Maximum Stable Matching with Matroids and Partial Orders ^††thanks: Some of the results in this paper have appeared in the 6th SIAM Symposium on Simplicity of Algorithms (SOSA 2023) [12]

The key tool for generalizing the $1.5$-approximation to matroid kernels is a result on the existence of a perfect matching of certain types of exchangeable pairs in a matroid. This is presented as Theorem 5 in Section 2. This result may be of independent interest, as it is a previously unknown elegant property of basis exchange graphs.

Using Theorem 5, we show in Section 3 that there is a $1.5$-approximation algorithm for max-kernel-po with interval orders. That is, we show the following theorem.

We remark that Theorem 1 implies a $1.5$-approximation algorithm for finding a maximum solution in the many-to-many stable matching problem on $G=(U,W;E)$ where each vertex $v$ has a matroid constraint and an interval order on $\delta_{G}(v)$. To reduce the problem to the setting in Theorem 1, we construct an interval order $\succ_{U}$ on the ground set $E$ consistent to the orders of all vertices in $U$ as follows. (The construction of $\succ_{W}$ is similar). Assign indices to elements in $U$ as $u_{1},u_{2},\dots,u_{|U|}$ arbitrarily, and then define $\succ_{U}$ by taking the union of binary relations in $\{\succ_{u}\}_{u\in U}$ and adding the relations $e\succ_{u}f$ for all $e\in\delta_{G}(u_{i})$ and $f\in\delta_{G}(u_{j})$ with $i<j$. We see that $\succ_{U}$ is an interval order, and its restriction to $\delta_{G}(u)$ is $\succ_{u}$ for any $u\in U$. Set $M_{1}=(E,\mathcal{I}_{U},\succ_{U})$ and $M_{2}=(E,\mathcal{I}_{W},\succ_{W})$, where $\mathcal{I}_{U}$ and $\mathcal{I}_{W}$ are respectively the direct sums of the matroids of vertices in $U$ and $V$. Then $(M_{1},M_{2})$-kernels correspond to stable matchings on $G$.

In the case of the stable marriage setting, in Section 4, we show an even stronger statement than Theorem 1: the output of the algorithm also $1.5$-approximates the LP optimum of the corresponding natural Integer Program. This complements the previously known lower bound of $1.5-o(1)$ of Iwama et al. [28] on the integrality gap of max-smti.

Our algorithm consists of three steps: (1) constructing an instance of the matroid kernel problem with total orders on the extended ground set obtained by replacing each element by three parallel elements, (2) running Fleiner’s algorithm on the new instance, and (3) projecting the solution to the original ground set. The running time of the algorithm is quadratic in $|S|$ (provided that the independence oracles of matroids are available), and linear in $|S|$ in case of partition matroids, which corresponds to the many-to-many stable matching problem with parallel edges allowed. Since interval orders can model $\alpha$-stability [1, 37] and near-stability [10], our algorithm implies $1.5$-approximation for these stability notions.

The approximation ratio attained by our algorithm is best possible under UGC. As explained at the end of Section 3.2, this fact easily follows from the result of Askalidis et al. [2], who proved that max-smiss, the maximum stable marriage problem with incomplete lists under social stability, is inapproximable within $\frac{3}{2}-\delta$ under UGC. As max-smiss can be seen as a special case of the maximum stable marriage problem with interval orders, the following proposition follows.

We then show that the $1.5$-approximability shown in Theorem 1 cannot be extended to the setting with general partial orders, even in the stable marriage case. In Section 4, we investigate max-smpo, the version of maximum stable marriage where preferences can be arbitrary partial orders. Note that for this problem and also for the more general max-kernel-po, $2$-approximation is easily attained; as explained before, a stable matching (a matroid kernel) can be found by running the (matroidal version of) Gale–Shapley algorithm with total orders consistent with the original partial orders. Since stability implies that the output matching (common independent set) is maximal, its size is at least half of the optimum. We show that for general partial orders, we cannot beat this trivial approximation ratio even for the stable marriage case under UGC.

