Prison Maze System: Computational Complexity Analysis

System Parameters

6 subsystems (education, exercise, counseling, etc.)

Each subsystem: 0-100 points

Target: ≥480 total points

Constraint: Prisoner can customize 3 subsystems with prosecutor

Movement: Can traverse through the 6 components

Complexity Classification

This problem is likely NP-Complete

Why NP-Complete?

Reduction from Subset Sum: Finding which combination of 6 subsystems (with customizable point ranges) sums to at least 480 points is essentially a subset sum variant.

State Space: With 6 subsystems × 101 possible scores each = 101⁶ ≈ 1.06×10¹² possible states

Verification: Easy to verify if a path achieves ≥480 points (polynomial time)

Optimization: Hard to find the optimal path through the maze

What Makes It Hard

• Combinatorial explosion: 6! = 720 possible orderings of subsystems
• Point optimization: Need to maximize points within each subsystem
• Strategic customization: Choosing which 3 subsystems to optimize with prosecutor
• Constraint satisfaction: Must hit exact threshold (480+)

For a Computer: Solvable but Challenging

Small instances (current system): Easily solvable

With only 6 subsystems, brute force could check all 720 orderings × point combinations

Scaled version: Would become intractable

If expanded to 50 subsystems, problem becomes practically unsolvable for exact solutions

Practical Solutions

Dynamic Programming: Could solve exactly for current size

Heuristic Approaches: Greedy algorithms, genetic algorithms

Integer Programming: Formulate as optimization problem

Approximation: Good-enough solutions that guarantee ≥480 points

Philosophical Interpretation

The system represents a constrained optimization problem where:

• Prisoner = Algorithm seeking optimal path
• Prosecutor customization = Limited control over problem parameters
• 480-point threshold = Constraint satisfaction
• Infractions/crimes = Constraints and penalties

The prisoner's dilemma becomes a computational optimization problem.