The sliding tile puzzle, roots to polynomials, and $\textbf{P}$ vs. $\textbf{NP}$ complexity
Roy Burson
TL;DR
This work addresses how solution-space structure influences time complexity in the P vs NP landscape, using the sliding tile puzzle and polynomial root finding as focal exemplars. It develops a matrix-based representation and a general-solution mapping $f$ for sliding tiles, deriving bounds such as $\Psi(\mathbf{A}) \le \log_4(n^2!)$ and highlighting that solution verification can be polynomial-time while discovery depends on navigating potentially exponential space. It then builds an algebraic-system framework for roots of polynomials, detailing the cost of evaluating versus finding roots and comparing several root-finding algorithms, while posing the question of polynomial-time solvability in terms of circuit and algebraic formulation complexity. The paper thereby illuminates how verification efficiency and the difficulty of constructing a general solution govern the practical separation between solving and checking NP-hard problems, and it outlines broad avenues for extending these insights to higher-dimensional puzzles, heuristic strategies, and cryptographic applications.
Abstract
This work explores the relationship between solution space and time complexity in the context of the $\textbf{P}$ vs. $\textbf{NP}$ problem, particularly through the lens of the sliding tile puzzle and root finding algorithms. We focus on the trade-off between finding a solution and verifying it, highlighting how understanding the structure of the solution space can inform the complexity of these problems. By examining the relationship between the number of possible configurations and the time complexity required to traverse this space we demonstrate that the minimal time to verify a solution is often smaller than the time required to discover it. Our results suggest that the efficiency of solving $\textbf{NP}$-complete problems is not only determined by the ability to find solutions but also by how effectively we can navigate and characterize the solution space. This study contributes to the ongoing discourse on computational complexity, particularly in understanding the interplay between solution space size, algorithm design, and the inherent challenges of finding versus verifying solutions.