Asymptotic Approximation Scheme Research Articles

Constant-Ratio Approximation for Robust Bin Packing with Budgeted Uncertainty

We consider robust variants of the bin packing problem with uncertain item sizes. Specifically we consider two uncertainty sets previously studied in the literature. The first is budgeted uncertainty (the $U^\Gamma$ model), in which at most $\Gamma$ items deviate, each reaching its peak value, while other items assume their nominal values. The second uncertainty set, the $U^\Omega$ model, bounds the total amount of deviation in each scenario. We show that a variant of the Next-cover algorithm is a $2$ approximation for the $U^\Omega$ model, and another variant of this algorithm is a $2\Gamma$ approximation for the $U^\Gamma$ model. Unlike the classical bin packing problem, it is shown that (unless $\mathcal{P}=\mathcal{NP}$) no asymptotic approximation scheme exists for the $U^\Gamma$ model, for $\Gamma=1$. This motivates the question of the existence of a constant approximation factor algorithm for the $U^\Gamma$ model. Our main result is to answer this question by proving a (polynomial-time) $4.5$ approximation algorithm, based on a dynamic-programming approach.

An Improved Approximation Scheme for Variable-Sized Bin Packing

The Variable-Sized Bin Packing Problem (abbreviated as VSBPP or VBP) is a well-known generalization of the NP-hard Bin Packing Problem (BP) where the items can be packed in bins of M given sizes. The objective is to minimize the total capacity of the bins used. We present an asymptotic approximation scheme (AFPTAS) for VBP and BP with performance guarantee Aź(I)≤(1+ź)OPT(I)+Olog21ź$A_{\varepsilon }(I) \leq (1+ \varepsilon )OPT(I) + \mathcal {O}\left ({\log ^{2}\left (\frac {1}{\varepsilon }\right )}\right )$ for any problem instance I and any ź>0. The additive term is much smaller than the additive term of already known AFPTAS. The running time of the algorithm is O1ź6log1ź+log1źn$\mathcal {O}\left ({ \frac {1}{\varepsilon ^{6}} \log \left ({\frac {1}{\varepsilon }}\right ) + \log \left ({\frac {1}{\varepsilon }}\right ) n}\right )$ for BP and O1ź6log21ź+M+log1źn$\mathcal {O}\left ({ \frac {1}{{\varepsilon }^{6}} \log ^{2}\left ({\frac {1}{\varepsilon }}\right ) + M + \log \left ({\frac {1}{\varepsilon }}\right )n}\right )$ for VBP, which is an improvement to previously known algorithms. Many approximation algorithms have to solve subproblems, for example instances of the Knapsack Problem (KP) or one of its variants. These subproblems - like KP - are in many cases NP-hard. Our AFPTAS for VBP must in fact solve a generalization of KP, the Knapsack Problem with Inversely Proportional Profits (KPIP). In this problem, one of several knapsack sizes has to be chosen. At the same time, the item profits are inversely proportional to the chosen knapsack size so that the largest knapsack in general does not yield the largest profit. We introduce KPIP in this paper and develop an approximation scheme for KPIP by extending Lawler's algorithm for KP. Thus, we are able to improve the running time of our AFPTAS for VBP.

Polynomial-Time Approximation Schemes for Circle and Other Packing Problems

We consider the problem of packing a set of circles into a minimum number of unit square bins. To obtain rational solutions, we use augmented bins of height $$1+\gamma $$ , for some arbitrarily small number $$\gamma > 0$$ . For this problem, we obtain an asymptotic approximation scheme (APTAS) that is polynomial on $$\log 1/\gamma $$ , and thus $$\gamma $$ may be given as part of the problem input. For the special case that $$\gamma $$ is constant, we give a (one dimensional) resource augmentation scheme, that is, we obtain a packing into bins of unit width and height $$1+\gamma $$ using no more than the number of bins in an optimal packing without resource augmentation. Additionally, we obtain an APTAS for the circle strip packing problem, whose goal is to pack a set of circles into a strip of unit width and minimum height. Our algorithms are the first approximation schemes for circle packing problems, and are based on novel ideas of iteratively separating small and large items, and may be extended to a wide range of packing problems that satisfy certain conditions. These extensions comprise problems with different kinds of items, such as regular polygons, or with bins of different shapes, such as circles and spheres. As an example, we obtain APTAS’s for the problems of packing d-dimensional spheres into hypercubes under the $$L_p$$ -norm.

