Optimal (0,1)-matrix completion with majorization ordered objectives

We propose and examine two inherently symmetric (0,1)-matrix completion problems with majorization ordered objectives, which provides a unique perspective for electric vehicle charging, portfolio optimization, and multi-agent cooperation. Our work elevates the seminal study by Gale and Ryser from feasibility to optimality in partial order programming (POP), referring to optimization with partially ordered objectives. Solving such integer POP (iPOP) problems is challenging because of the integer requirements and the fact that two objective values may not be comparable. Nevertheless, we prove the essential uniqueness of all optimal objective values and identify two particular ones for each iPOP problem. Furthermore, for every optimal objective value of each iPOP problem, we respectively develop a “peak-shaving” and a “valley-filling” algorithm to construct an associated optimal (0,1)-matrix via a series of sorting processes. We show that the resulting algorithms have linear time complexities and numerically verify their efficiency compared to the commonly used order-preserving method for POP.