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In 2005, David Pisinger asked the question "where are the hard knapsack problems?"
Noting that the classical benchmark test instances were limited in difficulty due to their selected structure, he proposed a set of new test instances for the 0-1 knapsack problem with characteristics that made them more challenging for dynamic programming and branch-and-bound algorithms. This important work highlighted the influence of diversity in test instances to draw reliable conclusions about algorithm performance. In this paper, we revisit the question in light of recent methodological advances-in the form of Instance Space Analysis-enabling the strengths and weaknesses of algorithms to be visualised and assessed across the broadest possible space of test instances. We show where the hard instances lie, and objectively assess algorithm performance across the instance space to articulate the strengths and weaknesses of algorithms. Furthermore, we propose a method to fill the instance space with diverse and challenging new test instances with controllable properties to support greater insights into algorithm selection, and drive future algorithmic innovations.

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This paper demonstrates that the performance of various outlier detection methods is sensitive to both the characteristics of the dataset, and the data normalization scheme employed. To understand these dependencies, we formally prove that normalization affects the nearest neighbor structure, and density of the dataset; hence, affecting which observations could be considered outliers. Then, we perform an instance space analysis of combinations of normalization and detection methods. Such analysis enables the visualization of the strengths and weaknesses of these combinations. Moreover, we gain insights into which method combination might obtain the best performance for a given dataset.

Full-text available

Article

- Sep 2020

This paper presents a method to generate diverse and challenging new test instances for continuous black-box optimization. Each instance is represented as a feature vector of exploratory landscape analysis measures. By projecting the features into a two-dimensional instance space, the location of existing test instances can be visualized, and their similarities and differences revealed. New instances are generated through genetic programming which evolves functions with controllable characteristics. Convergence to selected target points in the instance space is used to drive the evolutionary process, such that the new instances span the entire space more comprehensively. We demonstrate the method by generating two-dimensional functions to visualize its success, and ten-dimensional functions to test its scalability. We show that the method can recreate existing test functions when target points are co-located with existing functions, and can generate new functions with entirely different characteristics when target points are located in empty regions of the instance space. Moreover, we test the effectiveness of three state-of-the-art algorithms on the new set of instances. The results demonstrate that the new set is not only more diverse than a well-known benchmark set, but also more challenging for the tested algorithms. Hence, the method opens up a new avenue for developing test instances with controllable characteristics, necessary to expose the strengths and weaknesses of algorithms, and drive algorithm development.

This paper tackles the issue of objective performance evaluation of machine learning classifiers, and the impact of the choice of test instances. Given that statistical properties or features of a dataset affect the difficulty of an instance for particular classification algorithms, we examine the diversity and quality of the UCI repository of test instances used by most machine learning researchers. We show how an instance space can be visualized, with each classification dataset represented as a point in the space. The instance space is constructed to reveal pockets of hard and easy instances, and enables the strengths and weaknesses of individual classifiers to be identified. Finally, we propose a methodology to generate new test instances with the aim of enriching the diversity of the instance space, enabling potentially greater insights than can be afforded by the current UCI repository.

It is common practice to evaluate the strength of forecasting methods using collections of well-studied time series datasets, such as the M3 data. The question is, though, how diverse and challenging are these time series, and do they enable us to study the unique strengths and weaknesses of different forecasting methods? This paper proposes a visualisation method for collections of time series that enables a time series to be represented as a point in a two-dimensional instance space. The effectiveness of different forecasting methods across this space is easy to visualise, and the diversity of the time series in an existing collection can be assessed. Noting that the diversity of the M3 dataset has been questioned, this paper also proposes a method for generating new time series with controllable characteristics in order to fill in and spread out the instance space, making our generalisations of forecasting method performances as robust as possible.

