The abstract theory presented is based on Cook's Theorem and the framework of NP-completeness reductions within computational complexity theory. It establishes that specific combinatorial optimization problems, such as Clique, Independent Set, Vertex Cover, and their restricted variants (e.g., Degree ≤ 3), are computationally equivalent to SAT in terms of hardness under polynomial-time many-one reductions. The core principle relies on the formal definitions where a problem is NP-complete if it resides within class NP and remains hard for all problems in that class via reducibility proofs involving graph transformations like edge complementation or degree-reduction splits, thereby proving that P ≠ NP unless an efficient algorithm exists for these hardest instances.
The abstract theory presented is based on Cook's Theorem and the framework of NP-completeness reductions within computational complexity theory. It establishes that specific combinatorial optimizatio…