Improved approximation algorithms for multidimensional bin packing problems

In this paper we introduce a new general approximation method for set covering problems, based on the combination of randomized rounding of the (near-) optimal solution of the linear programming (LP) relaxation, leading to a partial integer solution and the application of a well-behaved approximation algorithm to complete this solution. If the value of the solution returned by the latter can be bounded in a suitable way, as is the case for the most relevant generalizations of bin packing, the method leads to improved approximation guarantees, along with a proof of tighter integrality gaps for the LP relaxation. For d-dimensional vector packing, we obtain a polynomial-time randomized algorithm with asymptotic approximation guarantee arbitrarily close to $\ln d + 1$. For $d=2$, this value is $1.693\dots$; i.e., we break the natural 2 “barrier” for this case. Moreover, for small values of d this is a notable improvement over the previously known $O(\ln d)$ guarantee by Chekuri and Khanna [SIAM J. Comput., 33 (2004), pp. 837–851]. For two-dimensional bin packing with and without rotations, we obtain polynomial-time randomized algorithms with asymptotic approximation guarantee $1.525\dots$, improving upon previous algorithms with asymptotic performance guarantees arbitrarily close to 2 by Jansen and van Stee [On strip packing with rotations, in Proceedings of the 37th Annual ACM Symposium on the Theory of Computing, 2005, pp. 755–761] for the problem with rotations and $1.691\ldots$ by Caprara [Math. Oper. Res., 33 (2008), pp. 203–215] for the problem without rotations. The previously unknown key property used in our proofs follows from a retrospective analysis of the implications of the landmark bin packing approximation scheme by Fernandez de la Vega and Lueker [Combinatorica, 1 (1981), pp. 349–355]. We prove that their approximation scheme is “subset oblivious,” which leads to numerous applications.

Many of the currently best-known approximation algorithms for NP-hard optimization problems are based on Linear Programming (LP) and Semi-definite Programming (SDP) relaxations. Given its power, this class of algorithms seems to contain the most favourable candidates for outperforming the current state-of-the-art approximation guarantees for NP-hard problems, for which there still exists a gap between the inapproximability results and the approximation guarantees that we know how to achieve in polynomial time. In this thesis, we address both the power and the limitations of these relaxations, as well as the connection between the shortcomings of these relaxations and the inapproximability of the underlying problem. In the first part, we study the limitations of LP relaxations of well-known graph problems such as the Vertex Cover problem and the Independent Set problem. We prove that any small LP relaxation for the aforementioned problems, cannot have an integrality gap strictly better than $2$ and $\omega(1)$, respectively. Furthermore, our lower bound for the Independent Set problem also holds for any SDP relaxation. Prior to our work, it was only known that such LP relaxations cannot have an integrality gap better than $1.5$ for the Vertex Cover Problem, and better than $2$ for the Independent Set problem. In the second part, we study the so-called knapsack cover inequalities that are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield LP relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. We address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. In the last part, we show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. This connection is inspired by a family of integrality gap instances of a certain LP relaxation. Assuming the hardness of an optimization problem on k-partite graphs, we obtain a hardness of $2-\varepsilon$ for the problem of minimizing the makespan for scheduling with preemption on identical parallel machines, and a super constant inapproximability for the problem of scheduling on related parallel machines. Prior to this result, it was only known that the first problem does not admit a PTAS, and the second problem is NP-hard to approximate within a factor strictly better than 2, assuming the Unique Games Conjecture.

Approximation and online algorithms for multidimensional bin packing: A survey

We present several polynomial-time approximation algorithms for the one-dimensional bin-packing problem. using a subroutine to solve a certain linear programming relaxation of the problem. Our main results are as follows: There is a polynomial-time algorithm A such that A(I) ≤ OPT(I) + O(log2 OPT(I)). There is a polynomial-time algorithm A such that, if m(I) denotes the number of distinct sizes of pieces occurring in instance I, then A(I) ≤ OPT(I) + O(log2 m(I)). There is an approximation scheme which accepts as input an instance I and a positive real number ε, and produces as output a packing using as most (1 + ε) OPT(I) + O(ε-2) bins. Its execution time is O(ε-c n log n), where c is a constant. These are the best asymptotic performance bounds that have been achieved to date for polynomial-time bin-packing. Each of our algorithms makes at most O(log n) calls on the LP relaxation subroutine and takes at most O(n log n) time for other operations. The LP relaxation of bin packing was solved efficiently in practice by Gilmore and Gomory. We prove its membership in P, despite the fact that it has an astronomically large number of variables.