A Harmonic Algorithm for the 3D Strip Packing Problem

In the three-dimensional (3D) strip packing problem, we are given a set of 3D rectangular items and a 3D box $B$. The goal is to pack all the items in $B$ such that the height of the packing is minimized. We consider the most basic version of the problem, where the items must be packed with their edges parallel to the edges of $B$ and cannot be rotated. Building upon Caprara's work for the two-dimensional (2D) bin packing problem, we obtain an algorithm that, given any $\epsilon>0$, achieves an approximation of $T_{\infty}+\epsilon\approx1.69103+\epsilon$, where $T_{\infty}$ is the well-known number that occurs naturally in the context of bin packing. Our key idea is to establish a connection between bin packing solutions for an arbitrary instance $I$ and the strip packing solutions for the corresponding instance obtained from $I$ by applying the harmonic transformation to certain dimensions. Based on this connection, we also give a simple alternate proof of the $T_{\infty}+\epsilon$ approximation for 2D bin packing due to Caprara. In particular, we show how his result follows from a simple modification of the asymptotic approximation scheme for 2D strip packing due to Kenyon and Rémila.

Structure of Polynomial-Time Approximation

Approximation schemes are commonly classified as being either a polynomial-time approximation scheme (ptas) or a fully polynomial-time approximation scheme (fptas). To properly differentiate between approximation schemes for concrete problems, several subclasses have been identified: (optimum-)asymptotic schemes (ptas∞, fptas∞), efficient schemes (eptas), and size-asymptotic schemes. We explore the structure of these subclasses, their mutual relationships, and their connection to the classic approximation classes. We prove that several of the classes are in fact equivalent. Furthermore, we prove the equivalence of eptas to so-called convergent polynomial-time approximation schemes. The results are used to refine the hierarchy of polynomial-time approximation schemes considerably and demonstrate the central position of eptas among approximation schemes.We also present two ways to bridge the hardness gap between asymptotic approximation schemes and classic approximation schemes. First, using notions from fixed-parameter complexity theory, we provide new characterizations of when problems have a ptas or fptas. Simultaneously, we prove that a large class of problems (including all MAX-SNP-complete problems) cannot have an optimum-asymptotic approximation scheme unless P=NP, thus strengthening results of Arora et al. (J. ACM 45(3):501–555, 1998). Secondly, we distinguish a new property exhibited by many optimization problems: pumpability. With this notion, we considerably generalize several problem-specific approaches to improve the effectiveness of approximation schemes with asymptotic behavior.

Asymptotic Approximation Schemes for the Concave Cost Bin Packing Problem

Asymptotic Approximation Schemes for the Concave Cost Bin Packing Problem

An approximation scheme for the two-stage, two-dimensional knapsack problem

We present an approximation scheme for the two-dimensional version of the knapsack problem which requires packing a maximum-area set of rectangles in a unit square bin, with the further restrictions that packing must be orthogonal without rotations and done in two stages. Achieving a solution which is close to the optimum modulo a small additive constant can be done by taking wide inspiration from an existing asymptotic approximation scheme for two-stage two-dimensional bin packing. On the other hand, getting rid of the additive constant to achieve a canonical approximation scheme appears to be widely nontrivial.

A sublinear-time approximation scheme for bin packing

The bin packing problem is defined as follows: given a set of n items with sizes 0 < w 1 , w 2 , … , w n ≤ 1 , find a packing of these items into a minimum number of unit-size bins possible. We present a sublinear-time asymptotic approximation scheme for the bin packing problem; that is, for any ϵ > 0 , we present an algorithm A ϵ that has sampling access to the input instance and outputs a value k such that C opt ≤ k ≤ ( 1 + ϵ ) ⋅ C opt + 1 , where C opt is the cost of an optimal solution. It is clear that uniform sampling by itself will not allow a sublinear-time algorithm in this setting; a small number of items might constitute most of the total weight and uniform samples will not hit them. In this work we use weighted samples, where item i is sampled with probability proportional to its weight: that is, with probability w i / ∑ i w i . In the presence of weighted samples, the approximation algorithm runs in O ̃ ( n ⋅ poly ( 1 / ϵ ) ) + g ( 1 / ϵ ) time, where g ( x ) is an exponential function of x . When both weighted sampling and uniform sampling are allowed, O ̃ ( n 1 / 3 ⋅ poly ( 1 / ϵ ) ) + g ( 1 / ϵ ) time suffices. In addition to an approximate value to C opt , our algorithm can also output a constant-size “template” of a packing that can later be used to find a near-optimal packing in linear time.

