A physicist's approach to number partitioning

The statistical physics approach to the number partioning problem, a classical NP-hard problem, is both simple and rewarding. Very basic notions and methods from statistical mechanics are enough to obtain analytical results for the phase boundary that separates the ``easy-to-solve'' from the ``hard-to-solve'' phase of the NPP as well as for the probability distributions of the optimal and sub-optimal solutions. In addition, it can be shown that solving a number partioning problem of size $N$ to some extent corresponds to locating the minimum in an unsorted list of \bigo{2^N} numbers. Considering this correspondence it is not surprising that known heuristics for the partitioning problem are not significantly better than simple random search.Comment: 35 pages, to appear in J. Theor. Comp. Science, typo corrected in eq.1

Bin Packing under Random-Order: Breaking the Barrier of 3/2

Best-Fit is one of the most prominent and practically used algorithms for the bin packing problem, where a set of items with associated sizes needs to be packed in the minimum number of unit-capacity bins. Kenyon [SODA '96] studied online bin packing under random-order arrival, where the adversary chooses the list of items, but the items arrive one by one according to an arrival order drawn uniformly randomly from the set of all permutations of the items. Kenyon's seminal result established an upper bound of $1.5$ and a lower bound of $1.08$ on the random-order ratio of Best-Fit, and it was conjectured that the true ratio is $\approx 1.15$. The conjecture, if true, will also imply that Best-Fit (on randomly permuted input) has the best performance guarantee among all the widely-used simple algorithms for (offline) bin packing. This conjecture has remained one of the major open problems in the area, as highlighted in the recent survey on random-order models by Gupta and Singla [Beyond the Worst-Case Analysis of Algorithms '20]. Recently, Albers et al. [Algorithmica '21] improved the upper bound to $1.25$ for the special case when all the item sizes are greater than $1/3$, and they improve the lower bound to $1.1$. Ayyadevara et al. [ICALP '22] obtained an improved result for the special case when all the item sizes lie in $(1/4, 1/2]$, which corresponds to the $3$-partition problem. The upper bound of $3/2$ for the general case, however, has remained unimproved. In this paper, we make the first progress towards the conjecture, by showing that Best-Fit achieves a random-order ratio of at most $1.5 - \varepsilon$, for a small constant $\varepsilon>0$. Furthermore, we establish an improved lower bound of $1.144$ on the random-order ratio of Best-Fit, nearly reaching the conjectured ratio

Algorithms and data structures for three-dimensional packing

Cutting and packing problems are increasingly prevalent in industry. A well utilised freight vehicle will save a business money when delivering goods, as well as reducing the environmental impact, when compared to sending out two lesser-utilised freight vehicles. A cutting machine that generates less wasted material will have a similar effect. Industry reliance on automating these processes and improving productivity is increasing year-on-year. This thesis presents a number of methods for generating high quality solutions for these cutting and packing challenges. It does so in a number of ways. A fast, efficient framework for heuristically generating solutions to large problems is presented, and a method of incrementally improving these solutions over time is implemented and shown to produce even higher packing utilisations. The results from these findings provide the best known results for 28 out of 35 problems from the literature. This framework is analysed and its effectiveness shown over a number of datasets, along with a discussion of its theoretical suitability for higher-dimensional packing problems. A way of automatically generating new heuristics for this framework that can be problem specific, and therefore highly tuned to a given dataset, is then demonstrated and shown to perform well when compared to the expert-designed packing heuristics. Finally some mathematical models which can guarantee the optimality of packings for small datasets are given, and the (in)effectiveness of these techniques discussed. The models are then strengthened and a novel model presented which can handle much larger problems under certain conditions. The thesis finishes with a discussion about the applicability of the different approaches taken to the real-world problems that motivate them

Alternative Approaches for Analysis of Bin Packing and List Update Problems

In this thesis we introduce and evaluate new algorithms and models for the analysis of online bin packing and list update problems. These are two classic online problems which are extensively studied in the literature and have many applications in the real world. Similar to other online problems, the framework of competitive analysis is often used to study the list update and bin packing algorithms. Under this framework, the behavior of online algorithms is compared to an optimal offline algorithm on the worst possible input. This is aligned with the traditional algorithm theory built around the concept of worst-case analysis. However, the pessimistic nature of the competitive analysis along with unrealistic assumptions behind the proposed models for the problems often result in situations where the existing theory is not quite useful in practice. The main goal of this thesis is to develop new approaches for studying online problems, and in particular bin packing and list update, to guide development of practical algorithms performing quite well on real-world inputs. In doing so, we introduce new algorithms with good performance (not only under the competitive analysis) as well as new models which are more realistic for certain applications of the studied problems. For many online problems, competitive analysis fails to provide a theoretical justification for observations made in practice. This is partially because, as a worst-case analysis method, competitive analysis does not necessarily reflect the typical behavior of algorithms. In the case of bin packing problem, the Best Fit and First Fit algorithms are widely used in practice. There are, however, other algorithms with better competitive ratios which are rarely used in practice since they perform poorly on average. We show that it is possible to optimize for both cases. In doing so, we introduce online bin packing algorithms which outperform Best Fit and First Fit in terms of competitive ratio while maintaining their good average-case performance. An alternative for analysis of online problems is the advice model which has received significant attention in the past few years. Under the advice model, an online algorithm receives a number of bits of advice about the unrevealed parts of the sequence. Generally, there is a trade-off between the size of the advice and the performance of online algorithms. The advice model generalizes the existing frameworks in which an online algorithm has partial knowledge about the input sequence, e.g., the access graph model for the paging problem. We study list update and bin packing problems under the advice model and answer several relevant questions about the advice complexity of these problems. Online problems are usually studied under specific settings which are not necessarily valid for all applications of the problem. As an example, online bin packing algorithms are widely used for server consolidation to minimize the number of active servers in a data center. In some applications, e.g., tenant placement in the Cloud, often a `fault-tolerant' solution for server consolidation is required. In this setting, the problem becomes different and the classic algorithms can no longer be used. We study a fault-tolerant model for the bin packing problem and analyze algorithms which fit this particular application of the problem. Similarly, the list update problem was initially proposed for maintaining self-adjusting linked lists. However, presently, the main application of the problem is in the data compression realm. We show that the standard cost model is not suitable for compression purposes and study a compression cost model for the list update problem. Our analysis justifies the advantage of the compression schemes which are based on Move-To-Front algorithm and might lead to improved compression algorithms

