Explain the application of the Quantum Approximate Optimization Algorithm (QAOA) to solve the Max-Cut problem on a non-bipartite graph. Your explanation should target an audience with a background in linear algebra and classical computer science but limited quantum knowledge. Break down the following components: 1) The construction of the problem Hamiltonian (Cost function) and the Mixer Hamiltonian, 2) The parameterization of the quantum circuit using alternating unitary operators, 3) The classical optimization loop used to find optimal parameters (p=1 depth), and 4) A comparison of the theoretical approximation ratio guarantees versus classical heuristics. Use LaTeX-style notation for equations.