Online Bin Packing Research Articles

A pattern-based algorithm with fuzzy logic bin selector for online bin packing problem

The online bin packing problem is a well-known optimization challenge that finds application in a wide range of real-world scenarios. In the paper, we propose a novel algorithm called FuzzyPatternPack(FPP), which leverages fuzzy inference and pattern-based predictions of the distribution of item sizes in online bin packing. In comparison to traditional heuristics like BestFit(BF) and FirstFit(FF), as well as the more recent PatternPack(PaP) and ProfilePacking(PrP) algorithm based on online predictions, FPP demonstrates competitive and superior performance in solving various benchmark problems. Particularly, it excels in addressing problems with evolving distributions, making it a promising solution for real-world applications where the item sizes may change over time. This research unveils the promising potential of employing fuzzy logic to effectively address uncertainty in scheduling and planning problems.

Mathematical discoveries from program search with large language models

Large language models (LLMs) have demonstrated tremendous capabilities in solving complex tasks, from quantitative reasoning to understanding natural language. However, LLMs sometimes suffer from confabulations (or hallucinations), which can result in them making plausible but incorrect statements1,2. This hinders the use of current large models in scientific discovery. Here we introduce FunSearch (short for searching in the function space), an evolutionary procedure based on pairing a pretrained LLM with a systematic evaluator. We demonstrate the effectiveness of this approach to surpass the best-known results in important problems, pushing the boundary of existing LLM-based approaches3. Applying FunSearch to a central problem in extremal combinatorics—the cap set problem—we discover new constructions of large cap sets going beyond the best-known ones, both in finite dimensional and asymptotic cases. This shows that it is possible to make discoveries for established open problems using LLMs. We showcase the generality of FunSearch by applying it to an algorithmic problem, online bin packing, finding new heuristics that improve on widely used baselines. In contrast to most computer search approaches, FunSearch searches for programs that describe how to solve a problem, rather than what the solution is. Beyond being an effective and scalable strategy, discovered programs tend to be more interpretable than raw solutions, enabling feedback loops between domain experts and FunSearch, and the deployment of such programs in real-world applications.

Optimal Online Bin Packing Algorithms for Some Cases with Two Item Sizes

Optimal Online Bin Packing Algorithms for Some Cases with Two Item Sizes

Automated Algorithm Selection: from Feature-Based to Feature-Free Approaches

We propose a novel technique for algorithm-selection, applicable to optimisation domains in which there is implicit sequential information encapsulated in the data, e.g., in online bin-packing. Specifically we train two types of recurrent neural networks to predict a packing heuristic in online bin-packing, selecting from four well-known heuristics. As input, the RNN methods only use the sequence of item-sizes. This contrasts to typical approaches to algorithm-selection which require a model to be trained using domain-specific instance features that need to be first derived from the input data. The RNN approaches are shown to be capable of achieving within 5% of the oracle performance on between 80.88 and 97.63% of the instances, depending on the dataset. They are also shown to outperform classical machine learning models trained using derived features. Finally, we hypothesise that the proposed methods perform well when the instances exhibit some implicit structure that results in discriminatory performance with respect to a set of heuristics. We test this hypothesis by generating fourteen new datasets with increasing levels of structure, and show that there is a critical threshold of structure required before algorithm-selection delivers benefit.

Online Bin Packing of Squares and Cubes

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-the-art results by providing counter-examples to the current best algorithms and their analysis, where the claimed bounds were 2.1187 for square packing and 2.6161 for cube packing.

Comparing the costs of Any Fit algorithms for bin packing

An online bin packing instance is a list of item sizes each of which is a rational number in (0,1], the goal is to partition the items into bins where the total size of items in a bin is at most 1, and the total number of bins is minimized. Here we analyze the ratio of two online algorithms for sequences where the total size of items is large and prove an old open conjecture of Johnson.

Identify Patterns in Online Bin Packing Problem: An Adaptive Pattern-Based Algorithm

Bin packing is a typical optimization problem with many real-world application scenarios. In the online bin packing problem, a sequence of items is revealed one at a time, and each item must be packed into a bin immediately after its arrival. Inspired by duality in optimization, we proposed pattern-based adaptive heuristics for the online bin packing problem. The idea is to predict the distribution of items based on packed items, and to apply this information in packing future arrival items in order to handle uncertainty in online bin packing. A pattern in bin packing is a combination of items that can be packed into a single bin. Patterns selected according to past items are adopted and periodically updated in scheduling future items in the algorithm. Symmetry in patterns and the stability of patterns in the online bin packing problem are discussed. We have implemented the algorithm and compared it with the Best-Fit in a series of experiments with various distribution of items to show its effectiveness.

