Karp Algebraic Reduction Manifold Architecture (KARMA): A Geometric Framework for NP-Complete Problem Equivalence

In 1972, computer scientist Richard Karp made a remarkable discovery: twenty-one very different problems—from routing networks and planning schedules to packing items efficiently—are all equally difficult in a deep mathematical sense. These problems are now called NP-complete, and for more than fifty years researchers have shown their connection by carefully transforming one problem into another step by step. While this approach proves that the problems are related, it often hides the bigger picture of why they share the same level of difficulty. This paper proposes a new way of understanding these problems through geometry. We introduce the Karp Algebraic Reduction Manifold Architecture (KARMA), aframework that places all 21 problems inside a single mathematical “landscape.” In this landscape, each problem describes a different region of the same terrain of computational difficulty, and moving from one problem to another becomes like traveling smoothly across this terrain. The framework naturally groups the problems into three families—graph-theoretic, set-theoretic, and number-theoretic problems. In this geometric interpretation, distances represent how difficult it is to transform one problem into another, while the curvature of the landscape reflects their inherent computational hardness. By revealing this hidden geometric structure, the KARMA framework provides a new perspective on computational complexity. Instead of studying hard problems individually, researchers can explore the entire landscape of computational difficulty at once, potentially inspiring new algorithms, better hardness predictions, and intelligent systems that can automatically reason about problem transformations.