Asymptotic Approximation Ratio Research Articles

The Near Exact Bin Covering Problem

AbstractWe present a new generalization of the bin covering problem that is known to be a strongly NP-hard problem. In our generalization there is a positive constant $$\varDelta $$ Δ , and we are given a set of items each of which has a positive size. We would like to find a partition of the items into bins. We say that a bin is near exact covered if the total size of items packed into the bin is between 1 and $$1+\varDelta $$ 1 + Δ . Our goal is to maximize the number of near exact covered bins. If $$\varDelta =0$$ Δ = 0 or $$\varDelta >0$$ Δ > 0 is given as part of the input, our problem is shown here to have no approximation algorithm with a bounded asymptotic approximation ratio (assuming that $$P\ne NP$$ P ≠ N P ). However, for the case where $$\varDelta >0$$ Δ > 0 is seen as a constant, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) that is our main contribution.

There is no APTAS for 2-dimensional vector bin packing: Revisited

We study the Vector Bin Packing and the Vector Bin Covering problems, multidimensional generalizations of the Bin Packing and the Bin Covering problems, respectively. In the Vector Bin Packing, we are given a set of d-dimensional vectors from [0,1]d and the aim is to partition the set into the minimum number of bins such that for each bin B, each component of the sum of the vectors in B is at most 1. Woeginger [17] claimed that the problem has no APTAS for dimensions greater than or equal to 2. We note that there was a slight oversight in the original proof. In this work, we give a revised proof using some additional ideas from [3,8]. In fact, we show that it is NP-hard to get an asymptotic approximation ratio better than 600599.An instance of Vector Bin Packing is called δ-skewed if every item has at most one dimension greater than δ. As a natural extension of our general d-Dimensional Vector Bin Packing result we show that for ε∈(0,12500) it is NP-hard to obtain a (1+ε)-approximation for δ-Skewed Vector Bin Packing if δ>20ε.In the Vector Bin Covering problem given a set of d-dimensional vectors from [0,1]d, the aim is to obtain a family of disjoint subsets (called bins) with the maximum cardinality such that for each bin B, each component of the sum of the vectors in B is at least 1. Using ideas similar to our Vector Bin Packing result, we show that for Vector Bin Covering there is no APTAS for dimensions greater than or equal to 2. In fact, we show that it is NP-hard to get an asymptotic approximation ratio better than 998997.

On the Theoretical Gap of Channel Hopping Sequences With Maximum Rendezvous Diversity in the Multichannel Rendezvous Problem

In the literature, there are several well-known periodic channel hopping (CH) sequences that can achieve maximum rendezvous diversity in a cognitive radio network (CRN). For a CRN with N channels, it is known that the period of such a CH sequence is at least N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . The asymptotic approximation ratio, defined as the ratio of the period of a CH sequence to the lower bound N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> when N → ∞, is still 2.5 for the best known CH sequence in the literature. An open question in the multichannel rendezvous problem is whether it is possible to construct a periodic CH sequence that has the asymptotic approximation ratio of 1. In this paper, we tighten the theoretical gap by proposing CH sequences, called IDEAL-CH, that have the asymptotic approximation ratio of 2. For a weaker requirement that only needs the two users to rendezvous on one commonly available channel in a period, we propose channel hopping sequences, called ORTHO-CH, with period (2 p+1) p, where p is the smallest prime not less than N.

Optimal Pricing for MHR and λ-regular Distributions

We study the performance of anonymous posted-price selling mechanisms for a standard Bayesian auction setting, where n bidders have i.i.d. valuations for a single item. We show that for the natural class of Monotone Hazard Rate (MHR) distributions, offering the same, take-it-or-leave-it price to all bidders can achieve an (asymptotically) optimal revenue. In particular, the approximation ratio is shown to be 1+ O (ln ln n / ln n ), matched by a tight lower bound for the case of exponential distributions. This improves upon the previously best-known upper bound of e /( e −1)≈ 1.58 for the slightly more general class of regular distributions. In the worst case (over n ), we still show a global upper bound of 1.35. We give a simple, closed-form description of our prices, which, interestingly enough, relies only on minimal knowledge of the prior distribution, namely, just the expectation of its second-highest order statistic. Furthermore, we extend our techniques to handle the more general class of λ-regular distributions that interpolate between MHR (λ =0) and regular (λ =1). Our anonymous pricing rule now results in an asymptotic approximation ratio that ranges smoothly, with respect to λ, from 1 (MHR distributions) to e /( e −1) (regular distributions). Finally, we explicitly give a class of continuous distributions that provide matching lower bounds, for every λ.