On bin packing with clustering and bin packing with delays

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 cannotexceed 1.In bin packing with clustering, items also have colors associated with them. A globally optimal solution can combine items of different colors into bins, while a clustered solution can only pack monochromatic bins. The goal is to compare a globally optimal solution to an optimal clustered solution, under certain constraints on the coloring provided with the input. We show close bounds on the worst-case ratio between these two costs, called the price of clustering, improving and simplifying previous results. Specifically, we show that the price of clustering does not exceed 1.93667, improving over the previous upper bound of 1.951, and that it is at least 1.93558, improving over the previous lower bound of 1.93344.In online bin packing with delays, items are presented over time. Items may wait to be packed, and an algorithm can create a new bin at any time, packing a subset of already existing unpacked items into it, under the condition that the bin is valid. A created bin cannot be used again in the future, and all items have to be packed into bins eventually. The objective is to minimize the number of used bins plus the sum of waiting costs of all items, called delays. We build on previous work and modify a simple phase-based algorithm. We combine the modification with a careful analysis to improve the previously known competitive ratio from 3.951 to below 3.1551.

Virtual network function placement with bounded migrations

With the penetration of Network Function Virtualization (NFV), network functions, traditionally deployed as proprietary physical equipment like firewalls, Network Address Translations (NATs), are gradually being implemented as software and deployed on standardized hardware. One of the crucial challenges in this paradigm is how to place the software implemented network functions to minimize the number of used physical servers. In this paper, we study the problem of how to optimally place Virtual Network Functions (VNFs) in networks where it is allowed to migrate already placed VNFs to decrease used servers. We first formulate the offline problem as an Integer Linear Programming (ILP) problem, and then propose a semi-online algorithm to solve the online variant of the problem. We name the proposed algorithm Semi-onlIne Vnf plAcement (SIVA). In particular, SIVA is based on a bin packing algorithm that solves online bin packing problem while taking care of migrations. According to our theoretical analysis, SIVA migrates at most $$\lambda$$ VNFs each step, and it has Asymptotic Competitive Ratio (ACR) of 3/2 if $$k \rightarrow \infty$$ , where $$\lambda = k \cdot |N|$$ , k is a tunable parameter, and $$|N|$$ is the the number of supported VNF types. We conduct extensive numerical simulations to evaluate the performances of SIVA. The experiment results validate the theoretical analysis and show that SIVA outperforms the state-of-the-art algorithms by achieving near-optimal performance with minor VNF migrations.

A New Lower Bound for Classic Online Bin Packing

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.

Analysis of Lower Bounds for Online Bin Packing with Two Item Sizes

Analysis of Lower Bounds for Online Bin Packing with Two Item Sizes

Asymptotic Approximation Ratios for Certain Classes of Online Bin Packing Algorithms

Asymptotic Approximation Ratios for Certain Classes of Online Bin Packing Algorithms

Bin-Packing-Based Online Dynamic Bandwidth and Wavelength Allocation Algorithm in Super-PON

Super passive optical network (Super-PON) is a next-generation Ethernet PON (NG-EPON) candidate that is envisaged to provide high data rate and low latency. For NG-EPON, there are two types of algorithms that manages bandwidth and wavelength scheduling, namely offline and online. The latter method is more scalable and efficient than the former method. Several online algorithms exist that propose schemes to manage wavelength and bandwidth scheduling. However, these algorithms lack in efficient wavelength utilization and switching. In this work, we propose a novel online bin-packing based dynamic bandwidth and wavelength allocation (DBWA) algorithm for Super-PON, namely updated best fit bin-packing (UBF-BP). This algorithm limits the wavelength switching per cycle, and uses the modified version of best fit bin-packing (BF-BP) technique for optimal wavelength allocation. Simulation results show that the proposed DBWA algorithm has a low complexity and overcomes the inefficiencies of wavelength utilization and switching. This results in lower network delay and higher channel utilization than the state-of-the-art DBWA. Furthermore, to verify the correctness of the proposed algorithm, we propose an analytical model and validate these simulation results with the analytical results.

Exact distributional analysis of online algorithms with lookahead

In online optimization, input data is revealed sequentially. Optimization problems in practice often exhibit this type of information disclosure as opposed to standard offline optimization where all information is known in advance. We analyze the performance of algorithms for online optimization with lookahead using a holistic distributional approach. To this end, we first introduce the performance measurement method of counting distribution functions. Then, we derive analytical expressions for the counting distribution functions of the objective value and the performance ratio in elementary cases of the online bin packing and the online traveling salesman problem. For bin packing, we also establish a relation between algorithm processing and the Catalan numbers. The paper shows that an exact analysis is strongly interconnected to the combinatorial structure of the problem and algorithm under consideration. Results further indicate that the value of lookahead heavily relies on the problem itself. The analysis also shows that exact distributional analysis could be used in order to discover key effects and identify related root causes in relatively simple problem settings. These insights can then be transferred to the analysis of more complex settings where the introduced performance measurement approach has to be used on an approximative basis (e.g., in a simulation-based optimization).

