A well studied special case of bin packing is the 3-partition problem, where n items of size > 1/4 have to be packed in a minimum number of bins of capacity one. The famous Karmarkar-Karp algorithm transforms a fractional solution of a suitable LP relaxation for this problem into an integral solution that requires at most O(log n) additional bins.
The three-permutations-problem of Beck is the following. Given any 3 permutations on n symbols, color the symbols red and blue, such that in any interval of any of those permutations, the number of red and blue symbols is roughly the same. The necessary difference is called the discrepancy.
We establish a surprising connection between bin packing and Beck's problem: The additive integrality gap of the 3-partition linear programming relaxation can be bounded by the discrepancy of 3 permutations.
Reversely, making use of a recent example of 3 permutations, for which a discrepancy of Omega(log n) is necessary, we prove the following: The O(log^2 n) upper bound on the additive gap for bin packing with arbitrary item sizes cannot be improved by any technique that is based on rounding up items. This lower bound holds for a large class of algorithms including the Karmarkar-Karp procedure.
We continue the study of two recently introduced bin packing type problems, called bin packing with clustering, and online bin packing with delays. A bin packing input consists of items of sizes not larger than 1, and the goal is to partition or pack them into bins, where the total size of items of every valid bin cannot exceed 1. In bin packing with clustering, items also have colors associated with them. A globally optimal solution can combine items of different colors in bins, while a clustered solution can only pack monochromatic bins. The goal is to compare a globally optimal solution to …
The problem considered is the splittable bin packing with cardinality constraint. It is a variant of the bin packing problem where items are allowed to be split into parts but the number of parts in each bin is at most a given upper bound. Two versions of the splittable bin packing with cardinality constraint have been studied in the literature. Among these variants we consider the dual one where the objective is to minimize the maximum bin size while packing (may be fractional) the items to a given set of bins. We exhibit an EPTAS for the dual problem when the cardinality upper bound is part …
In the d-dimensional online bin packing problem, d-dimensional cubes of positive sizes no larger than 1 are presented one by one to be assigned to positions in d-dimensional unit cube bins. In this work, we provide improved upper bounds on the asymptotic competitive ratio for square and cube bin packing problems, where our bounds do not exceed 2.0885 and 2.5735 for square and cube packing, respectively. To achieve these results, we adapt and improve a previously designed harmonic-type algorithm, and apply a different method for defining weight functions. We detect deficiencies in the state-of-…
We consider several previously studied online variants of bin packing and prove new and improved lower bounds on the asymptotic competitive ratios for them. For that, we use a method of fully adaptive constructions. In particular, we improve the lower bound for the asymptotic competitive ratio of online square packing significantly, raising it from roughly 1.68 to above 1.75.
We consider a known variant of bin packing called {\it cardinality constrained bin packing}, also called {\it bin packing with cardinality constraints} (BPCC). In this problem, there is a parameter k\geq 2, and items of rational sizes in [0,1] are to be packed into bins, such that no bin has more than k items or total size larger than 1. The goal is to minimize the number of bins. A recently introduced concept, called the price of clustering, deals with inputs that are presented in a way that they are split into clusters. Thus, an item has two attributes which are its size and its cluster. The…
We improve the lower bound on the asymptotic competitive ratio of any online algorithm for bin packing to above 1.54278. We demonstrate for the first time the advantage of branching and the applicability of full adaptivity in the design of lower bounds for the classic online bin packing problem. We apply a new method for weight based analysis, which is usually applied only in proofs of upper bounds. The values of previous lower bounds were approximately 1.5401 and 1.5403.
We analyze a recently introduced concept, called the price of clustering, for variants of bin packing called open-end bin packing problems (OEBP). Input items have sizes, and they also belong to a certain number of types. The new concept deals with the comparison of optimal solutions for the cases where items of distinct types can and cannot be packed together, respectively. The problem is related to greedy bin packing algorithms and to batched bin packing, and we discuss some of those concepts as well. We analyze max-OEBP, where a packed bin is valid if by excluding its largest item, the tota…
Tusnády's problem asks to bound the discrepancy of points and axis-parallel boxes in $\mathbb{R}^d$. Algorithmic bounds on Tusnády's problem use a canonical decomposition of Matoušek for the system of points and axis-parallel boxes, together with other techniques like partial coloring and / or random-walk based methods. We use the notion of \emph{shallow cell complexity} and the \emph{shallow packing lemma}, together with the chaining technique, to obtain an improved decomposition of the set system. Coupled with an algorithmic technique of Bansal and Garg for discrepancy minimization, which we…
We prove a size-sensitive version of Haussler's Packing lemma~\cite{Haussler92spherepacking} for set-systems with bounded primal shatter dimension, which have an additional {\em size-sensitive property}. This answers a question asked by Ezra~\cite{Ezra-sizesendisc-soda-14}. We also partially address another point raised by Ezra regarding overcounting of sets in her chaining procedure. As a consequence of these improvements, we get an improvement on the size-sensitive discrepancy bounds for set systems with the above property. Improved bounds on the discrepancy for these special set systems als…
We consider the Ordered Open End Bin Packing problem. Items of sizes in $(0,1]$ are presented one by one, to be assigned to bins in this order. An item can be assigned to any bin for which the current total size strictly below $1$. This means also that the bin can be overloaded by its last packed item. We improve lower and upper bounds on the asymptotic competitive ratio in the online case. Specifically, we design the first algorithm whose asymptotic competitive ratio is strictly below $2$ and it is close to the lower bound. This is in contrast to the best possible absolute approximation ratio…
The harmonic function was first introduced by Lee and Lee (JACM 1985) for analyzing their online bin-packing algorithm. Subsequently, it has been used to obtain approximation algorithms for many different packing problems. Here we slightly generalize the harmonic function and give alternative proofs of its important properties.
