Human Performance on Hard Non-Euclidean Graph Problems: Vertex Cover

Abstract

Abstract:Recent studies on a computationally hard visual optimization problem, the Traveling Salesperson Problem (TSP), indicate that humans are capable of finding close to optimal solutions in near-linear time. The current study is a preliminary step in investigating human performance on another hard problem, the Minimum Vertex Cover Problem, in which solvers attempt to find a smallest set of vertices that ensures that every edge in an undirected graph is incident with at least one of the selected vertices. We identify appropriate measures of performance, examine features of problem instances that impact performance, and describe strategies typically employed by participants to solve instances of the Vertex Cover problem.

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

“…Studies of human performance on instances of the optimization version of E-TSP have found that humans typically find solutions that are close to optimal (Best & Simon, 2003;Dry, Lee, Vickers, & Hughes, 2006;Dry, Preiss, & Wagemans, 2012;Graham, Joshi, & Pizlo, 2000;Kong & Schunn, 2007;Ormerod & Chronicle, 1999;Saalweachter & Pizlo, 2008), taking close to linear time (in terms of the number of points in the input) to complete (Dry et al, 2006(Dry et al, , 2012Graham et al, 2000). A study of the minimum vertex cover problem (a computationally hard problem that is not Euclidean in nature) (Carruthers, Masson, & Stege, 2012) focused on properties of instances that impact human performance, and what strategies participants might adopt when tackling instances. Performance on instances of this problem was in line with performance results reported on the E-TSP, ranging, depending on the factor, from roughly 4% to 10% above optimal.…”

On Evaluating Human Problem Solving of Computationally Hard Problems

“…One possible explanation for the quality and speed of human performance on instances of the E-TSP is that humans are able to leverage perceptual processes when searching for a solution (van Rooij et al, 2006). Preliminary work has been done on the human performance on the Minimum Vertex Cover Problem (Carruthers et al, 2012), investigating the quality of human performance on a problem where perceptual processes are less likely to be useful in the same way they are on E-TSP. Results of this study focused on properties of instances of Minimum Vertex Cover that impact human performance, and what strategies participants adopt when tackling instances.…”

“…However, a handful of other problems have been investigated, including: Generalized Steiner Tree problem (Burns, Lee, & Vickers, 2006), and the Minimum Vertex Cover Problem (Carruthers, Masson, & Stege, 2012).…”

“…Furthermore, for problems that are known to have approximation algorithms, or that are in FPT, researchers may look to the literature for techniques that humans may be able to identify and apply. For example, when solving Vertex Cover instances, people appear able to identify and apply two known reduction rules (Carruthers, et al, 2012), which may help to explain human performance, at least for instances where these reduction rules are available.…”

“…Vertex Cover was selected as a non-Euclidean alternative to E-TSP (Carruthers et al, 2012), but, as mentioned above, also has the desirable quality of being in the class FPT when parameterized by its vertex cover size; thus for small enough k (in its decision version), Vertex Cover's time complexity grows exponential only in k, and is polynomial in terms of its input size (Downey & Fellows, 1999) 17 . A natural question arises: is there a difference in human performance on computational problems that are FPT for natural parameterizations versus those that are (likely) not 18 ?…”