Fig 1.
(a) Task. The task was composed of three main stages: the items stage (3 s), the solving stage (22 s) and the response stage (2 s). Initially, participants were presented with a set of items of different values and weights. The green circle at the center of the screen indicated the time remaining in this stage of the trial. This first stage lasted 3 seconds. Then, both capacity constraint and target profit were shown at the center of the screen. The objective of the task is to decide whether there exists a subset of items for which (1) the sum of weights is lower or equal to the capacity constraint and (2) the sum of values yields at least the target profit. This stage lasted 22 seconds. Finally, participants had 2 seconds to make either a ‘YES’ or ‘NO’ response using the response button box. A fixation cross was shown during the inter-trial interval (jittered between 8 and 12 seconds). (b) Relation between TCC and human performance in the knapsack decision task. Each dot represents an instance; human performance corresponds the proportion of participants that solved the instance correctly. Instances are categorized according to their constrainedness region (αp) and their TCC. In the underconstrained region (low TCC) the satisfiability probability is close to one, while in the overconstrained region (low TCC) the probability is close to zero. The region with high TCC corresponds to a region in which the probability is close to 0.5. Additionally, instances are categorized according to their solution (satisfiability) which is represented by their color. The box plots represent the median, the interquartile range (IQR) and the whiskers extend to a maximum length of 1.5*IQR. (c) Pictorial representation of complexity and proof hardness as operationalized in relation to the properties of instances (αp and satisfiability).
Fig 2.
Neural correlates of TCC and satisfiability.
(a) Brain activation effect estimates (β) for the high vs. low TCC contrast (βhighTCC − βlowTCC). A positive contrast represents a higher BOLD signal for instances with high TCC compared to low TCC. Significant cluster-wise FWE-corrected (p < 0.05) clusters (with an uncorrected threshold of p < 0.001) are presented for each of the contrasts estimated using the Boxcar analysis. Each panel represents a different period in the solving stage. No significant clusters were found for period S1 nor for the response stage parameters. (b) Brain activation effect estimates (β) for the unsatisfiable vs. satisfiable contrast (βunsatisfiable − βsatisfiable). A positive contrast represents a higher BOLD signal for unsatisfiable instances. Significant cluster-wise FWE-corrected (p < 0.05) clusters (with an uncorrected threshold of p < 0.001) are presented for each of the contrasts estimated using the Boxcar analysis. Each panel represents a different period in the solving stage. No significant clusters were found in the response stage.
Fig 3.
Temporal dynamics of BOLD in regions of interest.
Mean effect estimate (β) for each ROI over time in trial. The effect at each time point represents the mean βFIR over all of the voxels from each ROI: right AI, dACC, and right IPS cluster extending to the angular gyrus. In the top row of panels, the βFIR’s characterize the coefficients of an FIR regression with four conditions: satisfiability×TCC. The βFIR parameters are aligned to the BOLD signal, which has a lag with respect to the task time. To correct for this, the gray time-markers represent the task events by assuming a 5-second BOLD signal lag. In the second row, the TCC contrast (βhigh − βlow) is presented. The third row displays the satisfiability contrast (βunsat − βsat). The bottom row shows the interaction effect between TCC and satisfiability ([βhighTCC,unsat − βhighTCC,sat] − [βlowTCC,unsat − βlowTCC,sat]). Red asterisks represent significance at the 0.05 level. Significance levels in the gray shaded regions are suggestive only; they represent the time period and contrast from which the ROIs were selected.
Table 1.
Significant cluster-wise FWE-corrected (p < 0.05) clusters (using an uncorrected threshold of p < 0.001) from the High TCC—low TCC contrast. Coordinates are in MNI space.
Table 2.
Significant cluster-wise FWE-corrected (p < 0.05) clusters (using an uncorrected threshold of p < 0.001) from the Unsatisfiable-Satisfiable contrast. Coordinates are in MNI space.
Fig 4.
Mean effect estimate (β) of each ROI against time in trial. The effect at each time point represents the mean βFIR over all of the voxels from each ROI: right AI (a), dACC (b), and right IPS cluster extending to the angular gyrus (c). The βFIR’s characterize the coefficients of an FIR regression with four conditions: accuracy×TCC in the top panels, and accuracy×satisfiability in the bottom panels. The βFIR parameters are aligned to the BOLD signal, which has a lag with respect to the task time. The gray vertical lines represent the task events assuming a 5-second BOLD signal lag. Below the mean ROI effects, the second and fourth rows of figures show the accuracy contrasts (βcorrect − βincorrect) for different levels of TCC or satisfiability. Red stars represent significance at a 0.05 significance level.
Fig 5.
The knapsack decision problem belongs to the class NP-Complete (NPC) because it satisfies the dual-qualifying criteria of NP and NP-hard. It is NP, given that it fulfills the NP defining condition: a YES-certificate of a satisfiable instance can be verified in polynomial time (P). It is NP-hard since it is at least as hard as any other problem in NP. It is conjectured that P≠NP, which entails that the NPC problems are not solvable in polynomial time (i.e., they are harder and require more computational resources—time—to solve than problems in P). Within the class of NPC problems, there are instances that are harder than others. A key discriminator factoring instances by the respective computational resources needed for their resolution is their typical-case complexity (TCC). The class noted as co-NP-Complete (co-NPC) comprehends problems such as the co-knapsack. The aim of this problem is to determine if the existence of a subset of items that satisfy the constraints is infeasible. Every satisfiable knapsack instance has a counterpart unsatisfiable co-knapsack instance. It is conjectured that co-NPC is not in NP, thereby implying that verifying a proof of non-existence for an unsatisfiable knapsack instance is not in P; it is harder.
Fig 6.
Instances were sampled using a 2x2 factorial design, ensuring that each participant answered an equal number of instances for each of the four possible categories of TCC and proof hardness. Each category presents a distinct profile of proof hardness (the computational difficulty of validating a solution) and complexity (the computational difficulty of solving the instance).