On the approximability of the stable matching problem with ties of size two
Robert Chiang, Kanstantsin Pashkovich
备注：
Added a detailed comparison with the approaches from the papers "On the approximability of the stable marriage problem with one-sided ties." by Bauckholt, Pashkovich, and Sanita and "Improved approximation algorithms for two variants of the stable marriage problem with ties." by Huang and Kavitha. (See Appendix)
The stable matching problem is one of the central problems of algorithmic game theory. If participants are allowed to have ties, the problem of finding a stable matching of maximum cardinality is an NP-hard problem, even when the ties are of size two. Moreover, in this setting it is UGC-hard to provide an approximation for the maximum cardinality stable matching problem with a constant factor smaller than 4/3. In this paper, we give a tight analysis of an approximation algorithm given by Huang and Kavitha for the maximum cardinality stable matching problem with ties of size two, demonstrating an improved 4/3-approximation factor.
Finding a stable matching is one of the central problems in algorithmic game theory. If participants are allowed to have ties and incomplete preferences, computing a stable matching of maximum cardinality is known to be NP-hard. In this paper we present a $(3L-2)/(2L-1)$-approximation algorithm for the stable matching problem with ties of size at most $L$ and incomplete lists. Our result matches the known lower bound on the integrality gap for the associated LP formulation.
Pareto Optimal Matchings in Many-to-Many Markets with Ties
我们认为帕累托最优匹配（POM）在一个多对多的市场申请人和课程，申请人有偏好，其中可能包括领带，在个别课程和字典的偏好，在套课程。由于这是迄今为止在文献中检查的最一般的设置，我们的工作统一和推广几个已知的结果。具体来说，我们的特点POM和介绍的\emph{Generalized Serial Dictatorship Mechanism with Ties (GSDT)}，有效地处理关系，通过网络流的属性。我们表明，GSDT可以生成所有POM使用不同的优先级排序的申请人，但它满足真实性只有某些这样的排序。这个缺点并不是我们的机制所特有的;我们表明，在我们的环境中生成所有POM的任何机制都容易受到战略操纵。这与一对一的情况（有或没有关系）相反，对于这种情况，生成所有POM的真实机制确实存在。
We present a polynomial-time $\frac{3}{2}$-approximation algorithm for the problem of finding a maximum-cardinality stable matching in a many-to-many matching model with ties and laminar constraints on both sides. We formulate our problem using a bipartite multigraph whose vertices are called workers and firms, and edges are called contracts. Our algorithm is described as the computation of a stable matching in an auxiliary instance, in which each contract is replaced with three of its copies and all agents have strict preferences on the copied contracts. The construction of this auxiliary ins…
When computing stable matchings, it is usually assumed that the preferences of the agents in the matching market are fixed. However, in many realistic scenarios, preferences change over time. Consequently, an initially stable matching may become unstable. Then, a natural goal is to find a matching which is stable with respect to the modified preferences and as close as possible to the initial one. For Stable Marriage/Roommates, this problem was formally defined as Incremental Stable Marriage/Roommates by Bredereck et al. [AAAI '20]. As they showed that Incremental Stable Roommates and Incremen…
Motivated by the serious problem that hospitals in rural areas suffer from a shortage of residents, we study the Hospitals/Residents model in which hospitals are associated with lower quotas and the objective is to satisfy them as much as possible. When preference lists are strict, the number of residents assigned to each hospital is the same in any stable matching because of the well-known rural hospitals theorem; thus there is no room for algorithmic interventions. However, when ties are introduced to preference lists, this will no longer apply because the number of residents may vary over s…
We introduce the problem of adapting a stable matching to forced and forbidden pairs. Specifically, given a stable matching $M_1$, a set $Q$ of forced pairs, and a set $P$ of forbidden pairs, we want to find a stable matching that includes all pairs from $Q$, no pair from $P$, and that is as close as possible to $M_1$. We study this problem in four classical stable matching settings: Stable Roommates (with Ties) and Stable Marriage (with Ties). As our main contribution, we employ the theory of rotations for Stable Roommates to develop a polynomial-time algorithm for adapting Stable Roommates m…
In the stable marriage problem (SM), a mechanism that always outputs a stable matching is called a stable mechanism. One of the well-known stable mechanisms is the man-oriented Gale-Shapley algorithm (MGS). MGS has a good property that it is strategy-proof to the men's side, i.e., no man can obtain a better outcome by falsifying a preference list. We call such a mechanism a man-strategy-proof mechanism. Unfortunately, MGS is not a woman-strategy-proof mechanism. Roth has shown that there is no stable mechanism that is simultaneously man-strategy-proof and woman-strategy-proof, which is known a…
We consider a many-to-one variant of the stable matching problem. More concretely, we consider the variant of the stable matching problem where one side has a matroid constraint. Furthermore, we consider the situation where the preference of each agent may contain ties. In this setting, we consider the problem of checking the existence of a strongly stable matching, and finding a strongly stable matching if a strongly stable matching exists. We propose a polynomial-time algorithm for this problem.
