# Improved approximation algorithms for multidimensional bin packing problems

## Abstract

In this paper we introduce a new general framework for set covering problems, based on the combination of randomized rounding of the (near-)optimal solution of the Linear Programming (LP) relaxation, leading to a partial integer solution, and the application of a well-behaved approximation algorithm to complete this solution. If the value of the solution returned by the latter can be bounded in a suitable way, as is the case for the most relevant generalizations of bin packing, the method leads to improved approximation guarantees, along with a proof of tighter integrality gaps for the LP relaxation. Applying our general framework we obtain a polynomial-time randomized algorithm for d-dimensional vector packing with approximation guarantee arbitrarily close to In d + 1. For d = 2, this value is 1.693 ..., i.e., we break the natural 2 "barrier" for this case. Moreover, for small values of d this is a notable improvement over the previously-known O(ln d) guarantee by Chekuri and Khanna [5]. For 2-dimensional bin packing with and without rotations, we construct algorithms with performance guarantee arbitrarily close to 1.525..., improving upon previous algorithms with performance guarantee of 2 + e by Jansen and Zhang [12]forthe problem with rotations and 1.691... by Caprara [2] for the problem without rotations. The previously-unknown key property used in our proofs follows from a retrospective analysis of the implications of the landmark bin packing approximation scheme by Fernandez de la Vega and Lueker [7]. We prove that their approximation scheme is "subset oblivious", which leads to numerous applications. Another byproduct of our paper is an algorithm that solves a well-known configuration LP for 2-dimensional bin packing within a factor of(1 + ε) for any e > 0. Interestingly, we do it without using an approximate separation oracle, which would correspond to a well-known geometric 2dimensional knapsack. Although separation and optimization are equivalent [10] and the existence of an approximation scheme for the separation problem remains open, we are able to design an approximation scheme for the configuration LP since its objective function is unweighed.