We revisit the classic online bin packing problem. In this problem, items of positive sizes no larger than 1 are presented one by one to be packed into subsets called "bins" of total sizes no larger than 1, such that every item is assigned to a bin before the next item is presented. We use online partitioning of items into classes based on sizes, as in previous work, but we also apply a new method where items of one class can be packed into more than two types of bins, where a bin type is defined according to the number of such items grouped together. Additionally, we allow the smallest class …
In this work, we consider online vector bin packing. It is known that no algorithm can have a competitive ratio of $o(d/\log^2 d)$ in the absolute sense, though upper bounds for this problem were always shown in the asymptotic sense. Since variants of bin packing are traditionally studied with respect to the asymptotic measure and since the two measures are different, we focus on the asymptotic measure and prove new lower bounds on the asymptotic competitive ratio. The existing lower bounds prior to this work were much smaller than $3$ even for very large dimensions. We significantly improve t…
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 θ(1) approximation scheme for certain set of inputs for any dime…
We study the following variant of the classic {\em bin packing} problem. Given a set of items of various sizes, partitioned into groups, find a packing of the items in a minimum number of identical (unit-size) bins, such that no two items of the same group are assigned to the same bin. This problem, known as {\em bin packing with clique-graph conflicts}, has natural applications in storing file replicas, security in cloud computing and signal distribution. Our main result is an {\em asymptotic polynomial time approximation scheme (APTAS)} for the problem, improving upon the best known ratio of…
We study the 2-dimensional vector packing problem, which is a generalization of the classical bin packing problem where each item has 2 distinct weights and each bin has 2 corresponding capacities. The goal is to group items into minimum number of bins, without violating the bin capacity constraints. We propose a Θ}(n)-time approximation algorithm that is inspired by the O(n^2) algorithm proposed by Chang, Hwang, and Park.
We introduce the strongly NP-complete pagination problem, an extension of BIN PACKING where packing together two items may make them occupy less volume than the sum of their individual sizes. To achieve this property, an item is defined as a finite set of symbols from a given alphabet: while, in BIN PACKING, any two such sets would be disjoint, in PAGINATION, they can share zero, one or more symbols. After formulating the problem as an integer linear program, we try to approximate its solutions with several families of algorithms: from straightforward adaptations of classical BIN PACKING heuri…
We consider the bin packing problem with $d$ different item sizes and revisit the structure theorem given by Goemans and Rothvoß[6] about solutions of the integer cone. We present new techniques on how solutions can be modified and give a new structure theorem that relies on the set of vertices of the underlying integer polytope. As a result of our new structure theorem, we obtain an algorithm for the bin packing problem with running time $|V|^{2^{O(d)}} \cdot enc(I)^{O(1)}$, where $V$ is the set of vertices of the integer knapsack polytope and $enc(I)$ is the encoding length of the bin packin…
In the online bin packing problem, a sequence of items is revealed one at a time, and each item must be packed into an available bin instantly upon its arrival. In this paper, we revisit the problem under a setting where the total number of items T is known in advance, also known as the closed online bin packing problem. Specifically, we study both the stochastic model and the random permutation model. We develop and analyze an adaptive algorithm that solves an offline bin packing problem at geometric time intervals and uses the offline optimal solution to guide online packing decisions. Under…
The Bin Packing Problem is one of the most important optimization problems. In recent years, due to its NP-hard nature, several approximation algorithms have been presented. It is proved that the best algorithm for the Bin Packing Problem has the approximation ratio 3/2 and the time order O(n), unless P=NP. In this paper, first, a 3/2-approximation algorithm is presented, then a modification to FFD algorithm is proposed to decrease time order to linear. Finally, these suggested approximation algorithms are compared with some other approximation algorithms. The experimental results show the sug…
Since the Bin Packing Problem (BPP) is one of the main NP-hard problems, a lot of approximation algorithms have been suggested for it. It has been proven that the best algorithm for BPP has the approximation ratio of 3/2 and the time order of O(n), unless P=NP. In the current paper, a linear 3/2-approximation algorithm is presented. The suggested algorithm not only has the best possible theoretical factors, approximation ratio, space order, and time order, but also outperforms the other approximation algorithms according to the experimental results, therefore, we are able to draw the conclusio…
The 2D Online Bin Packing is a fundamental problem in Computer Science and the determination of its asymptotic competitive ratio has attracted great research attention. In a long series of papers, the lower bound of this ratio has been improved from 1.808, 1.856 to 1.907 and its upper bound reduced from 3.25, 3.0625, 2.8596, 2.7834 to 2.66013. In this paper, we rewrite the upper bound record to 2.5545. Our idea for the improvement is as follows. In SODA 2002 \cite{SS03}, Seiden and van Stee proposed an elegant algorithm called $H \otimes B$, comprised of the {\em Harmonic algorithm} $H$ and th…