The classical Stable Roommates problem is to decide whether there exists a matching of an even number of agents such that no two agents which are not matched to each other would prefer to be with each other rather than with their respectively assigned partners. We investigate Stable Roommates with complete (i.e., every agent can be matched with any other agent) or incomplete preferences, with ties (i.e., two agents are considered of equal value to some agent) or without ties. It is known that in general allowing ties makes the problem NP-complete. We provide algorithms for Stable Roommates tha…
A super-stable matching, which was introduced by Irving, is a solution concept in a variant of the stable matching problem in which the preferences may contain ties. Irving proposed a polynomial-time algorithm for the problem of finding a super-stable matching if a super-stable matching exists. In this paper, we consider a matroid generalization of a super-stable matching. We call our generalization of a super-stable matching a super-stable common independent set. This can be considered as a generalization of the matroid generalization of a stable matching for strict preferences proposed by Fl…
We study stable matching problems where agents have multilayer preferences: There are $\ell$ layers each consisting of one preference relation for each agent. Recently, Chen et al. [EC '18] studied such problems with strict preferences, establishing four multilayer adaptions of classical notions of stability. We follow up on their work by analyzing the computational complexity of stable matching problems with multilayer approval preferences. We consider eleven stability notions derived from three well-established stability notions for stable matchings with ties and the four adaptions proposed …
Sabanci University（萨班哲大学）；Faculty of Engineering and Natural Sciences（工程与自然科学学院）
The Stable Roommates problem with Ties and Incomplete lists (SRTI) is a matching problem characterized by the preferences of agents over other agents as roommates, where the preferences may have ties or be incomplete. SRTI asks for a matching that is stable and, sometimes, optimizes a domain-independent fairness criterion (e.g., Egalitarian). However, in real-world applications (e.g., assigning students as roommates at a dormitory), we usually consider a variety of domain-specific criteria depending on preferences over the habits and desires of the agents. With this motivation, we introduce a …
We study the House Allocation problem (also known as the Assignment problem), i.e., the problem of allocating a set of objects among a set of agents, where each agent has ordinal preferences (possibly involving ties) over a subset of the objects. We focus on truthful mechanisms without monetary transfers for finding large Pareto optimal matchings. It is straightforward to show that no deterministic truthful mechanism can approximate a maximum cardinality Pareto optimal matching with ratio better than 2. We thus consider randomised mechanisms. We give a natural and explicit extension of the cla…
We study the Reaching Stable Marriage via Divorces (DivorceSM) problem of deciding, given a Stable Marriage instance and an initial matching $M$ , whether there exists a stable matching which is reachable from $M$ by divorce operations as introduced by Knuth [12]. Towards answering an open question of Manlove [13] and Cechlárová et al. [3], we show that for incomplete preferences without ties, DivorceSM is NP-hard. Our hardness reduction also implies that the problem remains parameterized intractable for the number $κ$ of allowed divorce operations. It remains NP-hard even if the maximum lengt…
Ex-post Stability under Two-Sided Matching: Complexity and Characterization
School of Computer Science and Engineering, UNSW Sydney（新南威尔士大学计算机科学与工程学院）；Institute of Economics, HUN-REN KRTK（匈牙利科学院经济研究所）；Department of Operations Research, ELTE（布达佩斯罗兰大学运筹学系）
In the Hospital Residents problem with lower and upper quotas ($HR-Q^U_L$), the goal is to find a stable matching of residents to hospitals where the number of residents matched to a hospital is either between its lower and upper quota or zero [Biró et al., TCS 2010]. We analyze this problem from a parameterized perspective using several natural parameters such as the number of hospitals and the number of residents. Moreover, we present a polynomial-time algorithm that finds a stable matching if it exists on instances with maximum lower quota two. Alongside $HR-Q^U_L$, we also consider two clo…
We study (coalitional) exchange stability, which Alcalde [Economic Design, 1995] introduced as an alternative solution concept for matching markets involving property rights, such as assigning persons to two-bed rooms. Here, a matching of a given Stable Marriage or Stable Roommates instance is called coalitional exchange-stable if it does not admit any exchange-blocking coalition, that is, a subset S of agents in which everyone prefers the partner of some other agent in S. The matching is exchange-stable if it does not admit any exchange-blocking pair, that is, an exchange-blocking coalition o…
Many important stable matching problems are known to be NP-hard, even when strong restrictions are placed on the input. In this paper we seek to identify structural properties of instances of stable matching problems which will allow us to design efficient algorithms using elementary techniques. We focus on the setting in which all agents involved in some matching problem can be partitioned into k different types, where the type of an agent determines his or her preferences, and agents have preferences over types (which may be refined by more detailed preferences within a single type). This si…
Sabanci University（萨班哲大学）；University of Glasgow（格拉斯哥大学）
The Stable Roommates problem (SR) is characterized by the preferences of agents over other agents as roommates: each agent ranks all others in strict order of preference. A solution to SR is then a partition of the agents into pairs so that each pair shares a room, and there is no pair of agents that would block this matching (i.e., who prefers the other to their roommate in the matching). There are interesting variations of SR that are motivated by applications (e.g., the preference lists may be incomplete (SRI) and involve ties (SRTI)), and that try to find a more fair solution (e.g., Egalit…
