The Role of the Goal in Solving Hard Computational Problems: Do People Really Optimize?

Abstract

The role that the mental, or internal, representation plays when people are solving hard computational problems has largely been overlooked to date, despite the reality that this internal representation drives problem solving. In this work we investigate how performance on versions of two hard computational problems differs based on what internal representations can be generated. Our findings suggest that problem solving performance depends not only on the objective difficulty of the problem, and of course the particular problem instance at hand, but also on how feasible it is to encode the goal of the given problem. A further implication of these findings is that previous human performance studies using NP-hard problems may have, surprisingly, underestimated human performance on instances of problems of this class. We suggest ways to meaningfully frame human performance results on instances of computationally hard problems in terms of these problems' computational complexity, and present a novel framework for interpreting results on problems of this type. The framework takes into account the limitations of the human cognitive system, in particular as it applies to the generation of internal representations of problems of this class.

Heuristics for Comparing the Lengths of Completed E-TSP Tours: Crossings and Areas

“…It also established that participants were able to reliably detect diff erences in areas of these magnitudes, thereby confi rming that area diff erences could provide a feasible heuristic for making length comparisons. Carruthers (2015) proposed that when problem solvers are faced with a problem where it is infeasible to identify whether or not the goal has been reached, they may modify the goal into one that is feasible to identify. In the case of the E-TSP, an example of a possible modifi ed goal is "a valid tour" (p. 49).…”

“…In the psychology of problem solving and decision making, a longstanding answer to similar questions has been that problem solvers use heuristic procedures (Gigerenzer & Goldstein, 1996;Kahneman & Tversky, 1972;Simon, 1983;Tversky & Kahneman, 1973). However, before assuming that heuristics explain performance, Carruthers (2015) proposes that identifying the problem as encoded is important, "because the encoded problem drives performance" (p. 47). The present article explores this proposal by designing a task that meets the definition of having an indeterminable goal and showing that people, first, act as if they believe they can achieve the goal and, second, appear to use heuristics in trying to do so.…”

“…The letter of invitation to this special issue indicated that it was partly motivated by the PhD dissertation of Sarah Carruthers (2015) which, among other things, proposed that when people are tasked with finding optimal solutions to instances of hard computational problems, the goal of the task may not be welldefined and therefore not encodable. For a goal to be well-defined, it must be possible to determine when it has been achieved (p. 41), which may not be the case in many hard optimization problems.…”

“…The results confirmed that they can, thereby supporting the feasibility of using differences in area as a heuristic to judge relative lengths. The results were discussed in terms of Carruthers's (2015) proposal of goal modification and the suggestion is made that applying heuristics of the type identified may represent a specific form of goal modification. …”

The Role of Problem Representation in Producing Near-Optimal TSP Tours

“…By easier we mean faster, smaller error measure, or more effective with respect to cognitive/computational resources such as memory. Carruthers (2015) recently showed, using two examples, that humans can reformulate problems and work on the reformulated problem when the original problem exceeds their cognitive resources. Similarly, one of us has shown that when subjects are presented with a 15-tile puzzle, they move around one or at most two tiles at a time, putting them in the correct location and treating the remaining tiles that are still out of place as equivalent (Pizlo & Li, 2005).…”