Solving fully dynamic bin packing problem for virtual machine allocation in the cloud environment by the futuristic greedy algorithm

Many scientists apply fully dynamic bin packing problem solving for resource allocation of virtual machines in cloud environments. The goal of problem-solving is to reduce the number of allocated hosts (bins) and virtual machines (items) migration rates for reducing energy consumption. This study demonstrates a greedy futuristic algorithm (proposed algorithm) for fully dynamic bin packaging with an average asymptotic approximation ratio of 1.231, better than other existing algorithms. The proposed algorithm identifies inappropriate local selections using special futuristic conditions to prevent them as much as possible. Eventually, suitable choices determine and discard the improper ones. The proposed algorithm illustrates an asymptotic approximation ratio of (t/ (t-1)) OPT, where the value of t depends on the distribution of the arrived and departed items. Also, OPT is the number of bins by an optimal solution. Finally, in experiments of datasets using a maximum utilization of 80% of each bin, the average migration rate is 0.338. Using the proposed method for allocating resources in the cloud environment can allocate hosts to a virtual machine using almost optimal use. This allocation can reduce the cost of maintaining and purchasing hosts. Also, this method can reduce the migration rate of virtual machines. As a result, decreasing migration improves the energy consumption cost in the cloud environment.

Approximation Algorithms forD-optimal Design

Experimental design is a classical statistics problem, and its aim is to estimate an unknown vector from linear measurements where a Gaussian noise is introduced in each measurement. For the combinatorial experimental design problem, the goal is to pick a subset of experiments so as to make the most accurate estimate of the unknown parameters. In this paper, we will study one of the most robust measures of error estimation—the D-optimality criterion, which corresponds to minimizing the volume of the confidence ellipsoid for the estimation error. The problem gives rise to two natural variants depending on whether repetitions of experiments are allowed or not. We first propose an approximation algorithm with a 1/e-approximation for the D-optimal design problem with and without repetitions, giving the first constant-factor approximation for the problem. We then analyze another sampling approximation algorithm and prove that it is asymptotically optimal. Finally, for D-optimal design with repetitions, we study a different algorithm proposed by the literature and show that it can improve this asymptotic approximation ratio. All the sampling algorithms studied in this paper are shown to admit polynomial-time deterministic implementations.

Approximation Algorithms for Training One-Node ReLU Neural Networks

Training a one-node neural network with the ReLU activation function via optimization, which we refer to as the ON-ReLU problem, is a fundamental problem in machine learning. In this paper, we begin by proving the NP-hardness of the ON-ReLU problem. We then present an approximation algorithm to solve the ON-ReLU problem, whose running time is $\mathcal {O}(n^k)$ where $n$ is the number of samples, and $k$ is a predefined integral constant as an algorithm parameter. We analyze the performance of this algorithm under two regimes and show that: (1) given any arbitrary set of training samples, the algorithm guarantees an $(n/k)$ -approximation for the ON-ReLU problem – to the best of our knowledge, this is the first time that an algorithm guarantees an approximation ratio for arbitrary data scenario; thus, in the ideal case (i.e., when the training error is zero) the approximation algorithm achieves the globally optimal solution for the ON-ReLU problem; and (2) given training sample with Gaussian noise, the same approximation algorithm achieves a much better asymptotic approximation ratio which is independent of the number of samples $n$ . Extensive numerical studies show that our approximation algorithm can perform better than the gradient descent algorithm. Our numerical results also show that the solution of the approximation algorithm can provide a good initialization for gradient descent, which can significantly improve the performance.

QoS-Constrained Semi-Persistent Scheduling of Machine-Type Communications in Cellular Networks

The dramatic growth of machine-to-machine (M2M) communication in cellular networks brings the challenge of satisfying the quality of service (QoS) requirements of a large number of M2M devices with limited radio resources. In this paper, we propose an optimization framework for the semi-persistent scheduling of M2M transmissions based on the exploitation of their periodicity with the goal of reducing the overhead of the signaling required for connection initiation and scheduling. The goal of the optimization problem is to minimize the number of frequency bands used by the M2M devices to allow fair resource allocation of newly joining M2M and human-to-human communications. The constraints of the problem are delay and periodicity requirements of the M2M devices. We first prove that the optimization problem is NP-hard and then propose a polynomial-time heuristic algorithm employing a fixed priority assignment according to the QoS characteristics of the devices. We show that this heuristic algorithm provides an asymptotic approximation ratio of 2.33 to the optimal solution for the case where the delay tolerances of the devices are equal to their periods. Through extensive simulations, we demonstrate that the proposed algorithm performs better than the existing algorithms in terms of frequency band usage and schedulability.

