Improved Approximation for Vector Bin Packing

We study the d-dimensional vector bin packing problem, a well-studied generalization of bin packing arising in resource allocation and scheduling problems. Here we are given a set of d-dimensional vectors v1,..., vn in [0,1]d, and the goal is to pack them into the least number of bins so that for each bin B, the sum of the vectors in it is at most 1 in every dimension, i.e., ||ΣvieB vi||∞ ≤ 1. For the 2-dimensional case we give an asymptotic approximation guarantee of 1 + ln(1.5) + e a (1.405 + e), improving upon the previous bound of 1 + ln 2 + e a (1.693 + e). We also give an almost tight (1.5 + e) absolute approximation guarantee, improving upon the previous bound of 2 [23]. For the d-dimensional case, we get a 1.5 + ln([EQUATION]) + e a 0.807 + ln(d + 1) + e guarantee, improving upon the previous (1+ln d+e) guarantee [2]. Here (1 + ln d) was a natural barrier as rounding-based algorithms can not achieve better than d approximation. We get around this by exploiting various structural properties of (near)-optimal packings, and using multi-objective multi-budget matching based techniques and expanding the Round & Approx framework to go beyond rounding-based algorithms. Along the way we also prove several results that could be of independent interest.