We show that given a SM instance G as input we can find a largest collection of pairwise edge-disjoint stable matchings of G in time linear in the input size. This extends two classical results:
1. The Gale-Shapley algorithm, which can find at most two ("extreme") pairwise edge-disjoint stable matchings of G in linear time, and
2. The polynomial-time algorithm for finding a largest collection of pairwise edge-disjoint perfect matchings (without the stability requirement) in a bipartite graph, obtained by combining König's characterization with Tutte's f-factor algorithm.
Moreover, we also give an algorithm to enumerate all maximum-length chains of disjoint stable matchings in the lattice of stable matchings of a given instance. This algorithm takes time polynomial in the input size for enumerating each chain. We also derive the expected number of such chains in a random instance of Stable Matching.
For the Stable Roomates problem, we show that determining if a given instance admits at most two disjoint stable matchings or three disjoint stable matchings is NP-complete.
We consider the following question: given an $(X,Y)$-bigraph $G$ and a set $S \subset X$, does $G$ contain two disjoint matchings $M_1$ and $M_2$ such that $M_1$ saturates $X$ and $M_2$ saturates $S$? When $|S|\geq |X|-1$, this question is solvable by finding an appropriate factor of the graph. In contrast, we show that when $S$ is allowed to be an arbitrary subset of $X$, the problem is NP-hard.
Reconfigurable optical topologies promise to improve the performance in datacenters by dynamically optimizing the physical network in a demand-aware manner. State-of-the-art optical technologies allow to establish and update direct connectivity (in the form of edge-disjoint matchings) between top-of-rack switches within microseconds or less. However, to fully exploit temporal structure in the demand, such fine-grained reconfigurations also require fast algorithms for optimizing the interconnecting matchings. Motivated by the desire to offload a maximum amount of demand to the reconfigurable ne…
The weighted $\mathcal{T}$-free $2$-matching problem is the following problem: given an undirected graph $G$, a weight function on its edge set, and a set $\mathcal{T}$ of triangles in $G$, find a maximum weight $2$-matching containing no triangle in $\mathcal{T}$. When $\mathcal{T}$ is the set of all triangles in $G$, this problem is known as the weighted triangle-free $2$-matching problem, which is a long-standing open problem. A main contribution of this paper is to give a first polynomial-time algorithm for the weighted $\mathcal{T}$-free $2$-matching problem under the assumption that $\ma…
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…
An instance of a strongly stable matching problem (SSMP) is an undirected bipartite graph $G=(A \cup B, E)$, with an adjacency list of each vertex being a linearly ordered list of ties, which are subsets of vertices equally good for a given vertex. Ties are disjoint and may contain one vertex. A matching $M$ is a set of vertex-disjoint edges. An edge $(x,y) \in E \setminus M$ is a {\em blocking edge} for $M$ if $x$ is either unmatched or strictly prefers $y$ to its current partner in $M$, and $y$ is either unmatched or strictly prefers $x$ to its current partner in $M$ or is indifferent betwee…
We study the stable matching problem in non-bipartite graphs with incomplete but strict preference lists, where the edges have weights and the goal is to compute a stable matching of minimum or maximum weight. This problem is known to be NP-hard in general. Our contribution is two fold: a polyhedral characterization and an approximation algorithm. Previously Chen et al. have shown that the stable matching polytope is integral if and only if the subgraph obtained after running phase one of Irving's algorithm is bipartite. We improve upon this result by showing that there are instances where thi…
We consider the problem of stable matching with dynamic preference lists. At each time step, the preference list of some player may change by swapping random adjacent members. The goal of a central agency (algorithm) is to maintain an approximately stable matching (in terms of number of blocking pairs) at all times. The changes in the preference lists are not reported to the algorithm, but must instead be probed explicitly by the algorithm. We design an algorithm that in expectation and with high probability maintains a matching that has at most $O((log (n))^2)$ blocking pairs.
In this article we study the stable marriage game induced by the men-proposing Gale-Shapley algorithm. Our setting is standard: all the lists are complete and the matching mechanism is the men-proposing Gale-Shapley algorithm. It is well known that in this setting, men cannot cheat, but women can. In fact, Teo, Sethuraman and Tan \cite{TST01}, show that there is a polynomial time algorithm to obtain, for a given strategy (the set of all lists) $Q$ and a woman $w$, the best partner attainable by changing her list. However, what if the resulting matching is not stable with respect to $Q$? Obviou…
We introduce the \textit{multi-view pattern matching} problem, where a text can have multiple views. Each view is a string of the same size and drawn from disjoint alphabets. The pattern is drawn from the union of all alphabets. The algorithm we present is an extension of the Horspool algorithm, and in our experiments on synthetic data it shows an $3 \times$ improvement over the naive baseline.
