Measure Of Problem Difficulty Research Articles

A novel feature-based approach to characterize algorithm performance for the traveling salesperson problem

Meta-heuristics are frequently used to tackle NP-hard combinatorial optimization problems. With this paper we contribute to the understanding of the success of 2-opt based local search algorithms for solving the traveling salesperson problem (TSP). Although 2-opt is widely used in practice, it is hard to understand its success from a theoretical perspective. We take a statistical approach and examine the features of TSP instances that make the problem either hard or easy to solve. As a measure of problem difficulty for 2-opt we use the approximation ratio that it achieves on a given instance. Our investigations point out important features that make TSP instances hard or easy to be approximated by 2-opt.

An fMRI study of the Tower of London: A look at problem structure differences

The aim of the current study was to explore the effects of problem structure, namely goal hierarchy and number of optimal solution paths, on the neural architecture that supports problem-solving and planning. Here, six-move problems with both an unambiguous and ambiguous goal hierarchy and single and multiple optimal solution paths were examined. In the task used, participants were encouraged to generate a solution plan before execution. The behavioral results revealed that problem-solving time and accuracy were both affected by both problem parameters. The fMRI activation results revealed three major findings. First, the right prefrontal cortex revealed a significantly different activation pattern than the other regions examined. This was the only region that revealed a larger response during the execution phase than the planning phase. Second, the effect of goal hierarchy was strongest during the execution phase. Finally, while there was no main effect of number of optimal solution paths, this parameter interacted with goal hierarchy in a number of regions across the brain. The present study also suggests that the minimum number of moves may not be the best measure of problem difficulty and that greater care be taken in the selection of TOL problems for both experimental studies as well as clinical assessment.

A Note on Problem Difficulty Measures in Black-Box Optimization: Classification, Realizations and Predictability

Various methods have been defined to measure the hardness of a fitness function for evolutionary algorithms and other black-box heuristics. Examples include fitness landscape analysis, epistasis, fitness-distance correlations etc., all of which are relatively easy to describe. However, they do not always correctly specify the hardness of the function. Some measures are easy to implement, others are more intuitive and hard to formalize. This paper rigorously defines difficulty measures in black-box optimization and proposes a classification. Different types of realizations of such measures are studied, namely exact and approximate ones. For both types of realizations, it is proven that predictive versions that run in polynomial time in general do not exist unless certain complexity-theoretical assumptions are wrong.

An analysis of how training data complexity affects the nearest neighbor classifiers

The k-nearest neighbors (k-NN) classifier is one of the most popular supervised classification methods. It is very simple, intuitive and accurate in a great variety of real-world domains. Nonetheless, despite its simplicity and effectiveness, practical use of this rule has been historically limited due to its high storage requirements and the computational costs involved. On the other hand, the performance of this classifier appears strongly sensitive to training data complexity. In this context, by means of several problem difficulty measures, we try to characterize the behavior of the k-NN rule when working under certain situations. More specifically, the present analysis focuses on the use of some data complexity measures to describe class overlapping, feature space dimensionality and class density, and discover their relation with the practical accuracy of this classifier.

A comparison of predictive measures of problem difficulty in evolutionary algorithms

This paper studies a number of predictive measures of problem difficulty, among which epistasis variance and fitness distance correlation are the most widely known. Our approach is based on comparing the reference class of a measure to a number of known easy function classes. First, we generalize the reference classes of fitness distance correlation and epistasis variance, and construct a new predictive measure that is insensitive to nonlinear fitness scaling. We then investigate the relations between the reference classes of the measures and a number of intuitively easy classes. We also point out the need to further identify which functions are easy for a given class of evolutionary algorithms in order to design more efficient hardness indicators for them. We finally restrict attention to the genetic algorithm (GA), and consider both GA-easy and GA-hard fitness functions, and give experimental evidence that the values of the measures, based on random samples, can be completely unreliable and entirely uncorrelated to the convergence quality and convergence speed of GA instances using either proportional or ranking selection.

The development of children's mental multiplication skills

Simple multiplication skills of subjects from third grade through college were studied. Data from Experiment 1 indicated that to solve simple multiplication problems, even third graders rely heavily on memory retrieval rather than on reconstructive procedures such as counting. Reaction time was predicted better by normative measures of problem difficulty or strength at all ages than by structural variables associated with the problems (e.g., size of multipliers). Furthermore, confusion problems, those with an incorrect answer that is multiplicatively related to the problem (e.g., 4 × 8 = 24), affected error rates beginning with fifth grade and reaction times beginning with seventh grade; by late elementary school, then, retrieval involves automatic activation of related information. In Experiment 2, subjects in all grades demonstrated automaticity of fact retrieval on simple problems, although children under seventh grade did not demonstrate automatic facilitation on more difficult problems. The data are consistent with the position that arithmetic processing in simple multiplication is at least partially automatic, even by the third grade, and with current models based on network or associative knowledge structures in long-term memory.

A Basic Experimental Study on Characteristics of On-line Human Information Processing Associated with Man-Machine Interface

Regarding human factors research on man-machine interface, a basic psychological experiment was conducted to observe psycho-physiological characteristics of on-line human cognitive behavior when cognitive tasks on learning and pattern classification were given to subjects by personal computer using a simple state transition model. During the experiment, three different types of subjects' data were recorded : (i) eye movement data by eye mark recorder, (ii) physio-electric signals by polygraph and (iii) verbal reports. Those subjects' data were analyzed with respect to : (i) the related human cognitive characteristics concerning problem solving strategy, measures of problem difficulty and mental image effect, (ii) observed eye movement characteristics such as saccade, attention, pupil reaction and blinking, etc., and (iii) obtained characteristics of skin potential response and heart rate. It was found that the application of psycho-physiological measurement would serve to objective and detailed analysis of on-line cognitive process.

Planning as search: A quantitative approach

We present the thesis that planning can be viewed as problem-solving search using subgoals, macro-operators, and abstraction as knowledge sources. Our goal is to quantify problem-solving performance using these sources of knowledge. New results include the identification of subgoal distance as a fundamental measure of problem difficulty, a multiplicative time-space tradeoff for macro-operators, and an analysis of abstraction which concludes that abstraction hierarchies can reduce exponential problems to linear complexity.

Structural Models for Predicting the Difficulty of Multiplication Problems

The purpose of this investigation was to explore the relationship between problem characteristics and problem difficulty in simple, 2-factor multiplication problems and to determine whether an objective, parametric measure of problem difficulty could be found.The difficulty levels of 168 randomly generated multiplication problems were obtained by testing 238 fifth-grade children. Variables concerning the number of operations, digits carried, and magnitude of the digits were defined, and a least-squares procedure was used to construct several structural models. The predictive power was quite good, with these models accounting for 60-78% of the observed variance of problem difficulty.Possibilities for extension of this technique to other types of problems were also discussed.