This paper presents a method for the objective assessment of an algorithm's strengths and weaknesses. Instead of examining only the performance of one or more algorithms on a benchmark set, or generating custom problems that maximize the performance difference between two algorithms, our method quantifies both the nature of the test instances and the algorithm performance. Our aim is to gather information about possible phase transitions in performance, i.e., the points in which a small change in problem structure produces algorithm failure. The method is based on the accurate estimation and characterization of the algorithm footprints, i.e., the regions of instance space in which good or exceptional performance is expected from an algorithm. A footprint can be estimated for each algorithm and for the overall portfolio. Therefore, we select a set of features to generate a common instance space, which we validate by constructing a sufficiently accurate prediction model. We characterize the footprints by their area and density. Our method identifies complementary performance between algorithms, quantifies the common features of hard problems, and locates regions where a phase transition may lie.

Full-text available

Article

- Sep 1995

The competitive nature of most algorithmic experimentation is a source of problems that are all too familiar to the research
community. It is hard to make fair comparisons between algorithms and to assemble realistic test problems. Competitive testing
tells us which algorithm is faster but not why. Because it requires polished code, it consumes time and energy that could
be better spent doing more experiments. This article argues that a more scientific approach of controlled experimentation,
similar to that used in other empirical sciences, avoids or alleviates these problems. We have confused research and development;
competitive testing is suited only for the latter.

Deductive algorithmic science has reached a high level of sophistication, but its worst-case and average-case results seldom tell us how well an algorithm is actually going to work in practice. I argue that an empirical science of algorithms is a viable alternative. I respond to misgivings about an empirical approach, including the prevalent notion that only a deductive treatment can be ''theoretical'' or sophisticated. NP-completeness theory, for instance, is interesting partly because it has significant, if unacknowledged, empirical content. An empirical approach requires not only rigorous experimental design and analysis, but also the invention of empirically-based explanatory theories. I give some examples of recent work that partially achieves this aim.

A framework is developed to explore the connection between
effective optimization algorithms and the problems they are solving. A
number of “no free lunch” (NFL) theorems are presented which
establish that for any algorithm, any elevated performance over one
class of problems is offset by performance over another class. These
theorems result in a geometric interpretation of what it means for an
algorithm to be well suited to an optimization problem. Applications of
the NFL theorems to information-theoretic aspects of optimization and
benchmark measures of performance are also presented. Other issues
addressed include time-varying optimization problems and a priori
“head-to-head” minimax distinctions between optimization
algorithms, distinctions that result despite the NFL theorems' enforcing
of a type of uniformity over all algorithms

Article

- Nov 2016
- EUR J OPER RES

Post-optimal analysis is the task of understanding the behavior of the solution of a problem due to changes in the data. Frequently, post-optimal analysis is as important as obtaining the optimal solution itself. Post-optimal analysis for linear programming problems is well established and widely used. However, for integer programming problems the task is much more computationally demanding, and various approaches based on branch-and-bound or cutting planes have been presented. In the present paper we study how much coefficients in the original problem can vary without changing the optimal solution vector, the so-called tolerance analysis. We show how to perform exact tolerance analysis for the 0-1 knapsack problem with integer coefficients in amortized time O(clog n) for each item, where n is the number of items, and c is the capacity of the knapsack. Amortized running times report the time used for each item, when calculating tolerance limits of all items. Exact tolerance limits are the widest possible intervals, while approximate tolerance limits may be suboptimal. We show how various upper bounds can be used to determine approximate tolerance limits in time O(log n) or O(1) per item using the Dantzig bound and Dembo-Hammer bound, respectively. The running times and quality of the tolerance limits of all exact and approximate algorithms are experimentally compared, showing that all tolerance limits can be found in less than a second. The approximate bounds are of good quality for large-sized instances, while it is worth using the exact approach for smaller instances.