Min-Sum Bin Packing

We study min-sum bin packing (MSBP). This is a bin packing problem, where the cost of an item is the index of the bin into which it is packed. The problem is equivalent to a batch scheduling problem we define, where the total completion time is to be minimized. The problem is NP-hard in the strong sense. We show that it is not harder than this by designing a polynomial time approximation scheme for it. We also show that several natural algorithms which are based on well-known bin packing heuristics (such as First Fit Decreasing) fail to achieve an asymptotic finite approximation ratio, whereas Next Fit Increasing has an absolute approximation ratio of at most 2, and an asymptotic approximation ratio of at most 1.6188. We design a new heuristic that applies Next Fit Increasing on the relatively small items and adds the larger items using First Fit Decreasing, and show that its asymptotic approximation ratio is at most 1.5604.

The tight asymptotic approximation ratio of First Fit for bin packing with cardinality constraints

In bin packing with cardinality constraints (BPCC), there is an upper bound k≥2 on the number of items that can be packed into each bin, additionally to the standard constraint on the total size of items. We study the algorithm First Fit (FF), acting on a list of items, packing each item into the minimum indexed bin that contains at most k−1 items and has sufficient space for the item. We present a complete analysis of its asymptotic approximation ratio for all values of k. Many years after FF for BPCC was introduced, its tight asymptotic approximation ratio is finally found.

The tight absolute bound of First Fit in the parameterized case

The tight asymptotic approximation ratio of the classical bin packing algorithm called First Fit was known for many years, also in the parameterized case, when any item has size of at most 1/d, where d is integer. But the tight absolute bound was found only recently, and only for d=1. Here we give the tight absolute approximation ratio of First Fit for any d≥2. In fact, we do more. For any value of OPT (the number of bins in an optimum solution) we determine that exactly how large FF (the number of bins created by First Fit) can be in the worst case.The proof is very simple, since we can apply the lower bound construction of case d=1, in a slightly modified form.

Batched bin packing revisited

We revisit the batched bin packing problem. In this model, items come in K consecutive batches, and the items of the earlier batches must be packed without any knowledge of later batches. We give the first approximation algorithm for the case $$K=2$$K=2, with tight asymptotic approximation ratio of 1.5833, while the known lower bound of the model is 1.378. With the application of this result, we are also able to provide an improved algorithm for the recently defined graph-bin packing problem in a special case, where we improve the upper bound from 3 to 2.5833.

Two-dimensional strip packing with unloading constraints

In this paper we present approximation algorithms for the two-dimensional strip packing problem with unloading constraints. In this problem, we are given a strip S of width 1 and unbounded height, and n items of C different classes, each item ai with height h(ai), width w(ai) and class c(ai). As in the strip packing problem, we have to pack all items, minimizing the height used, but now we have the additional constraint that items of higher classes cannot block the way out of items of lower classes. For the case in which horizontal and vertical movements for removing the items are allowed, we design an algorithm whose asymptotic performance bound is 3. For the case in which only vertical movements are allowed, we design a bin packing based algorithm with an asymptotic approximation ratio of 5.745. Moreover, we also design approximation algorithms for restricted cases of both versions of the problem. These problems have practical applications in dealing with routing problems with loading/unloading constraints.

Two-dimensional strip packing problem with load balancing, load bearing and multi-drop constraints

In the oriented Two-Dimensional Strip Packing Problem (2SP), one has to pack a set of rectangular items into a rectangular strip and minimizes the overall strip height used to pack all items. This paper deals with the 2SP under two practical situations. In the first, feasible packings must respect the load balancing and the multi-drop constraints. That is, the center of gravity of the packing at each moment must lie in a safety region, even after a subset of items is unloaded. In the second situation, the load balancing constraint is combined with the load bearing constraint. In a packing that respect the load bearing constraint, the bearing capacity of each item must be respected. That is, there is a maximum tolerable weight that each item can bear. For both situations, we present approximate 0–1 integer linear programming models and heuristics, based on level-packing algorithms and packing on corner points. The level-packing heuristic has an asymptotic approximation ratio bounded by 1.75, when the number of orders is bounded by a constant. The heuristics have proven to be helpful when combined with the integer models. In addition, many computational experiments validate the integer models and show that they are suitable to deal with problems where the number of possible positions to arrange the items in the bin is small.

Bin covering with a general profit function: approximability results

In the recent paper (Benkő et al. 2010) we introduced a new problem that we call Bin Packing/Covering with Delivery, or BP/CD for short. Mainly we mean under this expression that we look for not only a good, but a “good and fast” packing or covering. In the present paper we investigate the offline case. For the analysis, a novel view on “offline optimum” is introduced, which appears to be relevant concerning all problems where a final solution is ordering-dependent. We prove that if the item sizes are not allowed to be arbitrarily close to zero, then an optimal offline solution can be found in polynomial time. On the other hand, for unrestricted problem instances, no polynomial-time algorithm can achieve an asymptotic approximation ratio better than 6/7 if \(P\ne NP\).

