Relevancy in Problem Solving: A Computational Framework

When computer scientists discuss the computational complexity of, for example, finding the shortest path from building A to building B in some town or city, their starting point typically is a formal description of the problem at hand, e.g., a graph with weights on every edge where buildings correspond to vertices, routes between buildings to edges, and route-distances to edge-weights. Given such a formal description, either tractability or intractability of the problem is established, by proving that the problem either enjoys a polynomial time algorithm or is NP-hard. However, this problem description is in fact an abstraction of the actual problem of being in A and desiring to go to B: it focuses on the relevant aspects of the problem (e.g., distances between landmarks and crossings) and leaves out a lot of irrelevant details. This abstraction step is often overlooked, but may well contribute to the overall complexity of solving the problem at hand. For example, it appears that “going from A to B” is rather easy to abstract: it is fairly clear that the distance between A and the next crossing is relevant, and that the color of the roof of B is typically not. However, when the problem to be solved is “make X love me”, where the current state is (assumed to be) “X doesn’t love me”, it is hard to agree on all the relevant aspects of this problem. In this paper a computational framework is presented in order to formally investigate the notion of relevance in finding a suitable problem representation. It is shown that it is in itself intractable in general to find a minimal relevant subset of all problem dimensions that might or might not be relevant to the problem. Starting from a computational complexity stance, this paper aims to contribute a computational framework of ‘relevancy’ in problem solving, in order to be able to separate ‘easy to abstract’ from ‘hard to abstract’ problems. This framework is then used to discuss results in the literature on representation, (insight) problem solving and individual differences in the abstraction task, e.g., when experts in a particular domain are compared with novice problem solvers

Complexity of Discrete Energy Minimization Problems

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.Comment: ECCV'16 accepte

Intentional Communication: Computationally Easy or Difficult?

Human intentional communication is marked by its flexibility and context sensitivity. Hypothesized brain mechanisms can provide convincing and complete explanations of the human capacity for intentional communication only insofar as they can match the computational power required for displaying that capacity. It is thus of importance for cognitive neuroscience to know how computationally complex intentional communication actually is. Though the subject of considerable debate, the computational complexity of communication remains so far unknown. In this paper we defend the position that the computational complexity of communication is not a constant, as some views of communication seem to hold, but rather a function of situational factors. We present a methodology for studying and characterizing the computational complexity of communication under different situational constraints. We illustrate our methodology for a model of the problems solved by receivers and senders during a communicative exchange. This approach opens the way to a principled identification of putative model parameters that control cognitive processes supporting intentional communication

Ignorance is Bliss: A Complexity Perspective on Adapting Reactive Architectures

Abstract-We study the computational complexity of adapting a reactive architecture to meet task constraints. This computational problem has application in a wide variety of fields, including cognitive and evolutionary robotics and cognitive neuroscience. We show that-even for a rather simple world and a simple task-adapting a reactive architecture to perform a given task in the given world is N P -hard. This result implies that adapting reactive architectures is computationally intractable regardless the nature of the adaptation process (e.g., engineering, development, evolution, learning, etc.) unless very special conditions apply. In order to find such special conditions for tractability, we have performed parameterized complexity analyses. One of our main findings is that architectures with limited sensory and perceptual abilities are efficiently adaptable