Online algorithms for 1-space bounded multi dimensional bin packing and hypercube packing

In this paper, we study 1-space bounded multi-dimensional bin packing and hypercube packing. A sequence of items arrive over time, each item is a d-dimensional hyperbox (in bin packing) or hypercube (in hypercube packing), and the length of each side is no more than 1. These items must be packed without overlapping into d-dimensional hypercubes with unit length on each side. In d-dimensional space, any two dimensions i and j define a space P ij. When an item arrives, we must pack it into an active bin immediately without any knowledge of the future items, and 90 {ring operator}-rotation on any plane P ij is allowed. The objective is to minimize the total number of bins used for packing all these items in the sequence. In the 1-space bounded variant, there is only one active bin for packing the current item. If the active bin does not have enough space to pack the item, it must be closed and a new active bin is opened. For d-dimensional bin packing, an online algorithm with competitive ratio 4 d is given. Moreover, we consider d-dimensional hypercube packing, and give a 2 d+1-competitive algorithm. These two results are the first study on 1-space bounded multi dimensional bin packing and hypercube packing. © 2012 The Author(s).published_or_final_versionSpringer Open Choice, 28 May 201

Bounds for online bounded space hypercube packing

In hypercube packing, we receive a sequence of hypercubes that need to be packed into unit hypercubes which are called bins. Items arrive online and each item must be placed within its bin without overlapping with other items in that bin. The goal is to minimize the total number of bins used. We present lower and upper bounds for online bounded space hypercube packing in dimensions 2,...,

Optimal online bounded space multidimensional packing

We solve an open problem in the literature by providing an online algorithm for multidimensional bin packing that uses only bounded space. We show that it is optimal among bounded space algorithms for any dimension $d>1$. Its asymptotic performance ratio is $(Pi_{infty})^d$, where $Pi_{infty}approx1.691$ is the asymptotic performance ratio of the one-dimensional algorithm harm. A modified version of this algorithm for the case where all items are hypercubes is also shown to be optimal. Its asymptotic performance ratio is sublinear in $d$. Additionally, for the special case of packing squares in two-dimensional bins, we present a new unbounded space online algorithm with asymptotic performance ratio of at most $2.271$. We also present an approximation algorithm for the offline problem with approximation ratio of $16/11$. This improves upon all earlier approximation algorithms for this problem, including the algorithm from Caprara, Packing 2-dimensional bins in harmony, Proc. 43rd FOCS, 2002

Improved Lower Bounds for Online Hypercube Packing

Packing a given sequence of items into as few bins as possible in an online fashion is a widely studied problem. We improve lower bounds for packing hypercubes into bins in two or more dimensions, once for general algorithms (in two dimensions) and once for an important subclass, so-called Harmonic-type algorithms (in two or more dimensions). Lastly, we show that two adaptions of the ideas from the best known one-dimensional packing algorithm to square packing also do not help to break the barrier of 2

Approximate Convex Optimization by Online Game Playing

Lagrangian relaxation and approximate optimization algorithms have received much attention in the last two decades. Typically, the running time of these methods to obtain a $\epsilon$ approximate solution is proportional to $\frac{1}{\epsilon^2}$. Recently, Bienstock and Iyengar, following Nesterov, gave an algorithm for fractional packing linear programs which runs in $\frac{1}{\epsilon}$ iterations. The latter algorithm requires to solve a convex quadratic program every iteration - an optimization subroutine which dominates the theoretical running time. We give an algorithm for convex programs with strictly convex constraints which runs in time proportional to $\frac{1}{\epsilon}$. The algorithm does NOT require to solve any quadratic program, but uses gradient steps and elementary operations only. Problems which have strictly convex constraints include maximum entropy frequency estimation, portfolio optimization with loss risk constraints, and various computational problems in signal processing. As a side product, we also obtain a simpler version of Bienstock and Iyengar's result for general linear programming, with similar running time. We derive these algorithms using a new framework for deriving convex optimization algorithms from online game playing algorithms, which may be of independent interest