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] , 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., || ∑ vi∈B vi||∞ ≤ 1. For the 2-dimensional case we give an asymptotic approximation guarantee of 1 + ln(1.5)+ ≈ (1.405+ ), improving upon the previous bound of 1 + ln 2 + ≈ (1.693 + ). We also give an almost tight (1.5+ ) absolute approximation guarantee, improving upon the previous bound of 2 [23]. For the d-dimensional case, we get a 1.5 + ln( d+1 2 ) + ≈ 0.807 + ln(d + 1) + guarantee, improving upon the previous (1+ln d+ ) 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.