<![CDATA[This study aims to enhance dynamic programming techniques for efficiently solving the Traveling Salesman Problem, a fundamental combinatorial optimization challenge. Given its NP-hard classification, traditional exact algorithms become computationally infeasible as the problem size increases. The research revisits foundational dynamic programming principles, notably the Held-Karp algorithm, and identifies existing limitations. The study begins with a comprehensive literature review, followed by an analysis of the dynamic programming challenges specific to TSP. Novel algorithms are then developed, implemented, and rigorously tested against benchmark instances. Performance evaluation is conducted using metrics such as execution time, memory usage, and solution optimality across different problem sizes. Results demonstrate significant improvements in efficiency and scalability, with enhanced algorithms achieving optimal solutions in reduced time and computational resource usage. However, the exponential growth in complexity remains a challenge for larger instances. The study concludes with recommendations for future research, focusing on further algorithmic refinements and exploring hybrid approaches to address large-scale TSPs.]]>
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