Distributed Stable Matching Problems with Ties and Incomplete Lists

We consider the Stable Marriage Problem and the Stable Roommates Problem in presence of ties and incomplete preference lists. They can be solved by centralized algorithms, but this requires to make public preference lists, something that members would prefer to avoid for privacy reasons. This motivates a distributed formulation to keep privacy. We propose a distributed constraint approach that solves all the considered problems, keeping privacy. 1 The Stable Marriage Problem The Stable Marriage Problem (SM ) [5] involves n men and n women. Each man mi ranks women forming his preference list, and each woman wj ranks men forming hers. A matching M is a one-to-one mapping between the two sexes. M is stable when there is no blocking pair (m,w) such that m and w, no partners in M , prefer one another to his/her partner in M . A solution is a stable matching. There are several SM versions. With Incomplete Lists (SMI ) some people may consider unacceptable some members of the other sex. With Ties (SMT ), some people may consider equally acceptable some members of the other sex, so there is a tie among them. For SMT , three stability types have studied: weak, strong and super [6]. (m,w) is a weak blocking pair for M if m and w are not partners in M , and each of whom strictly prefers the other to his/her partner in M . (m,w) is a strong blocking pair for M if m and w are not partners in M , and one strictly prefers the other to his/her partner in M and the other is at least indifferent between them. (m,w) is a super blocking pair for M if m and w are not partners in M , and each of whom either strictly prefers the other to his/her partner in M or it is indifferent between them. With Ties and Incomplete Lists (SMTI), some person may consider as equally acceptable some members of the other sex, while others are unacceptable. The three stability types apply here. Solvability conditions, complexity and solving algorithms of each SM version appear in Table 1. For SMTI-weak, different solutions may exist with different lengths, so it is of interest to find the matching of maximum cardinality. This is SMTI-weak-max, an optimization problem that is NP-hard. Supported by the Spanish project TIN2005-09312-C03-01. F. Benhamou (Ed.): CP 2006, LNCS 4204, pp. 675–679, 2006. c © Springer-Verlag Berlin Heidelberg 2006 676 I. Brito and P. Meseguer Table 1. Solvability conditions, solving algorithm and complexity for the SM versions SM version ∃ solution? Size All solutions Algorithm Compl Length Partners SM always n same same EGS [5] poly SMI always ≤ n same same EGS [5] poly SMT-weak always n same same break ties + EGS [6] poly SMT-strong not always n same same STRONG [6] poly SMT-super not always n same same SUPER [6] poly SMTI-weak always ≤ n diff diff break ties + EGS [7] poly SMTI-strong not always ≤ n same same STRONG2 [7] poly SMTI-super not always ≤ n same same SUPER2 [7] poly SMTI-weak-max always ≤ n same diff break ties in all possible ways + EGS [7] NP-hard Table 2. Algorithms for solving SM and DisSM versions DisSM problem Centralized Extension to the distributed Algorithm case, keeping privacy DisSMT-weak break ties + EGS [6] break ties arbitrary + DisEGS [3] DisSMT-strong STRONG [6] No extension [2] DisSMT-super SUPER [6] Extension [2] DisSMTI-weak break ties + EGS [7] break ties arbitrary + DisEGS [3] DisSMTI-strong STRONG2 [7] No extension [2] DisSMTI-super SUPER2 [7] No extension [2] DisSMTI-weak-max break ties in all possible ways + EGS [7] Discussion [2] SM appears to be naturally distributed. Each person would like to keep his/her preference lists private, which is not possible in the centralized case. This motivates the Distributed Stable Marriage (DisSM ) [3], defined as SM plus a set of 2n agents. Each agent owns exactly one person. An agent knows all the information of its owned person, but it cannot access the information of people owned by other agents. A solution is a stable matching. Similarly, we define here the distributed versions with incomplete lists (DisSMI ), with ties (DisSMT ) and with ties and incomplete lists (DisSMTI ) 1. Is it possible to extend the centralized algorithms to the distributed setting keeping privacy? DisEGS is a distributed version of Extended Gale-Shapley (EGS ) that maintains privacy. It was used to solve DisSM and DisSMI [3]. Here we focus on DisSMT and DisSMTI, that jointly with the three stability types, produce six decision problems plus one optimization problem. The extension of centralized algorithms to the distributed case appear in Table 2. Only three out of the six decision problems can be solved by extending the centralized algorithms to the distributed case while keeping preferences private. Details appear in [2]. 2 Constraint Formulation In [4], SM is modeled and solved as a binary CSP with 2n variables. Variable domains are the preference lists. Constraints are defined between men and women: Cij is a table with all possible partial matchings involving man i and woman j. For any pair k, l (k ∈ Dom(i), l ∈ Dom(j)), the element Cij [k, l] represents the partial matching (mi, wk)(ml, wj); which could be: Allowed, Illegal, Blocked or Support. Tables with A, I, B, S are passed into 1/0 by A, S → 1, I, B → 0. 1 [9] proposes a solving method based on encryption techniques. Distributed Stable Matching Problems with Ties and Incomplete Lists 677 For instances with ties, we consider the different definitions of stability. This affects the usage of Blocked pair in Cij . The definition given in [4] is valid for weak stability. For strong and super stability, we replace B definition in [4] by, – Cij [k, l] = Blocked if (mi, wj) is a strong blocking pair for (ml, wk). – Cij [k, l] = Blocked if (mi, wj) is a super blocking pair for (ml, wk). Privacy is one of the main motivations for distributed CSP (DisCSP ). We differentiate between value privacy and constraint privacy [1]. Value privacy implies that agents are not aware of other agent’s values during the solving process and in the final solution. On constraint privacy, the Partially Known Constraints (PKC) model was presented. It assumes that when two agents i, j share a constraint Cij , none of them fully knows it. Each agent knows the part of the constraint that it is able to build, using its own information. We say that agent i knows Ci(j), and j knows C(i)j . Similarly to [3], we use the constraint formulation of Section 2 to solveDisSMT andDisSMTI by theDisFC algorithm [1], keeping privacy of preference lists using the PKC model. How can i build Ci(j)? Ordering rows of Ci(j) following his preference list, all elements in rows above wj are 1 (except m i column that are 0). All elements in rows below wj may be 1 or 0, depending on the ordering of columns (except m i column that are 0). Since xi does not know the preference list of yj , columns are ordered lexicographically, and the elements below wj row are ? (undecided). If there is a tie between wj and wj′ elements in wj′ row are # (tie). Ci(j) is, Ci(j) = m1 . . . mi . . . mn i1 1 . . . 1 0 1 . . . 1 . . . 1 . . . 1 0 1 . . . 1 wj 0 . . . 0 1 0 . . . 0 w j′ # . . . # 0 # . . . # ? . . . ? 0 ? . . . ?