Bridging Pattern-Aware Complexity with NP-Hard Optimization: A Unifying Framework and Empirical Study

NP-hard optimization problems like the Traveling Salesman Problem (TSP) defy efficient solutions in the worst case, yet real-world instances often exhibit exploitable patterns. We propose a novel patternaware complexity framework that quantifies and leverages structural regularities—e.g., clustering, symmetry—to reduce effective computational complexity across domains, including financial forecasting and LLM optimization. With rigorous definitions, theorems, and a meta-learning-driven solver pipeline, we introduce metrics like Pattern Utilization Efficiency (PUE) and achieve up to 79% solution quality gains in TSP benchmarks (22–2392 cities). Distinct from theoretical NP-hardness, our approach offers a unified, practical lens for pattern-driven efficiency.