We consider the stable matching problem when the preference lists are not given explicitly but are represented in a succinct way and ask whether the problem becomes computationally easier and investigate other implications. We give subquadratic algorithms for finding a stable matching in special cases of natural succinct representations of the problem, the $d$-attribute, $d$-list, geometric, and single-peaked models. We also present algorithms for verifying a stable matching in the same models. We further show that for $d = ω(\log n)$ both finding and verifying a stable matching in the $d$-att…
We study the problem of finding "fair" stable matchings in the Stable Marriage problem with Incomplete lists (SMI). For an instance $I$ of SMI there may be many stable matchings, providing significantly different outcomes for the sets of men and women. We introduce two new notions of fairness in SMI. Firstly, a regret-equal stable matching minimises the difference in ranks of a worst-off man and a worst-off woman, among all stable matchings. Secondly, a min-regret sum stable matching minimises the sum of ranks of a worst-off man and a worst-off woman, among all stable matchings. We present two…
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…
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…
We give the first polynomial-time algorithms on graphs of bounded maximum induced matching width (mim-width) for problems that are not locally checkable. In particular, we give $n^{\mathcal{O}(w)}$-time algorithms on graphs of mim-width at most $w$, when given a decomposition, for the following problems: Longest Induced Path, Induced Disjoint Paths and $H$-Induced Topological Minor for fixed $H$. Our results imply that the following graph classes have polynomial-time algorithms for these three problems: Interval and Bi-Interval graphs, Circular Arc, Permutation and Circular Permutation graphs,…
While the stable marriage problem and its variants model a vast range of matching markets, they fail to capture complex agent relationships, such as the affiliation of applicants and employers in an interview marketplace. To model this problem, the existing literature on matching with externalities permits agents to provide complete and total rankings over matchings based off of both their own and their affiliates' matches. This complete ordering restriction is unrealistic, and further the model may have an empty core. To address this, we introduce the Dichotomous Affiliate Stable Matching (DA…
Given an undirected graph and two disjoint vertex pairs $s_1,t_1$ and $s_2,t_2$, the Shortest two disjoint paths problem (S2DP) asks for the minimum total length of two vertex disjoint paths connecting $s_1$ with $t_1$, and $s_2$ with $t_2$, respectively. We show that for cubic planar graphs there are NC algorithms, uniform circuits of polynomial size and polylogarithmic depth, that compute the S2DP and moreover also output the number of such minimum length path pairs. Previously, to the best of our knowledge, no deterministic polynomial time algorithm was known for S2DP in cubic planar graphs…
We study a natural generalization of stable matching to the maximum weight stable matching problem and we obtain a combinatorial polynomial time algorithm for it by reducing it to the problem of finding a maximum weight ideal cut in a DAG. We give the first polynomial time algorithm for the latter problem; this algorithm is also combinatorial. The combinatorial nature of our algorithms not only means that they are efficient but also that they enable us to obtain additional structural and algorithmic results: - We show that the set, $\cal M'$, of maximum weight stable matchings forms a sublatti…
We study the problem of finding solutions to the stable matching problem that are robust to errors in the input and we obtain a polynomial time algorithm for a special class of errors. In the process, we also initiate work on a new structural question concerning the stable matching problem, namely finding relationships between the lattices of solutions of two "nearby" instances. Our main algorithmic result is the following: We identify a polynomially large class of errors, $D$, that can be introduced in a stable matching instance. Given an instance $A$ of stable matching, let $B$ be the random…
In $k$-hypergraph matching, we are given a collection of sets of size at most $k$, each with an associated weight, and we seek a maximum-weight subcollection whose sets are pairwise disjoint. More generally, in $k$-hypergraph $b$-matching, instead of disjointness we require that every element appears in at most $b$ sets of the subcollection. Our main result is a linear-programming based $(k-1+\tfrac{1}{k})$-approximation algorithm for $k$-hypergraph $b$-matching. This settles the integrality gap when $k$ is one more than a prime power, since it matches a previously-known lower bound. When the …
An instance of the super-stable matching problem with incomplete lists and ties is an undirected bipartite graph $G = (A \cup B, E)$, with an adjacency list being a linearly ordered list of ties. Ties are subsets of vertices equally good for a given vertex. An edge $(x,y) \in E \backslash M$ is a blocking edge for a matching $M$ if by getting matched to each other neither of the vertices $x$ and $y$ would become worse off. Thus, there is no disadvantage if the two vertices would like to match up. A matching $M$ is super-stable if there is no blocking edge with respect to $M$. It has previously…
We analyze the Disjoint Path Allocation problem (DPA) in the priority framework. Motivated by the problem of traffic regulation in communication networks, DPA consists of allocating edge-disjoint paths in a graph. While online algorithms for DPA have been thoroughly studied in the past, we extend the analysis of this optimization problem by considering the more powerful class of priority algorithms. Like an online algorithm, a priority algorithm receives its input only sequentially and must output irrevocable decisions for individual input items before having seen the input in its entirety. Ho…
We consider the following multi-component sparse PCA problem: given a set of data points, we seek to extract a small number of sparse components with disjoint supports that jointly capture the maximum possible variance. These components can be computed one by one, repeatedly solving the single-component problem and deflating the input data matrix, but as we show this greedy procedure is suboptimal. We present a novel algorithm for sparse PCA that jointly optimizes multiple disjoint components. The extracted features capture variance that lies within a multiplicative factor arbitrarily close to…
Let $G$ be a graph and suppose we are given, for each $v \in V(G)$, a strict ordering of the neighbors of $v$. A set of matchings ${\cal M}$ of $G$ is called internally stable if there are no matchings $M,M' \in {\cal M}$ such that an edge of $M$ blocks $M'$. The sets of stable (à la Gale and Shapley) matchings and of von Neumann-Morgenstern stable matchings are examples of internally stable sets of matching. In this paper, we study, in both the marriage and the roommate case, inclusionwise maximal internally stable sets of matchings. We call those sets internally closed. By building on known …
