Improved approximation for two-dimensional vector multiple knapsack

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-ln 2/2-epsilon)-approximation algorithm for 2VMK, for every fixed epsilon>0, thus improving the best known ratio of (1-1/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 approximate to m center dot ln 2 approximate to 0.693 center dot m of the bins, followed by a reduction to the (1-dimensional) Multiple Knapsack problem for assigning items to the remaining bins. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AIt raining, and similar technologies.