A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin—i.e., for each bin, a family of subsets of items that fit in to that bin. The goal is to pack items into bins to maximize the aggregate value. This class of problems includes the maximum generalized assignment problem (GAP)1 and a distributed caching problem (DCP) described in this paper. Given a β-approximation algorithm for finding the highest value packing of a single bin, we give A polynomial-time LP-rounding based ((1 − 1/e)β)-approximation algorithm. A simple polynomial-time local search (β/(β + 1) − ε)-approximation algorithm, for any ε > 0. Therefore, for all examples of SAP that admit an approximation scheme for the single-bin problem, we obtain an LP-based algorithm with (1 − 1/e − ε)-approximation and a local search algorithm with (½ - ε)-approximation guarantee. Furthermore, for cases in which the subproblem admits a fully polynomial approximation scheme (such as for GAP), the LP-based algorithm analysis can be strengthened to give a guarantee of 1 − 1/e. The best previously known approximation algorithm for GAP is a ½-approximation by Shmoys and Tardos and Chekuri and Khanna. Our LP algorithm is based on rounding a new linear programming relaxation, with a provably better integrality gap. To complement these results, we show that SAP and DCP cannot be approximated within a factor better than 1 − 1/e unless NP ⫅ DTIME(nO(log log n)), even if there exists a polynomial-time exact algorithm for the single-bin problem. We extend the (1 − 1/e)-approximation algorithm to a constant-factor approximation algorithms for a nonseparable assignment problem with applications in maximizing revenue for budget-constrained combinatorial auctions and the AdWords assignment problem. We generalize the local search algorithm to yield a ½ - ε approximation algorithm for the maximum k-median problem with hard capacities.

Summary. The bin packing problem is a fundamental and important optimization problem in theoretical computer science [4], [6]. An instance is a sequence of items, each being of positive size at most one. The task is to place all the items into bins so that the total size of items in each bin is at most one and the number of bins that contain at least one item is minimum. Approximation algorithms have been intensively studied. Algorithm NextFit would be the simplest one. The algorithm repeatedly does the following: If the first unprocessed item in the sequence can be placed, in terms of size, additionally to the bin into which the algorithm has placed an item the last time, place the item into that bin; otherwise place the item into an empty bin. Johnson [5] proved that the number of the resulting bins by algorithm NextFit is less than twice the number of the fewest bins that are needed to contain all items. In this article, we formalize in Mizar [1], [2] the bin packing problem as follows: An instance is a sequence of positive real numbers that are each at most one. The task is to find a function that maps the indices of the sequence to positive integers such that the sum of the subsequence for each of the inverse images is at most one and the size of the image is minimum. We then formalize algorithm NextFit, its feasibility, its approximation guarantee, and the tightness of the approximation guarantee.

Structured prediction is used in areas such as computer vision and natural language processing to predict structured outputs such as segmentations or parse trees. In these settings, prediction is performed by MAP inference or, equivalently, by solving an integer linear program. Because of the complex scoring functions required to obtain accurate predictions, both learning and inference typically require the use of approximate solvers. We propose a theoretical explanation to the striking observation that approximations based on linear programming (LP) relaxations are often tight on real-world instances. In particular, we show that learning with LP relaxed inference encourages integrality of training instances, and that tightness generalizes from train to test data.

We consider two variants of the classical bin packing problem in which items may be fragmented. This can potentially reduce the total number of bins needed for packing the instance. However, since fragmentation incurs overhead, we attempt to avoid it as much as possible. In bin packing with size increasing fragmentation (BP-SIF), fragmenting an item increases the input size (due to a header/footer of fixed size that is added to each fragment). In bin packing with size preserving fragmentation (BP-SPF), there is a bound on the total number of fragmented items. These two variants of bin packing capture many practical scenarios, including message transmission in community TV networks, VLSI circuit design and preemptive scheduling on parallel machines with setup times/setup costs.While both BP-SPF and BP-SIF do not belong to the class of problems that admit a polynomial time approximation scheme (PTAS), we show in this paper that both problems admit a dual PTAS and an asymptotic PTAS. We also develop for each of the problems a dual asymptotic fully polynomial time approximation scheme (AFPTAS). The AFPTASs are based on a non-trivial application of a fast combinatorial FPTAS for packing linear programs with negative entries, proposed recently by Garg and Khandekar [5].KeywordsApproximation SchemeParallel MachinePolynomial Time Approximation SchemeLarge ItemSmall ItemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We show that solving linear programming (LP) relaxations of many classical NP-hard combinatorial optimization problems is as hard as solving the general LP problem. Precisely, the general LP can be reduced in linear time to the LP relaxation of each of these problems. This result poses a fundamental limitation for designing efficient algorithms to solve the LP relaxations, because finding such an algorithm might improve the complexity of best known algorithms for the general LP. Besides linear-time reductions, we show that the LP relaxations of the considered problems are P-complete under log-space reduction, therefore also hard to parallelize.