Approximation quality of the hypervolume indicator

In order to allow a comparison of (otherwise incomparable) sets, many evolutionary multi-objective optimizers use indicator functions to guide the search and to evaluate the performance of search algorithms. The most widely used indicator is the hypervolume indicator. It measures the volume of the dominated portion of the objective space bounded from below by a reference point.Though the hypervolume indicator is very popular, it has not been shown that maximizing the hypervolume indicator of sets of bounded size is indeed equivalent to the overall objective of finding a good approximation of the Pareto front. To address this question, we compare the optimal approximation ratio with the approximation ratio achieved by two-dimensional sets maximizing the hypervolume indicator. We bound the optimal multiplicative approximation ratio of n points by 1+Θ(1/n) for arbitrary Pareto fronts. Furthermore, we prove that the same asymptotic approximation ratio is achieved by sets of n points that maximize the hypervolume indicator. However, there is a provable gap between the two approximation ratios which is even exponential in the ratio between the largest and the smallest value of the front.We also examine the additive approximation ratio of the hypervolume indicator in two dimensions and prove that it achieves the optimal additive approximation ratio apart from a small ratio ⩽n/(n−2), where n is the size of the population. Hence the hypervolume indicator can be used to achieve a good additive but not a good multiplicative approximation of a Pareto front. This motivates the introduction of a “logarithmic hypervolume indicator” which provably achieves a good multiplicative approximation ratio.

Two Dimensional Strip Packing with Unloading Constraints

Abstract In this paper we present approximation algorithms for the two dimensional strip packing problem with unloading constraints. In this problem, we are given a strip S of width 1 and unbounded height, and n items of C different classes, each item a i of height h ( a i ) , width w ( a i ) and class c ( a i ) . As in the strip packing problem with rotations, we have to pack all items minimizing the used height, but now we have the additional constraint that items of higher classes cannot block the way out of lower classes items. We design a bin packing based algorithm with asymptotic approximation ratio of 5.745. These problems have practical applications on routing problems with loading/unloading constraints.

Uniform unweighted set cover: The power of non-oblivious local search

We are given n base elements and a finite collection of subsets of them. The size of any subset varies between p to k ( p < k ). In addition, we assume that the input contains all possible subsets of size p . Our objective is to find a subcollection of minimum-cardinality which covers all the elements. This problem is known to be NP-hard. We provide two approximation algorithms for it, one for the generic case, and an improved one for the special case of ( p , k ) = ( 2 , 4 ) . The algorithm for the generic case is a greedy one, based on packing phases: at each phase we pick a collection of disjoint subsets covering i new elements, starting from i = k down to i = p + 1 . At a final step we cover the remaining base elements by the subsets of size p . We derive the exact performance guarantee of this algorithm for all values of k and p , which is less than H k , where H k is the k ’th harmonic number. However, the algorithm exhibits the known improvement methods over the greedy one for the unweighted k -set cover problem (in which subset sizes are only restricted not to exceed k ), and hence it serves as a benchmark for our improved algorithm. The improved algorithm for the special case of ( p , k ) = ( 2 , 4 ) is based on non-oblivious local search: it starts with a feasible cover, and then repeatedly tries to replace sets of size 3 and 4 so as to maximize an objective function which prefers big sets over small ones. For this case, our generic algorithm achieves an asymptotic approximation ratio of 1.5 + ϵ , and the local search algorithm achieves a better ratio, which is bounded by 1.458333 … + ϵ .

Bin packing with general cost structures

Following the work of Anily et al., we consider a variant of bin packing called bin packing with general cost structures (GCBP) and design an asymptotic fully polynomial time approximation scheme (AFPTAS) for this problem. In the classic bin packing problem, a set of one-dimensional items is to be assigned to subsets of total size at most 1, that is, to be packed into unit sized bins. However, in GCBP, the cost of a bin is not 1 as in classic bin packing, but it is a non-decreasing and concave function of the number of items packed in it, where the cost of an empty bin is zero. The construction of the AFPTAS requires novel techniques for dealing with small items, which are developed in this work. In addition, we develop a fast approximation algorithm which acts identically for all non-decreasing and concave functions, and has an asymptotic approximation ratio of 1.5 for all functions simultaneously.

Tight results for Next Fit and Worst Fit with resource augmentation

It is well known that the two simple algorithms for the classic bin packing problem, NF and WF both have an approximation ratio of 2 . However, WF seems to be a more reasonable algorithm, since it never opens a new bin if an existing bin can still be used. Using resource augmented analysis, where the output of an approximation algorithm, which can use bins of size b > 1 , is compared to an optimal packing into bins of size 1 , we give a complete analysis of the asymptotic approximation ratio of WF and of NF , and use it to show that WF is strictly better than NF for any 1 < b < 2 , while they have the same asymptotic performance guarantee for all b ≥ 2 , and for b = 1 .