We introduce a variant of the three-sided stable matching problem for a PhD market with students, advisors, and co-advisors. In our formalization, students have consistent (lexicographic) preferences over advisors and co-advisors, and the latter have preferences over students only (hence advisors and co-advisors are cooperative). A student must be matched to one advisor and one co-advisor, or not at all. In contrast to previous work, advisor-student and student-co-advisor pairs may not be mutually acceptable, e.g., a student may not want to work with an advisor or co-advisor and vice versa. We show that stable three-sided matchings always exist, and present the PhD algorithm, a three-sided matching algorithm with polynomial running time which uses any two-sided stable matching algorithm as matching engine. Borrowing from results on two-sided markets, we provide some approximate optimality results. We also present an extension to three-sided markets with quotas, where each student conducts several projects, and each project is supervised by one advisor and one co-advisor. As it is often the case in practice that the same student should not do more than one project with the same advisor or co-advisor, we modify our PhD algorithm for this setting by adapting the two-sided Gale--Shapley algorithm to many-to-many two-sided markets, in which the same pair can match at most once. We also generalize the three-sided market to an $n$-sided market consisting of $n-1$ two-sided markets. We extend the PhD algorithm to this multi-sided setting to compute a stable matching in polynomial time, and we discuss its extension to arbitrary quotas. Finally, we illustrate the challenges that arise when not all advisor-co-advisor pairs are compatible, and critically review the statements from [30, 29].
翻译:我们引入了与学生、顾问和共同顾问的博士市场三面稳定匹配问题的变体。 在我们的正规化过程中,学生可能不愿与顾问和共同顾问合作,而共同顾问对学生有一贯的(语言)偏好。 后者只对学生有偏好( 顾问和共同顾问是合作的 ) 。 学生必须与一位顾问和共同顾问匹配三面稳定匹配问题。 学生必须与一位顾问和一位共同顾问匹配, 而不是完全匹配问题。 与以往的工作相比, 顾问和学生共同顾问对博士市场来说,三面稳定匹配问题可能无法相互接受。 与以往不同, 顾问、 顾问和学生对30对立对立对立问题可能无法相互接受, 例如, 学生可能不想与顾问或共同顾问一起工作。 我们的三面匹配总是存在稳定的三面匹配算法, 并且经常用双面匹配的算法来调整我们的市场。