A linear programming (LP)-based framework is presented for obtaining converses for finite blocklength lossy joint source-channel coding problems. The framework applies for any loss criterion, generalizes certain previously known converses, and also extends to multi-terminal settings. The finite blocklength problem is posed equivalently as a nonconvex optimization problem and using a lift-and-project-like method, a close but tractable LP relaxation of this problem is derived. Lower bounds on the original problem are obtained by the construction of feasible points for the dual of the LP relaxation. A particular application of this approach leads to new converses, which recover and improve on the converses of Kostina and Verdu for finite blocklength lossy joint source-channel coding and lossy source coding. For finite blocklength channel coding, the LP relaxation recovers the converse of Polyanskiy, Poor and Verdu and leads to a new improvement on the converse of Wolfowitz, showing thereby that our LP relaxation is asymptotically tight with increasing blocklengths for channel coding, lossless source coding, and joint source-channel coding with the excess distortion probability as the loss criterion. Using a duality-based argument, a new converse is derived for finite blocklength joint source-channel coding for a class of source-channel pairs. Employing this converse, the LP relaxation is also shown to be tight for all blocklengths for the minimization of the expected average symbolwise Hamming distortion of a $q$ -ary uniform source over a $q$ -ary symmetric memoryless channel for any $q\in {\mathbb {N}}$ . The optimization formulation and the lift-and-project method are extended to networked settings and demonstrated by obtaining an improvement on a converse of Zhou et al. for the successive refinement problem for successively refinable source-distortion measure triplets.

We consider machine scheduling on unrelated parallel machines with the objective to minimize the schedule makespan. We assume that, in addition to its machine dependence, the processing time of any job is dependent on the usage of a discrete renewable resource, e.g. workers. A given amount of that resource can be distributed over the jobs in process at any time, and the more of that resource is allocated to a job, the smaller is its processing time. This model generalizes the classical unrelated parallel machine scheduling problem by adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos. On the basis of an integer linear programming formulation for a relaxation of the problem, we use LP rounding techniques to allocate resources to jobs, and to assign jobs to machines. Combined with Graham’s list scheduling, we show how to derive a 4-approximation algorithm. We also show how to tune our approach to yield a 3.75-approximation algorithm. This is achieved by applying the same rounding technique to a slightly modified linear programming relaxation, and by using a more sophisticated scheduling algorithm that is inspired by the harmonic algorithm for bin packing. We finally derive inapproximability results for two special cases, and discuss tightness of the integer linear programming relaxations.

Multi-capacity bin-packing is a generalization of the classical one-dimensional bin-packing problem in which the bin capacity and the item sizes are represented by d-dimensional vectors. Previous work in d-capacity bin-packing algorithms analyzed variants of single capacity bin-packing, extended to deal with the d capacities. These extended algorithms are oblivious to the additional capacity information, however, and do not scale well with increasing d. We provide new packing algorithms which use the additional capacity information to provide better packing and show how these algorithms might lead to better multi-resource allocation and scheduling solutions.

We consider the two-dimensional bin packing and strip packing problem, where a list of rectangles has to be packed into a minimal number of rectangular bins or a strip of minimal height, respectively. All packings have to be non-overlapping and orthogonal, i.e., axis-parallel. Our algorithm for strip packing has an absolute approximation ratio of 1.9396 and is the first algorithm to break the approximation ratio of 2 which was established more than a decade ago. Moreover, we present a polynomial time approximation scheme ($\mathcal{PTAS}$) for strip packing where rotations by 90 degrees are permitted and an algorithm for two-dimensional bin packing with an absolute worst-case ratio of 2, which is optimal provided $\mathcal{P} \not= \mathcal{NP}$.

In this paper, we study two variants of the bin packing /covering problems called Maximum Resource Bin Packing (MRBP) and Lazy Bin Covering (LBC) problems, and present new approximation algorithms for each of them. For the offline MRBP problem, the previous best known approximation ratio is $\frac{6}{5}=1.2$, achieved by the classical First-Fit-Increasing (FFI) algorithm [1]. In this paper, we give a new FFI-type algorithm with an approximation ratio of $\frac{80}{71}\approx 1.12676$. For the offline LBC problem, it has been shown in [2] that the classical First-Fit-Decreasing (FFD) algorithm achieves an approximation ratio of $\frac{71}{60}\approx 1.18333$. In this paper, we present a new FFD-type algorithm with an approximation ratio of $\frac{17}{15}\approx 1.13333$. Both algorithms are simple, run in near linear time (i.e., O(n logn)), and therefore are practical.

The shortest route representation of the dynamic multi-item multi-level capacitated lotsizing problem is appealing due to the tight bound provided by its Linear Programming (LP) relaxation. However, it suffers from two main drawbacks: Firstly, even solving the LP relaxation of problem instances with up to 16 time periods and 40 items with standard mathematical programming software might require a prohibitive amount of computer time. Secondly, the quadratic growth of the number of variables with the number of periods restricts the solution of problem instances with many periods. Both drawbacks will be addressed in this paper by proposing reformulations of the original shortest route model. Especially we will introduce a model formulation which allows theuser to find a tradeoff between model size and tightness of the lower bound obtained by the LP relaxation by specifying the number oflook ahead periods τ. Furthermore, we will provide an iterative procedure for determining those look ahead periods which result in the tightest LP relaxation. Theoretical insights as well as computational results are provided, too.