Quantum Speed-ups for Single-machine Scheduling Problems

Grover search is currently one of the main approaches to obtain quantum speed-ups for combinatorial optimization problems. The combination of Quantum Minimum Finding (obtained from Grover search) with dynamic programming has proved particularly efficient to improve the worst-case complexity of several NP-hard optimization problems. Specifically, for these problems, the classical dynamic programming complexity (ignoring the polynomial factors) in O* (c(n)) can be reduced by a bounded-error hybrid quantumclassical algorithm to O* (c(quant)(n)) for c(quant) < c. In this paper, we extend the resulting hybrid dynamic programming algorithm to three examples of single-machine scheduling problems: minimizing the total weighted completion time with deadlines, minimizing the total weighted completion time with precedence constraints, and minimizing the total weighted tardiness. The extension relies on the inclusion of a pseudo-polynomial term in the state space of the dynamic programming as well as an additive term in the recurrence.