A Two-Phase Quantum Algorithm for the Partial Max-CSP

In this paper, we present a novel two-phase quantum algorithm designed to analyze a variant of the Maximum Constraint Satisfaction Problem (Max-CSP), denominated Partial Binary Max-CSP. The goal of Partial Binary Max-CSP is to maximize the number of feasibly assigned variables while ensuring conflict-free constraints. The proposed method uses the Quantum Approximate Optimization Algorithm (QAOA) in two distinct steps. In the f irst step, QAOA is utilized to obtain an initial solution focused on satisfying as many constraints as possible. In cases where the problem is oversubscribed and conflicts remain, we use a second step: the unresolved conflicts are mapped into a Minimum Vertex Cover (MVC) problem, which is subsequently solved using a second QAOA. This two-phase approach ensures a conflict-free solution while minimizing the number of deactivated variables. Wedemonstrate the efficacy of our algorithm with an illustrative example involving the safe storage of dangerous materials, showing its potential application in real-world scenarios. This paper sets the foundation for further exploration of quantum algorithms in solving complex constraint satisfaction problems.