Tight Bounds for Clairvoyant Dynamic Bin Packing

In this article, we focus on the Clairvoyant Dynamic Bin Packing (DBP) problem, which extends the Classical Online Bin Packing problem in that items arrive and depart over time and the departure time of an item is known upon its arrival. The problem naturally arises when handling cloud-based networks. We focus specifically on the MinUsageTime objective function, which aims to minimize the overall usage time of all bins that are opened during the packing process. Earlier work has shown a O (log &mu / log log μ) upper bound on the algorithm’s competitiveness, where μ is defined as the ratio between the maximal and minimal durations of all items. We improve the upper bound by giving a O (√ log μ)-competitive algorithm. We then provide a matching lower bound of Ω (√ log μ) on the competitive ratio of any online algorithm, thus closing the gap with regard to this problem. We then focus on what we call the class of aligned inputs and give a O (log log μ)-competitive algorithm for this case, beating the lower bound of the general case by an exponential factor. Surprisingly enough, the analysis of our algorithm that we present is closely related to various properties of binary strings.

A lower bound for online rectangle packing

We slightly improve the known lower bound on the asymptotic competitive ratio for online bin packing of rectangles. We present a complete proof for the new lower bound, whose value is above 1.91.

The optimal absolute ratio for online bin packing

We present an online bin packing algorithm with absolute competitive ratio 5/3, which is optimal.

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

In this paper we develop new techniques for covering linear programs and covering integer linear programs to find an approximate solution with improved objective value close to an existing solution. The task of improving an approximate solution is closely related to a classical theorem of Cook et al. [Math. Programming, 34 (1986), pp. 251--264] in the sensitivity analysis for linear programs and integer linear programs. This result is often applied in the design of robust algorithms for online problems. We apply our new techniques to the online bin packing problem, where it is allowed to reassign already packed items. The migration factor measures the amount of repacked items. It is defined by the total size of reassigned items divided by the size of the arriving item. We obtain a robust asymptotic fully polynomial time approximation scheme (AFPTAS) for the online bin packing problem with migration factor bounded by a polynomial in $\frac{1}{\epsilon}$. To the best of our knowledge, this is the first (asymptotic) PTAS with polynomial migration for an NP-hard problem. As a byproduct we prove an approximate variant of the sensitivity theorem by Cook et al. [Math. Programming, 34 (1986), pp. 251--264] for linear programs.

Bin packing problem with scenarios

Scheduling over scenarios is one of the latest approaches in modelling scheduling problems including uncertainty. However, to the best of our knowledge, scenarios have never been applied to the bin packing problem, so here we introduce the bin packing problem with scenarios. In this model, we have a list of items with sizes between 0 and 1, and each item is assigned to one or more scenarios. In reality, the items of only one scenario will occur, but this chosen scenario is unknown at the time of packing, so the algorithms have to examine all scenarios. This means that the items have to be packed into bins such that for any scenario, the total size of the items in this scenario is at most 1 in each bin. The objective of the standard bin packing problem is to minimize the number of bins. Here, we introduce some extensions of the objective function to the scenario based model, and we present our competitive analysis of some online bin packing algorithms adapted to scenarios.

Online Bin Packing with Advice of Small Size

In this paper, we study the advice complexity of the online bin packing problem. In this well-studied setting, the online algorithm is supplemented with some additional information concerning the input. We improve upon both known upper and lower bounds of online algorithms for this problem. On the positive side, we first provide a relatively simple algorithm that achieves a competitive ratio arbitrarily close to 1.5, using constant-size advice. Our result implies that 16 bits of advice suffice to obtain a competitive ratio better than any online algorithm without advice, thus improving the previously known bound of O(log(n)) bits required to attain this performance. In addition, we introduce a more complex algorithm that still requires only constant-size advice, and has a competitive ratio arbitrarily close to 1.47012. This is the currently best performance of any online bin packing algorithm with sublinear advice. On the negative side, we extend a construction due to Boyar et al. (Algorithmica 74(1), 507–527 2016) so as to show that no online algorithm with sub-linear advice can be 7/6-competitive, improving on the lower bound of 9/8 from Boyar et al.