Verbesserte Approximationsalgorithmen für Packungs- und Ablaufplanungsprobleme

This thesis presents approximation algorithms for geometric packing and scheduling problems. First, improved AFPTAS for the Bin Packing Problem (BP) and its generalization, the Variable-sized Bin Packing Problem (VBP), are explained. Our algorithms have to solve the unbounded variant of the Knapsack Problem (KP) and of the Knapsack Problem with Inversely Proportional Profits (KPIP) as subproblems. In the normal 0-1 variant of KP, an item can be chosen only once. In the bounded variant, an individual bounded number of copies can be taken of every item. The unbounded variant (UKP) allows for an infinite number of copies of every item. KPIP is a generalization of KP in which we have not only one, but several knapsack sizes. The profit of an item is inversely proportional to the size of the knapsack into which it has been packed. This makes it non-trivial to choose the knapsack size that maximizes the profit over all knapsack sizes. Similar to KP, there are the 0-1, the bounded, and the unbounded variant of KPIP. We first present FPTAS for every of the three variants of KPIP. They are faster than the natural approach to separately solve for every knapsack size the corresponding Knapsack Problem. Second, we present an FPTAS for UKP that is faster and needs less storage space than previously known algorithms. Finally, we combine the approaches of the KPIP and of the UKP FPTAS to get an FPTAS for the Unbounded KPIP that has again a better time and space complexity. All these results improve the running time for our BP and VBP algorithms. As a corollary, we also improve the running time for a Strip Packing AFPTAS. Finally, we consider Scheduling on Unrelated Machines of which we study the special case with a constant number K of machine types: one job has the same processing time on every machine of the same type. We present a PTAS for this special case. The algorithm has a better running time than the previously known algorithm for general (but constant) K.Diese Dissertation stellt Approximationsalgorithmen für geometrische Packungs- und Ablaufplanungsprobleme (Packing and Scheduling Problems) vor. Zuerst werden verbesserte AFPTAS für das Behälterproblem (Bin Packing, BP) und seine Verallgemeinerung, das Behälterproblem mit verschiedenen Behältergrößen (Variable-sized Bin Packing, VBP), erklärt. Unsere Algorithmen müssen die unbeschränkte (unbounded) Variante des Rucksackproblems (Knapsack Problem, KP) und des Rucksackproblems mit invers proportionalen Profiten (Knapsack Problem with Inversely Proportional Profits, KPIP) als Unterprobleme lösen. Bei KP gibt es die Varianten 0-1, beschränkt und unbeschränkt. KPIP ist eine Verallgemeinerung des Rucksackproblems mit mehreren Rucksackgrößen, die in dieser Dissertation eingeführt wird. Wie bei KP gibt es bei KPIP ebenfalls die Varianten 0-1, beschränkt und unbeschränkt. Wir stellen zuerst FPTAS für alle drei Varianten von KPIP vor. Sie sind schneller als der natürliche Ansatz, für jede Rucksackgröße das entsprechende Rucksackproblem einzeln zu lösen. Danach stellen wir ein FPTAS für UKP vor, das schneller ist und weniger Speicherplatz benötigt als zuvor bekannte Algorithmen. Schließlich kombinieren wir den Ansatz für KPIP und für das unbeschränkte KPIP, um ein FPTAS für die unbeschränkte Variante von KPIP zu erhalten, das wiederum eine kleinere Zeit- und Speicherkomplexität besitzt. All diese Resultate verbessern die Laufzeit unserer BP- und VBP-Algorithmen. Als Korollar verbessern wir außerdem die Laufzeit eines AFPTAS für das geometrische Zuschnittproblem (Strip Packing). Schließlich betrachten wir das Ablaufplanungsproblem auf heterogenen Maschinen (Scheduling on Unrelated Machines), bei dem wir den Spezialfall mit einer konstanten Anzahl K an Maschinentypen untersuchen: Eine Aufgabe (Job) hat auf jeder Maschine desselben Typs die gleiche Ausführungszeit. Wir stellen für diesen Spezialfall ein PTAS vor. Der Algorithmus ist schneller als das zuvor bekannte Verfahren für allgemeines (aber konstantes) K

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum