Improved Approximation for Two-dimensional Vector Multiple Knapsack

TL;DR

This paper presents a new approximation algorithm for the 2-dimensional vector multiple knapsack problem, achieving a better ratio than previous methods by adapting a rounding framework for maximization.

Contribution

It introduces a novel $(1- rac{ ext{ln} 2}{2} - ext{ε})$-approximation algorithm for 2VMK, improving upon the prior $(1 - rac{1}{e}- ext{ε})$ ratio using a configuration LP and randomized rounding.

Findings

Achieves a $(1 - rac{ ext{ln} 2}{2} - ext{ε})$-approximation ratio.

Uses an adaptation of the Round&Approx framework for maximization.

Reduces the problem to 1D multiple knapsack for remaining bins.

Abstract

We study the uniform $2$-dimensional vector multiple knapsack (2VMK) problem, a natural variant of multiple knapsack arising in real-world applications such as virtual machine placement. The input for 2VMK is a set of items, each associated with a $2$-dimensional weight vector and a positive profit, along with $m$ $2$-dimensional bins of uniform (unit) capacity in each dimension. The goal is to find an assignment of a subset of the items to the bins, such that the total weight of items assigned to a single bin is at most one in each dimension, and the total profit is maximized. Our main result is a $(1- \frac{\ln 2}{2} - \varepsilon)$-approximation algorithm for 2VMK, for every fixed $\varepsilon > 0$, thus improving the best known ratio of $(1 - \frac{1}{e}-\varepsilon)$ which follows as a special case from a result of [Fleischer at al., MOR 2011]. Our algorithm relies on an adaptation of the Round$\&$Approx framework of [Bansal et al., SICOMP 2010], originally designed for set covering problems, to maximization problems. The algorithm uses randomized rounding of a configuration-LP solution to assign items to $\approx m\cdot \ln 2 \approx 0.693\cdot m$ of the bins, followed by a reduction to the ($1$-dimensional) Multiple Knapsack problem for assigning items to the remaining bins.