Structural properties of individual instances predict human effort and performance on an NP-Hard problem

Abstract

Life presents us with decisions of varying degrees of difficulty. Many of them are NP-hard, that is, they are computationally intractable. Two important questions arise: which properties of decisions drive extreme computational hardness and what are the effects of these properties on human-decision making? Here, we postulate that we can study the effects of computational complexity on human decision-making by studying the mathematical properties of individual instances of NP-hard problems. We draw on prior work in computational complexity theory, which suggests that computational difficulty can be characterized based on the features of instances of a problem. This study is the first to apply this approach to human decision-making. We measured hardness, first, based on typical-case complexity (TCC), a measure of average complexity of a random ensemble of instances, and, second, based on instance complexity (IC), a measure that captures the hardness of a single instance of a problem, regardless of the ensemble it came from. We tested the relation between these measures and (i) decision quality as well as (ii) time expended in a decision, using two variants of the 0-1 knapsack problem, a canonical and ubiquitous computational problem. We show that participants expended more time on instances with higher complexity but that decision quality was lower in those instances. These results suggest that computational complexity is an inherent property of the instances of a problem, which affect human and other kinds of computers.