The proof of Theorem 3 uses a result of Bansal and Khot [5] about the hardness of finding an independent set of size $\epsilon n$ in a graph that has two disjoint independent sets of size $(\frac{1}{2}-\epsilon)n$ each. We also show in Section 4 that the integrality gap of the LP relaxation of a natural IP of max-smpo has integrality gap at least $2$ for general partial orders while it is at most $1.5$ for interval orders as mentioned above. Our results suggest that maybe the class of interval orders is the right generalization for which a nontrivial approximation is still possible.

The stable marriage problem with ties and incomplete lists (smti) was first studied by Iwama et al. [25], who showed the NP-hardness of max-smti. Since then, various algorithms have been proposed to improve the approximation ratio [20, 26, 27, 30], and the current best ratio is $1.5$ by a polynomial-time algorithm of McDermid [33], where the same ratio is attained by linear-time algorithms of Paluch [35, 36] and Király [31, 32]. The $1.5$-approximability extends to the many-to-one matching setting [32] and the student-project allocation problem with ties [11]. As for the inapproximability of max-smti, Halldórsson et al. [19] showed that it is NP-hard to approximate it within some constant factor. Later, inapproximability results have been improved, especially assuming stronger complexity-theoretic conjectures. Yanagisawa [41] and Huang et al. [22] showed that assuming the Unique Games Conjecture, there is no $(\frac{4}{3}-\varepsilon)$-approximation for any $\varepsilon>0$, if P$\neq$NP. By a recent work by Dudycz, Manurangsi and Marcinkowski [13], it follows that, assuming the Strong Unique Games Conjecture or the Small Set Expansion Hypothesis, there cannot even be a $(\frac{3}{2}-\varepsilon)$-approximation algorithm for max-smti, if P$\neq$NP.

The stable marriage problem also has generalizations in which constraints are imposed on the possible sets of edges. Biró et al. [6] studied the college admissions problem with common quotas. It is a many-to-one matching problem between students and colleges, where not only individual colleges but also certain subsets of colleges have upper quotas. Huang [21] investigated the classified stable matching problem, a many-to-one matching model in which each individual college has upper and lower quotas for subsets of students. In the absence of lower quotas, it was shown that a stable matching exists in these models if the constraints have a laminar structure. Yokoi [42] considered a many-to-many matching model with ties and laminar constraints and presented a $1.5$-approximation algorithm. Its approximation analysis depends on the base orderability of laminar matroids and cannot extend to the general matroid setting.

The stable matching problem has been studied with many different types of preference structures. There are studies on super and strong stabilities with partial order preferences [23, 38]. Condon et al. [38] studied super stability with partial orders, where a blocking edge is defined as an edge $e=(u,w)$ such that each of $u$ and $w$ either strictly prefers the other to its current assignment, or are indifferent between them (i.e. the two are incomparable in the partial order). They showed that we can find a super stable matching in polynomial-time if one exists. Irving, Manlove, and Scott [23] studied strong stability, which differs from super stability in the aspect that an edge $e=(u,w)$ only blocks a matching if at least one of $u,w$ strictly prefers the other to their current assignment. They showed that with arbitrary partial orders in the preferences, it is NP-hard to decide if a strongly stable matching exists. With some modifications to their reduction from 3-SAT, it can also be shown that this hardness holds even in the case of semiorders, which is a special case of interval orders.

A closely related area is stable matchings with uncertain or changing preferences, where the preferences of the agents may be partially unknown, or may depend on some random factors and may change over time, which has also been the focus of interest lately [3, 9, 34, 7]. The usual purpose here is to find matchings that are stable with probability one, if there are any, or otherwise find matchings that are stable with maximum probability. In particular, as Aziz et al. [3] mentioned, uncertain preferences are strongly connected to partial order preferences. For example, if we suppose that there is a set of possible preference lists for each agent, then deciding if there is a matching that is stable with any possible choices for these preference lists can be reduced to finding a super stable matching where the agents have partial orders.

