Improved approximation for two-dimensional vector multiple knapsack

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 $(1-\frac{\ln 2}{2}-\epsilon )$-approximation algorithm for 2VMK, for every fixed $\epsilon >0$, thus improving the best known ratio of $(1-\frac{1}{e}-\epsilon )$ which follows as a special case from a result of Fleischer et al. (2011) [6].

Our algorithm relies on an adaptation of the Round&Approx framework of Bansal et al. (2010) [15], 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.