Article

- May 2015
- COMPUT OPER RES

Our confidence in the future performance of any algorithm, including optimization algorithms, depends on how carefully we select test instances so that the generalization of algorithm performance on future instances can be inferred. In recent work, we have established a methodology to generate a two-dimensional representation of the instance space, comprising a set of known test instances. This instance space shows the similarities and differences between the instances using measurable features or properties, and enables the performance of algorithms to be viewed across the instance space, where generalizations can be inferred. The power of this methodology is the insights that can be generated into algorithm strengths and weaknesses by examining the regions in instance space where strong performance can be expected. The representation of the instance space is dependent on the choice of test instances however. In this paper we present a methodology for generating new test instances with controllable properties, by filling observed gaps in the instance space. This enables the generation of rich new sets of test instances to support better the understanding of algorithm strengths and weaknesses. The methodology is demonstrated on graph coloring as a case study.

Article

- May 2014
- COMPUT OPER RES

This paper tackles the difficult but important task of objective algorithm performance assessment for optimization. Rather than reporting average performance of algorithms across a set of chosen instances, which may bias conclusions, we propose a methodology to enable the strengths and weaknesses of different optimization algorithms to be compared across a broader instance space. The results reported in a recent Computers and Operations Research paper comparing the performance of graph coloring heuristics are revisited with this new methodology to demonstrate (i) how pockets of the instance space can be found where algorithm performance varies significantly from the average performance of an algorithm; (ii) how the properties of the instances can be used to predict algorithm performance on previously unseen instances with high accuracy; and (iii) how the relative strengths and weaknesses of each algorithm can be visualized and measured objectively.

Article

- Oct 2010

This chapter discusses approaches to generating synthetic data for use in scientific experiments. In many diverse scientific fields, the lack of availability, high cost or inconvenience of the collection of real-world data motivates the generation of synthetic data. In many experiments, the method chosen to generate synthetic data can significantly affect the results of an experiment. Unfortunately, the scientific literature does not contain general protocols for how synthetic data should be generated. The purpose of this chapter is to rectify that deficiency. The protocol we propose is based on several generation principles. These principles motivate and organize the data generation process. The principles are operationalized by generation properties. Then, together with information about the features of the application and of the experiment, the properties are used to construct a data generation scheme. Finally, we suggest procedures for validating the synthetic data generated. The usefulness of our protocol is illustrated by a discussion of numerous applications of data generation from the optimization literature. This discussion identifies examples of both good and bad data generation practice as it relates to our protocol.

Article

- Mar 1970

This paper presents a more detailed analysis of a class of minimization algorithms, which includes as a special case the DFP (Davidon-Fletcher-Powell) method, than has previously appeared. Only quadratic functions are considered but particular attention is paid to the magnitude of successive errors and their dependence upon the initial matrix. On the basis of this a possible explanation of some of the observed characteristics of the class is tentatively suggested.

Article

- Oct 1997

It is well-known that many instances of the 0-1 knapsack problem can be effectively solved to optimality also for very large values of n (the number of binary variables), while other instances cannot be solved for n equal to only a few hundreds. We propose upper bounds obtained from the mathematical model of the problem by adding valid inequalities on the cardinality of an optimal solution, and relaxing it in a Lagrangian fashion. We then introduce a specialized iterative technique for determining the optimal Lagrangian multipliers in polynomial time. A branch-and-bound algorithm is finally developed. Computational experiments prove that several classes of hard instances are effectively solved even for large values of n.

Article

- Dec 1980

We consider a class of algorithms which use the combined powers of branch-and-bound, dynamic programming and rudimentary divisibility arguments for solving the zero-one knapsack problem. Our main result identifies a class of instances of the problem which are difficult to solve by such algorithms. More precisely, if reading the data takes t units of time, then the time required to solve the problem grows exponentially with the square root of t.

Article

- Feb 2000

This paper presents the results of an empirical study of the effects of coefficient correlation structure and constraint slackness settings on the performance of solution procedures on synthetic two-dimensional knapsack problems (2KP). The population correlation structure among 2KP coefficients, the level of constraint slackness, and the type of correlation (product moment or rank) are varied in this study. Representative branch-and-bound and heuristic solution procedures are used to investigate the influence of these problem parameters on solution procedure performance. Population correlation structure, and in particular the interconstraint component of the correlation structure, is found to be a significant factor influencing the performance of both the algorithm and the heuristic. In addition, the interaction between constraint slackness and population correlation structure is found to influence solution procedure performance.

