a $(2+\varepsilon)$-approximation algorithm be designed for the $(2,d)$ Knapsack problem with both geometric and vector constraints?
• RQ2 How can the Maximum Generalized Assignment Problem be generalized to incorporate global vector constraints while still admitting a PTAS?
• RQ3 Can corridor decomposition be adapted to handle multidimensional vector constraints in geometric knapsack problems?
• RQ4 Is it possible to achieve better approximation ratios than $2+\varepsilon$ using more refined packing structures like $L$-packings?
• RQ5 Can the proposed algorithm be extended to the more general $(d_G,d_V)$ Knapsack problem with $d_G$-dimensional items and $d_V$-dimensional weights?

Key findings

• A $(2+\varepsilon)$-approximation algorithm is achieved for the $(2,d)$ Knapsack problem in both the rotation and non-rotation variants.
• The Vector-Max-GAP problem is introduced and a PTAS is developed for it, which is a key technical contribution enabling the main result.
• The structural result via corridor decomposition allows the reduction of the problem to a constant number of container types, making exhaustive search feasible.
• The container packing problem is shown to be a special case of Vector-Max-GAP, and thus inherits a PTAS, which supports the algorithmic framework.
• The algorithmic framework can be used as a subroutine to obtain an approximation algorithm for the $(2,d)$ Bin Packing problem.
• The approach demonstrates that combining geometric decomposition with multidimensional knapsack techniques enables improved approximation guarantees in constrained packing problems.