Graph edge colouring: Tashkinov trees and Goldberg's conjecture

For the chromatic index χ ′ ( G ) of a (multi)graph G, there are two trivial lower bounds, namely the maximum degree Δ ( G ) and the density W ( G ) = max H ⊆ G , | V ( H ) | ⩾ 2 ⌈ | E ( H ) | / ⌊ | V ( H ) | / 2 ⌋ ⌉ . A famous conjecture due to Goldberg [M.K. Goldberg, On multigraphs of almost maximal chromatic class, Diskret. Analiz 23 (1973) 3–7 (in Russian)] and Seymour [P.D. Seymour, Some unsolved problems on one-factorization of graphs, in: J.A. Bondy, U.S.R. Murty (Eds.), Graph Theory and Related Topics, Academic Press, New York, 1979] says that every graph G satisfies χ ′ ( G ) ⩽ max { Δ ( G ) + 1 , W ( G ) } . This means that χ ′ ( G ) = W ( G ) for every graph G with χ ′ ( G ) ⩾ Δ ( G ) + 2 . The considered class of graphs J can be subdivided into an ascending sequence of classes ( J m ) m ⩾ 3 , and for m ⩽ 13 the conjecture is already proved. The “last” step was done by Favrholdt, Stiebitz and Toft [L.M. Favrholdt, M. Stiebitz, B. Toft, Graph edge colouring: Vizing's theorem and Goldberg's conjecture, DMF-2006-10-003, IMADA-PP-2006-20, University of Southern Denmark, preprint] in 2006, using and extending results from Tashkinov [V.A. Tashkinov, On an algorithm to colour the edges of a multigraph, Diskret. Analiz 7 (2000) 72–85 (in Russian)]. These methods are based on a colouring structure called Tashkinov tree. In this paper the same methods are used and extended to handle the “next” step m ⩽ 15 . This leads to the result χ ′ ( G ) ⩽ max { ⌊ 15 14 Δ ( G ) + 12 14 ⌋ , W ( G ) } for every graph G. Furthermore, the used methods also lead to several improvements of other known upper bounds for the chromatic index. In particular, an asymptotic approximation of the chromatic index can be obtained. We prove that for every ϵ > 0 and every graph G satisfying Δ ( G ) ⩾ 1 2 ϵ 2 the estimate χ ′ ( G ) ⩽ max { ( 1 + ϵ ) Δ ( G ) , W ( G ) } holds. This extends a result of Kahn [J. Kahn, Asymptotics of the chromatic index for multigraphs, J. Combin. Theory Ser. B 68 (1996) 233–254] as well as a result of Sanders and Steurer [P. Sanders, D. Steurer, An asymptotic approximation scheme for multigraph edge coloring, in: Proceedings of the Sixteenth ACM-SIAM Symposium on Discrete Algorithms (SODA05), em SIAM, 2005, pp. 897–906].

An asymptotic approximation scheme for the concave cost bin packing problem

Abstract We consider a generalized one-dimensional bin packing model in which the cost of a bin is a nondecreasing concave function of the utilization of the bin. We show that for any given positive constant ϵ , there exists a polynomial-time approximation algorithm with an asymptotic worst-case performance ratio of no more than 1 + ϵ .

Bin packing with controllable item sizes

We consider a natural resource allocation problem in which we are given a set of items, where each item has a list of pairs associated with it. Each pair is a configuration of an allowed size for this item, together with a nonnegative penalty, and an item can be packed using any configuration in its list. The goal is to select a configuration for each item so that the number of unit bins needed to pack the sizes plus the sum of penalties is minimized. This problem has applications in operating systems, bandwidth allocation, and discrete time–cost tradeoff planning problems. For the offline version of the problem we design an augmented asymptotic PTAS. That is, an asymptotic approximation scheme that uses bins of size slightly larger than 1. We further consider the online bounded space variant, where only a constant number of bins can be open simultaneously. We design a sequence of algorithms, where the sequence of their competitive ratios tends to the best possible asymptotic competitive ratio. Finally, we study the bin covering problem, in which a bin is covered if the sum of sizes allocated to it is at least 1. In this setting, penalties are interpreted as profits and the goal is to maximize the sum of profits plus the number of covered bins. We design an algorithm of best possible competitive ratio, which is 2. We generalize our results for online algorithms and unit sized bins to the case of variable sized bins, where there may be several distinct sizes of bins available for packing or covering, and get that the competitive ratios are again the same as for the more standard online problems.

