  Recently, Hadfield has proposed a novel Quantum Alternating Operator Ansatz
(QAOA+) to tackle Constrained Combinatorial Optimization Problems (CCOPs), and
it has wide applications. However, the large requirement of multi-qubit
controlled gates in QAOA+ limits its applications in solving larger-scale
CCOPs. To mitigate the resources overhead of QAOA+, we introduce an approach
termed Progressive Quantum Algorithm (PQA). In this paper, the concept and
performance of PQA are introduced focusing on the Maximal Independent Set (MIS)
problem. PQA aims to yield the solution of the target graph $G$ with fewer
resources by solving the MIS problem on a desired derived subgraph that has the
same MIS solution as $G$ but has a much smaller graph size. To construct such a
desired subgraph, PQA gradually and regularly expands the graph size starting
from a well-designed initial subgraph. After each expansion, PQA solves the MIS
problem on the current subgraph using QAOA+ and estimates whether the current
graph has the same MIS solution as the target graph. PQA repeats the graph
expansion and solving process until reaching the stop condition. In our
simulations, the performance of PQA is benchmarked on Erd\H{o}s-R\'enyi (ER)
and regular graphs. The simulation results suggest that PQA showcases higher
average approximation ratio (AAR) and significant quantum resource savings
compared with directly solves the original problem using QAOA+ (DS-QAOA+) at
the same level depth $p$. Remarkably, the AAR obtained by PQA is $12.9305\%$
($4.8645\%$) higher than DS-QAOA+ on ER (regular) graphs, and the average
number of multi-qubit gates (qubits) consumed by PQA is 1/3 (1/2) of that of
DS-QAOA+. The remarkable efficiency of PQA makes it possible to solve
larger-scale CCOPs on the current quantum devices.
