The construction of group search algorithm for the secretary problem

The object of this study is a generalized secretary problem in which candidates are divided into groups of varying sizes and are observed simultaneously within each group. The problem addressed is to determine the optimal order of reviewing such groups in order to maximize the probability of selecting the best candidate. The results obtained consist in the development of an efficient group ordering algorithm that combines theoretical findings based on Bruss’s odds theorem with numerical modeling using the Monte Carlo method. Owing to its specific features and distinctive advantages, the proposed approach increases the probability of selecting the best candidate by 8–15% compared to a random group order. This is achieved through two proven lemmas that restrict consideration only to those permutations in which the groups considered as candidates for stopping are ordered in non-increasing size, and the search begins with the largest group. Such an algorithm makes the problem computationally feasible even for a moderately large number of groups. The obtained results are explained by the fact that uneven distributions of group sizes introduce exploitable structural asymmetries. The proposed ordering strategy effectively leverages this unevenness, which is confirmed by numerical experiments demonstrating a positive correlation between the degree of unevenness and the probability of successful selection. The practical applicability of the results extends to scenarios involving online resource allocation, adaptive algorithms with real-time updates, and decision-support systems. The proposed framework can be efficiently implemented in adaptive recruitment platforms and similar applications, making it relevant not only for theoretical research on optimal stopping problems but also for practical use in operations research, economics, and artificial intelligence