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 approximation bounds for Vector Bin Packing

In this paper we propose an improved approximation scheme for the Vector Bin Packing problem (VBP), based on the combination of (near-)optimal solution of the Linear Programming (LP) relaxation and a greedy (modified first-fit) heuristic. The Vector Bin Packing problem of higher dimension (d \geq 2) is not known to have asymptotic polynomial-time approximation schemes (unless P = NP). Our algorithm improves over the previously-known guarantee of (ln d + 1 + epsilon) by Bansal et al. [1] for higher dimensions (d > 2). We provide a {\theta}(1) approximation scheme for certain set of inputs for any dimension d. More precisely, we provide a 2-OPT algorithm, a result which is irrespective of the number of dimensions d.Comment: 15 pages, 3 algorithm

Optimal Placement Algorithms for Virtual Machines

Cloud computing provides a computing platform for the users to meet their demands in an efficient, cost-effective way. Virtualization technologies are used in the clouds to aid the efficient usage of hardware. Virtual machines (VMs) are utilized to satisfy the user needs and are placed on physical machines (PMs) of the cloud for effective usage of hardware resources and electricity in the cloud. Optimizing the number of PMs used helps in cutting down the power consumption by a substantial amount. In this paper, we present an optimal technique to map virtual machines to physical machines (nodes) such that the number of required nodes is minimized. We provide two approaches based on linear programming and quadratic programming techniques that significantly improve over the existing theoretical bounds and efficiently solve the problem of virtual machine (VM) placement in data centers

Improved Hardness of Approximation for Geometric Bin Packing

The Geometric Bin Packing (GBP) problem is a generalization of Bin Packing where the input is a set of $d$-dimensional rectangles, and the goal is to pack them into unit $d$-dimensional cubes efficiently. It is NP-Hard to obtain a PTAS for the problem, even when $d=2$. For general $d$, the best known approximation algorithm has an approximation guarantee exponential in $d$, while the best hardness of approximation is still a small constant inapproximability from the case when $d=2$. In this paper, we show that the problem cannot be approximated within $d^{1-\epsilon}$ factor unless NP=ZPP. Recently, $d$-dimensional Vector Bin Packing, a closely related problem to the GBP, was shown to be hard to approximate within $\Omega(\log d)$ when $d$ is a fixed constant, using a notion of Packing Dimension of set families. In this paper, we introduce a geometric analog of it, the Geometric Packing Dimension of set families. While we fall short of obtaining similar inapproximability results for the Geometric Bin Packing problem when $d$ is fixed, we prove a couple of key properties of the Geometric Packing Dimension that highlight the difference between Geometric Packing Dimension and Packing Dimension.Comment: 10 page

A 2-approximation for 2D bin packing

We study the two|-dimensional geometrical bin packing problem (2DBP): given a list of rectangles, provide a packing of all these into the smallest possible number of $1\times1$ bins without rotating the rectangles. We present a $2$|-approximate algorithm, which improves over the previous best known ratio of $3$, matches the best results for the rotational case and also matches the known lower bound of approximability. Our approach makes strong use of a recently-discovered PTAS for a related knapsack problem and a new algorithm that can pack instances into \OPT+2 bins for any constant \OPT

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