Article

- Mar 1999

Two new algorithms recently proved to outperform all previous methods for the exact solution of the 0-1 Knapsack Problem. This paper presents a combination of such approaches, where, in addition, valid inequalities are generated and surrogate relaxed, and a new initial core problem is adopted. The algorithm is able to solve all classical test instances, with up to 10,000 variables, in less than 0.2 seconds on a HP9000-735/99 computer. The C language implementation of the algorithm is available on the internet.

Article

- May 1988

We present a new algorithm for the optimal solution of the 0-1 Knapsack problem, which is particularly effective for large-size problems. The algorithm is based on determination of an appropriate small subset of items and the solution of the corresponding "core problem": from this we derive a heuristic solution for the original problem which, with high probability, can be proved to be optimal. The algorithm incorporates a new method of computation of upper bounds and efficient implementations of reduction procedures. The corresponding Fortran code is available. We report computational experiments on small-size and large-size random problems, comparing the proposed code with all those available in the literature.

Article

- Sep 2005
- COMPUT OPER RES

The knapsack problem is believed to be one of the “easier” -hard problems. Not only can it be solved in pseudo-polynomial time, but also decades of algorithmic improvements have made it possible to solve nearly all standard instances from the literature. The purpose of this paper is to give an overview of all recent exact solution approaches, and to show that the knapsack problem still is hard to solve for these algorithms for a variety of new test problems. These problems are constructed either by using standard benchmark instances with larger coefficients, or by introducing new classes of instances for which most upper bounds perform badly. The first group of problems challenge the dynamic programming algorithms while the other group of problems are focused towards branch-and-bound algorithms. Numerous computational experiments with all recent state-of-the-art codes are used to show that (KP) is still difficult to solve for a wide number of problems. One could say that the previous benchmark tests were limited to a few highly structured instances, which do not show the full characteristics of knapsack problems.

Article

- Dec 2002
- EUR J OPER RES

A new branch-and-bound algorithm for the exact solution of the 0–1 Knapsack Problem is presented. The algorithm is based on solving an ‘expanding core’, which intially only contains the break item, but which is expanded each time the branch-and-bound algorithm reaches the border of the core. Computational experiments show that most data instances are optimally solved without sorting or preprocessing a great majority of the items. Detailed program sketches are provided, and computational experiments are reported, indicating that the algorithm presented not only is shorter, but also faster and more stable than any other algorithm hitherto proposed.

Article

- Dec 1976
- ADV COMPUT

Publisher Summary The problem of selecting an effective algorithm arises in a wide variety of situations. This chapter starts with a discussion on abstract models: the basic model and associated problems, the model with selection based on features, and the model with variable performance criteria. One objective of this chapter is to explore the applicability of the approximation theory to the algorithm selection problem. There is an intimate relationship here and that the approximation theory forms an appropriate base upon which to develop a theory of algorithm selection methods. The approximation theory currently lacks much of the necessary machinery for the algorithm selection problem. There is a need to develop new results and apply known techniques to these new circumstances. The final pages of this chapter form a sort of appendix, which lists 15 specific open problems and questions in this area. There is a close relationship between the algorithm selection problem and the general optimization theory. This is not surprising since the approximation problem is a special form of the optimization problem. Most realistic algorithm selection problems are of moderate to high dimensionality and thus one should expect them to be quite complex. One consequence of this is that most straightforward approaches (even well-conceived ones) are likely to lead to enormous computations for the best selection. The single most important part of the solution of a selection problem is the appropriate choice of the form for selection mapping. It is here that theories give the least guidance and that the art of problem solving is most crucial.

Article

- Aug 1997
- DISCRETE APPL MATH

The collapsing knapsack problem is a generalization of the ordinary knapsack problem, where the knapsack capacity is a non-increasing function of the number of items included. Whereas previous papers on the topic have applied quite involved techniques, the current paper presents and analyzes two rather simple approaches: One approach that is based on the reduction to a standard knapsack problem, and another approach that is based on a simple dynamic programming recursion. Both algorithms have pseudo-polynomial solution times, guaranteeing reasonable solution times for moderate coefficient sizes. Computational experiments are provided to expose the efficiency of the two approaches compared to previous algorithms.

