We study the costs and benefits of different quantum approaches to finding approximate solutions of constrained combinatorial optimization problems with a focus on Maximum Independent Set. In the Lagrange multiplier approach we analyze the dependence of the output on graph density and circuit depth. The Quantum Alternating Ansatz Approach is then analyzed and we examine the dependence on different choices of initial states. The Quantum Alternating Ansatz Approach, although powerful, is expensive in terms of quantum resources. A new algorithm based on a Dynamic Quantum Variational Ansatz (DQVA) is proposed that dynamically changes to ensure the maximum utilization of a fixed allocation of quantum resources. Our analysis and the new proposed algorithm can also be generalized to other related constrained combinatorial optimization problems.
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