A fast asymptotic approximation scheme for bin packing with rejection

“Bin packing with rejection” is the following problem: Given a list of items with associated sizes and rejection costs, find a packing into unit bins of a subset of the list such that the number of bins used plus the sum of rejection costs of unpacked items is minimized. We show that bin packing with rejection can be reduced to n multiple knapsack problems and, based on techniques for the multiple knapsack problem, we give a fast asymptotic polynomial time approximation scheme,“Reject&Pack”, with time complexity O ( n O ( ϵ − 2 ) ) . This improves a recent approximation scheme given by Epstein, which has time complexity O ( n O ( ( ϵ − 4 ) ϵ − 1 ) ) . We also show that Reject&Pack can be extended to variable-sized bin packing with rejection and give an asymptotic polynomial time approximation scheme.

Approximation Algorithms for Extensible Bin Packing

In a variation of bin packing called extensible bin packing, the number of bins to use is specified as part of the input, and bins may be extended to hold more than the usual unit capacity. The cost of a bin is one if it is not extended, and the size if it is extended. The goal is to pack a set of items of given sizes with minimum cost. Adapting ideas in [7, 8, ?], we give a fully polynomial asymptotic approximation scheme (FPTAAS) for extensible bin packing. We note that under a different scaling the problem could not admit an FPTAAS unless P = NP.

An Approximation Scheme for Bin Packing with Conflicts

In this paper we consider the following bin packing problem with conflicts. Given a set of items V = {1,..., n} with sizes s1,..., s ∈ (0,1) and a conflict graph G = (V, E), we consider the problem to find a packing for the items into bins of size one such that adjacent items (j, j′) ∈ E are assigned to different bins. The goal is to find an assignment with a minimum number of bins. This problem is a natural generalization of the classical bin packing problem. We propose an asymptotic approximation scheme for the bin packing problem with conflicts restricted to d-inductive graphs with constant d. This graph class contains trees, grid graphs, planar graphs and graphs with constant treewidth. The algorithm finds an assignment for the items such that the generated number of bins is within a factor of (1 + ∈) of optimal provided that the optimum number is sufficiently large. The running time of the algorithm is polynomial both in n and in $$\tfrac{1}{\varepsilon}$$ .

An asymptotic approximation applied to a simple model

An asymptotic approximation scheme based on a sequence of solutions is applied to a simple model problem of a harmonic oscillator coupled to a scalar field. We evaluate explicitly the evolution of the field off the initial hypersurface and discuss an appropriate choice of the initial data for the field in order for a sequence of solutions to have a Newtonian limit. The study may help to understand the more complicated situation in general relativity. We also discuss an alternative sequence of solutions that may be useful in studying instabilities that occur for large values of the coupling constant.

Gravitational radiation reaction in the Newtonian limit

The asymptotic approximation scheme based on the theory of the Newtonian limit developed in the preceding paper is applied to the gravitational radiation-reaction problem. All divergences encountered in previous approaches disappear: For any $\ensuremath{\epsilon}$ the asymptotic approximation is finite to all orders we have calculated, even beyond radiation-reaction order. This is because the divergent terms in previous work were misordered, and make finite contributions to coefficients of lower-order terms in the asymptotic expansion. The logarithmic divergences, in particular, turn up as an ${\ensuremath{\epsilon}}^{10}$ $\mathrm{ln}\ensuremath{\epsilon}$ term in the asymptotic expansion (i.e., between 2.5 and 3 post-Newtonian order) which shows that the relativistic sequence is not ${C}^{\ensuremath{\infty}}$ at $\ensuremath{\epsilon}=0$. This does not, however, affect the asymptotic convergence of the approximation. The radiation-reaction terms are used to calculate the period shortening of a nearly-Newtonian binary system directly from the equations of motion, avoiding the well-known difficulties associated with energy in general relativity. It is proved that the prediction derived from the standard quadrupole formula applies in the Newtonian limit. It is also shown that random data for the initial gravitational wave field do not affect the calculation of radiation reaction, even if their amplitude is of first post-Newtonian order.