The Computational Complexity of Mazes

This analysis places various types of mazes on the P versus NP computational complexity continuum.

The Crucial Distinction: Solving vs. Verifying

P (Polynomial Time): Problems for which a solution can be found by a deterministic algorithm in a reasonable (polynomial) amount of time relative to the input size.

NP (Nondeterministic Polynomial Time): Problems for which a proposed solution can be verified quickly (in polynomial time), but finding that solution might be very difficult.

1. Where Standard Mazes Live: In P

Finding a path through a standard, finite maze—even the most complex Hamster Maze or Borg Complex—is a P problem.

The Algorithm: You can use graph traversal algorithms like Depth-First Search (DFS) or Breadth-First Search (BFS).

Why it's in P: These algorithms are guaranteed to find a path from the start to the goal (if one exists) in time proportional to the number of junctions (nodes) and paths (edges) in the maze. This is polynomial time, specifically O(N + E). BFS even finds the shortest path.

Even for the "most difficult" human maze, a computer using BFS would solve it almost instantly. The difficulty for humans is our limited memory and poor spatial reasoning, not the mathematical complexity of the problem itself.

2. Where Mazes Get "NP-Hard": The Generalization

The connection to NP appears when we generalize the maze problem. The classic NP-Complete problem that relates to mazes is the Hamiltonian Path Problem.

The Problem: "Given a graph (a network of nodes and edges), does there exist a path that visits every node exactly once?"

The Maze Analogy: Imagine a maze where the goal is to walk through every single room (junction) exactly once before finding the exit. There is no known algorithm that can solve this generally faster than brute-forcing all possible paths.

So, while solving a normal maze is in P, solving a maze where you must visit every junction is NP-Hard.

3. The "Multi-Conclusion" and "Shortest Path" Nuance

This is where the P vs NP continuum gets interesting.

Finding a path: In P (using BFS/DFS).

Finding the shortest path: Also in P. BFS finds it for unweighted mazes, and Dijkstra's algorithm finds it for weighted ones (where paths have different "costs").

Finding a path that collects specific items (The "Travelling Salesperson" in a maze): This becomes NP-Hard. If you have to visit a set of specific points in the maze in any order, finding the shortest route that hits all of them is computationally monstrous as the maze grows.

The "Multi-Conclusion" mazes mentioned earlier are difficult because they force the human to implicitly solve a problem that is closer to this NP-Hard version. You have to evaluate all goals to find the best one, which is a much harder task than just finding a path to one.

Mapping the Mazes to the Continuum

Maze Type	Problem Type	Why	Computational Class
Standard Maze (Hamster, Hedge)	Find a path from A to B.	Solvable with BFS/DFS.	P (Easy for computers)
Maze with Weights	Find the shortest/fastest path.	Solvable with Dijkstra's Algorithm.	P (Easy for computers)
"Visit Every Junction" Maze	Find a Hamiltonian Path.	Requires checking all path permutations.	NP-Complete (Hard for everyone)
"Collect All Items" Maze	Solve a TSP-like problem within the maze.	Requires finding the optimal order to visit points.	NP-Hard (Hard for everyone)
Multi-Conclusion Maze (for a human)	Find the best goal.	Human must approximate a complex optimization.	Feels NP-Hard (The human is solving a harder problem)

Conclusion

On the P v NP continuum, the mazes themselves, as solvable structures, are P problems. A computer equipped with the right algorithm is never truly "challenged" by a standard maze.

However, the perception of difficulty for humans comes from two places:

1. Inefficient Algorithms: Humans don't run BFS in their heads. We use flawed heuristics (like the "right-hand rule") and have limited working memory, which makes even P problems feel impossibly hard.

2. Implicitly Harder Problems: When a maze doesn't have a single, clear goal, or requires optimization, it forces the human brain to tackle a problem that is much closer to an NP-Hard one. We are bad at this, and that's what creates the profound feeling of difficulty and complexity.

So, while the mazes technically reside in P, the cognitive tasks we assign ourselves when solving them often brush up against the much harder world of NP.