Another similar problem is robust stable matchings and locally nearly stable matchings [10]. Intuitively speaking, a robust stable matching is one that is stable, and remains stable even if each agent is allowed to make some swaps (i.e. switch two adjacent entries) in their preference lists, while a locally nearly stable matching is a matching that can be made stable if each agent is allowed to make some swaps in their list, or equivalently, there are no blocking edges, where both agents improve by a lot. Finding robust stable matchings reduces to finding super stable matchings, while finding a maximum size locally nearly stable matching reduces to finding a maximum size (weakly) stable matching with partial order preferences (in this case the arising orders are semiorders).

Let $M_{1}=(S,\mathcal{I}_{1},\succ_{1})$ and $M_{2}=(S,\mathcal{I}_{2},\succ_{2})$ be partially ordered matroids on the same ground set $S$ and suppose that $\succ_{1}$ and $\succ_{2}$ are interval orders. Our algorithm consists of three steps.

1. 1.

   The algorithm first creates a new instance by replacing each element of $S$ by three parallel elements, and by defining total orders on the extended ground set as explained below.

2. 2.

   For the obtained totally ordered matroids $M^{*}_{1}$, $M^{*}_{2}$, an $(M^{*}_{1},M^{*}_{2})$-kernel $A^{*}$ is computed.

3. 3.

   The algorithm returns a set $A$ that is the projection of $A^{*}$ to the original ground set $S$.

The first step can be done in $\mathcal{O}(|S|^{2})$ time as explained later, and the second step can be done in $\mathcal{O}(|S|^{2})$ time using Fleiner’s algorithm, which we will briefly describe later.

Here, we explain the precise construction of the totally ordered matroids $M^{*}_{1}$ and $M^{*}_{2}$. Let the extended ground set be $S^{*}\coloneqq\cup_{u\in S}\{x_{u},y_{u},z_{u}\}$. We define $M^{*}_{i}=(S^{*},\mathcal{I}^{*}_{i},\succ^{*}_{i})$ for $i=1,2$ as follows. The elements $x_{u},y_{u},z_{u}$ are parallel in each $M^{*}_{i}$, that is,

where $\pi(I^{*})=\{\,u\in S:I^{*}\cap\{x_{u},y_{u},z_{u}\}\neq\emptyset\,\}$. It is easy to see that $(S^{*},\mathcal{I}^{*}_{i})$ is a matroid for $i=1,2$. To define the total order $\succ^{*}_{1}$ on the extended ground set $S^{*}$, we first define a binary relation $\mathcal{R}^{*}_{1}\subseteq S^{*}\times S^{*}$ using $\succ_{1}$ as follows:

To define $\succ^{*}_{2}$, we define $\mathcal{R}^{*}_{2}$ from $\succ_{2}$ as follows, which is described in the same manner as $\mathcal{R}^{*}_{1}$ but the roles of $x_{u}$ and $z_{u}$ are interchanged.

For each $i=1,2$, we let $\succ^{*}_{i}$ be an arbitrary total order on $S^{*}$ such that $s\succ^{*}_{i}t$ holds for any $(s,t)\in\mathcal{R}^{*}_{i}$. The existence of such a total order is guaranteed by the following lemma.

The proof of this lemma, which uses the fact that $\succ_{i}$ is an interval order, is postponed to the end of this section. We now complete the construction of the new instance.

Since the digraph $D_{i}$ is acyclic and has $\mathcal{O}(|S|^{2})$ arcs, we can define a total order $\succ_{i}^{*}$ on $S^{*}$ consistent to $D_{i}$ by topological sort in $\mathcal{O}(|S|^{2})$ time. This $\succ_{i}^{*}$ indeed satisfies the required condition, i.e., $s\succ^{*}_{i}t$ holds for any $(s,t)\in\mathcal{R}^{*}_{i}$. This completes the construction of the totally ordered matroids $M^{*}_{1}$ and $M^{*}_{2}$.

