The neural dynamics associated with computational complexity
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.