Article

- May 2012
- COMPUT OPER RES

Discovering the conditions under which an optimization algorithm or search heuristic will succeed or fail is critical for understanding the strengths and weaknesses of different algorithms, and for automated algorithm selection. Large scale experimental studies – studying the performance of a variety of optimization algorithms across a large collection of diverse problem instances – provide the resources to derive these conditions. Data mining techniques can be used to learn the relationships between the critical features of the instances and the performance of algorithms. This paper discusses how we can adequately characterize the features of a problem instance that have impact on difficulty in terms of algorithmic performance, and how such features can be defined and measured for various optimization problems. We provide a comprehensive survey of the research field with a focus on six combinatorial optimization problems: assignment, traveling salesman, and knapsack problems, bin-packing, graph coloring, and timetabling. For these problems – which are important abstractions of many real-world problems – we review hardness-revealing features as developed over decades of research, and we discuss the suitability of more problem-independent landscape metrics. We discuss how the features developed for one problem may be transferred to study related problems exhibiting similar structures.

Article

- Aug 2007

The operations research literature contains numerous studies on the design and application of optimization and heuristic solution procedures. These studies identify a particular optimization problem, suggest a general solution procedure, and then customize that procedure to improve its efficiency and/or accuracy. In contrast, this paper shows how to use existing solution procedures more effectively. We develop a methodology for predicting the relative performance of alternative procedures, using easily computed problem characteristics. This methodology enables us, for any given data set, to preselect a solution procedure. We apply this preselection methodology to the 0-1 knapsack problem for which two successful optimization procedures, dynamic programming and branch-and-search, are available. Extensive computational testing indicates that substantial savings in average computation time are achieved. The benefits of our work include faster and cheaper identification of effective solution procedures, as well as an improved understanding of the relationship between problem characteristics and the performance of various procedures. Our methodology can be applied to many optimization problems to develop easily implemented guidelines for selecting appropriate solution procedures.

Article

- Dec 2002

Several types of large-sized 0-1 Knapsack Problems (KP) may be easily solved, but in such cases most of the computational effort is used for sorting and reduction. In order to avoid this problem it has been proposed to solve the so-called core of the problem: a Knapsack Problem defined on a small subset of the variables. The exact core cannot, however, be identified before KP is solved to optimality, thus, previous algorithms had to rely on approximate core sizes.
In this paper we present an algorithm for KP where the enumerated core size is minimal, and the computational effort for sorting and reduction also is limited according to a hierarchy. The algorithm is based on a dynamic programming approach, where the core size is extended by need, and the sorting and reduction is performed in a similar “lazy” way.
Computational experiments are presented for several commonly occurring types of data instances. Experience from these tests indicate that the presented approach outperforms any known algorithm for KP, having very stable solution times.

Instance space analysis: A toolkit for the 830 assessment of algorithmic power

- Jan 2020

- M Muñoz
- K Smith-Miles

M. Muñoz, K. Smith-Miles, Instance space analysis: A toolkit for the
830
assessment of algorithmic power, Source code is available at https://
github.com/andremun/InstanceSpace (2020).

MATILDA: Melbourne algorithm test instance library with data analytics

- Jan 2019

- K Smith-Miles
- M Muñoz

K. Smith-Miles, M. Muñoz, Neelofar, MATILDA: Melbourne algorithm
test instance library with data analytics, Available at https://matilda.
unimelb.edu.au (2019).

- Jan 2017

- B Meade
- L Lafayette
- G Sauter
- D Tosello

B. Meade, L. Lafayette, G. Sauter, D. Tosello, Spartan HPC-Cloud Hy-880
brid: Delivering Performance and Flexibility (4 2017). doi:10.4225/49/
58ead90dceaaa.