For completeness, here we describe how Fleiner’s algorithm works for our new instance $M_{1}^{*}$ and $M^{*}_{2}$. The algorithm first sets $R$ to be an empty set and repeats the following three steps: set $X$ to be the optimal base of $M^{*}_{1}$ with the ground set restricted to $S^{*}\setminus R$, set $Y$ to be the optimal base of $M^{*}_{2}$ with the ground set restricted to $X$, and update $R$ with $R\cup(X\setminus Y)$. The repetition stops if $Y=X$, and $Y$ is returned, which is an $(M^{*}_{1},M^{*}_{2})$-kernel (see [4] for this version of the description).

Here, we show that the output of the algorithm is indeed a matroid kernel (i.e., a stable common independent set) in the original instance. The approximation ratio is shown in the next section. We use $C^{1}_{I}(u)$ and $C^{2}_{I}(u)$ to denote fundamental circuits in $M_{1}$ and $M_{2}$, respectively.

We now provide the postponed proof.

We now show the approximation ratio of our algorithm. As in the previous section, we denote by $C^{1}_{I}(u)$ and $C^{2}_{I}(u)$ fundamental circuits in $M_{1}$ and $M_{2}$, respectively.

Thus, we have completed the proof of the $1.5$-approximability of max-kernel-po with interval orders.

We conclude this section by observing that Proposition 2, i.e., the UGC-hardness of $(1.5-\delta)$-approximation, follows from the result of Askalidis et al. [2] on the maximum stable marriage problem with incomplete lists under social stability (max-smiss). An input of the problem is a graph $G=(U,W;E)$, a total order $\succ_{v}$ on $\delta_{G}(v)$ for each $v\in U\cup W$, and a set $A\subseteq E$ whose elements are called acquainted pairs. For a matching $M\subseteq E$, an edge $e\in E$ socially blocks $M$ if it blocks $M$ in the classical sense and belongs to $A$. A matching is socially stable if there is no social blocking pair. The task of max-smiss is to find a maximum socially stable matching. Askalidis et al. [2] proved that, assuming UGC, it is NP-hard to approximate max-smiss within $\frac{3}{2}-\delta$ for any $\delta>0$.

We can reduce max-smiss to the stable marriage problem with interval orders as follows. Given an instance of max-smiss, from the total order $\succ_{v}$ of each $v\in U\cup W$, define a partial order $\succ^{\prime}_{v}$ by $e\succ^{\prime}_{v}f\Leftrightarrow e\succ_{v}f,e\in A$. Then $\succ^{\prime}_{v}$ is an interval order. Indeed, we can easily observe that, if $e\succ^{\prime}_{v}f$ and $g\succ^{\prime}_{v}h$ hold for distinct $e,f,g,h$, then $e\succ^{\prime}_{v}h$ or $g\succ^{\prime}_{v}f$. It is also not hard to see that a socially stable matching in the original instance is a stable matching with respect to the interval orders $\{\succ^{\prime}_{v}\}_{v\in U\cup W}$. Thus, the maximum stable marriage with interval orders and its generalization to max-kernel-po are inapproximable within $\frac{3}{2}-\delta$ under UGC.

Here we provide some structural observations on max-smpo. The first one shows that the main tool which is used to show the $1.5$-approximability in previous approaches for max-smti cannot be used for partial order preferences. The previous $1.5$-approximation algorithms for max-smti, such as one in [32], are designed so that, for the output stable matching $M$ and any stable matching $N$, there is no maximal alternating path in $M\cup N$ consisting of one $M$-edge and two $N$-edges. In the case of arbitrary partial orders, it may happen that for any stable matching $M$, there exists another stable matching $N$, such that a maximal alternating path with one $M$-edge and two $N$-edges exists, as the next example shows.

The second observation is about the integrality gap. Observe that max-smpo on $G=(U,W;E)$ can be modeled with the following integer programming problem (IP) with variables $\{p_{e}\}_{e\in E}$. Since we allow parallel edges, to distinguish edges connecting the same vertices, say $u$ and $w$, we use a notation $(u,w)_{i}$ where distinct edges have distinct subscripts $i$: