Let $G$ be a bipartite graph where every node has a strict ranking of its neighbors. For every node, its preferences over neighbors extend naturally to preferences over matchings. Matching $N$ is more popular than matching $M$ if the number of nodes that prefer $N$ to $M$ is more than the number that prefer $M$ to $N$. A maximum matching $M$ in $G$ is a "popular max-matching" if there is no maximum matching in $G$ that is more popular than $M$. Such matchings are relevant in applications where the set of admissible solutions is the set of maximum matchings and we wish to find a best maximum matching as per node preferences. It is known that a popular max-matching always exists in $G$. Here we show a compact extended formulation for the popular max-matching polytope. So when there are edge costs, a min-cost popular max-matching in $G$ can be computed in polynomial time. This is in contrast to the min-cost popular matching problem which is known to be NP-hard. We also consider Pareto-optimality, which is a relaxation of popularity, and show that computing a min-cost Pareto-optimal matching/max-matching is NP-hard.
We consider the problem of computing popular matchings in a bipartite graph G = (R U H, E) where R and H denote a set of residents and a set of hospitals respectively. Each hospital h has a positive capacity denoting the number of residents that can be matched to h. The residents and the hospitals specify strict preferences over each other. This is the well-studied Hospital Residents (HR) problem which is a generalization of the Stable Marriage (SM) problem. The goal is to assign residents to hospitals optimally while respecting the capacities of the hospitals. Stability is a well-accepted not…
We consider an extension of the {\em popular matching} problem in this paper. The input to the popular matching problem is a bipartite graph G = (A U B,E), where A is a set of people, B is a set of items, and each person a belonging to A ranks a subset of items in an order of preference, with ties allowed. The popular matching problem seeks to compute a matching M* between people and items such that there is no matching M where more people are happier with M than with M*. Such a matching M* is called a popular matching. However, there are simple instances where no popular matching exists. Here…
Let $G = (A \cup B,E)$ be a bipartite graph where the set $A$ consists of agents or main players and the set $B$ consists of jobs or secondary players. Every vertex has a strict ranking of its neighbors. A matching $M$ is popular if for any matching $N$, the number of vertices that prefer $M$ to $N$ is at least the number that prefer $N$ to $M$. Popular matchings always exist in $G$ since every stable matching is popular. A matching $M$ is $A$-popular if for any matching $N$, the number of agents (i.e., vertices in $A$) that prefer $M$ to $N$ is at least the number of agents that prefer $N$ to…
Several papers have achieved time $O(\sqrt n m)$ for cardinality matching, starting from first principles. This results in a long derivation. We simplify the task by employing well-known concepts for maximum weight matching. We use Edmonds' algorithm to derive the structure of shortest augmenting paths. We extend this to a complete algorithm for maximum cardinality matching in time $O(\sqrt n m)$.
We present a polylogarithmic local computation matching algorithm which guarantees a $(1-\eps)$-approximation to the maximum matching in graphs of bounded degree.
We consider the problem of computing a matching in a bipartite graph in the presence of one-sided preferences. There are several well studied notions of optimality which include pareto optimality, rank maximality, fairness and popularity. In this paper, we conduct an in-depth experimental study comparing different notions of optimality based on a variety of metrics like cardinality, number of rank-1 edges, popularity, to name a few. Observing certain shortcomings in the standard notions of optimality, we propose an algorithm which maximizes an alternative metric called the Area under Profile C…
We propose a new greedy algorithm for the maximum cardinality matching problem. We give experimental evidence that this algorithm is likely to find a maximum matching in random graphs with constant expected degree c>0, independent of the value of c. This is contrary to the behavior of commonly used greedy matching heuristics which are known to have some range of c where they probably fail to compute a maximum matching.
Suppose that each member of a set of agents has a preference list of a subset of houses, possibly involving ties and each agent and house has their capacity denoting the maximum number of correspondingly agents/houses that can be matched to him/her/it. We want to find a matching $M$, for which there is no other matching $M'$ such that more agents prefer $M'$ to $M$ than $M$ to $M'$. (What it means that an agent prefers one matching to the other is explained in the paper.) Popular matchings have been studied quite extensively, especially in the one-to-one setting. We provide a characterization …
In this paper, we have developed a fully-dynamic algorithm for maintaining cardinality of maximum-matching in a tree using the construction of top-trees. The time complexities are as follows: 1. Initialization Time: $O(n(log(n)))$ to build the Top-tree. 2. Update Time: $O(log(n))$ 3. Query Time: O(1) to query the cardinality of maximum-matching and $O(log(n))$ to find if a particular edge is matched.
We present a fully dynamic algorithm for maintaining approximate maximum weight matching in general weighted graphs. The algorithm maintains a matching ${\cal M}$ whose weight is at least $1/8 M^{*}$ where $M^{*}$ is the weight of the maximum weight matching. The algorithm achieves an expected amortized $O(\log n \log \mathcal C)$ time per edge insertion or deletion, where $\mathcal C$ is the ratio of the weights of the highest weight edge to the smallest weight edge in the given graph. Using a simple randomized scaling technique, we are able to obtain a matching whith expected approximation r…
We show that there exist linear-time algorithms that compute the strong chromatic index and a maximum induced matching of tree-cographs when the decomposition tree is a part of the input. We also show that there exists an efficient algorithm for the strong chromatic index of permutation graphs.
The problem of finding a maximum size matching in a graph (known as the maximum matching problem) is one of the most classical problems in computer science. Despite a significant body of work dedicated to the study of this problem in the data stream model, the state-of-the-art single-pass semi-streaming algorithm for it is still a simple greedy algorithm that computes a maximal matching, and this way obtains 1/2-approximation. Some previous works described two/three-pass algorithms that improve over this approximation ratio by using their second and third passes to improve the above mentioned …
An $(f,g)$-semi-matching in a bipartite graph $G=(U \cup V,E)$ is a set of edges $M \subseteq E$ such that each vertex $u\in U$ is incident with at most $f(u)$ edges of $M$, and each vertex $v\in V$ is incident with at most $g(v)$ edges of $M$. In this paper we give an algorithm that for a graph with $n$ vertices and $m$ edges, $n\le m$, constructs a maximum $(f,g)$-semi-matching in running time $O(m\cdot \min (\sqrt{\sum_{u\in U}f(u)}, \sqrt{\sum_{v\in V}g(v)}))$. Using the reduction of [5], our result on maximum $(f,g)$-semi-matching problem directly implies an algorithm for the optimal semi…
Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph problems. For general m-edge and n-vertex graphs, it is well-known to be solvable in $O(m \sqrt{n})$ time. We develop a linear-time algorithm to find maximum-cardinality matchings on cocomparability graphs, a prominent subclass of perfect graphs that contains interval graphs as well as permutation graphs. Our algorithm is based on the recently discovered Lexicographic Depth First Search (LDFS).
The {\em maximum cardinality} and {\em maximum weight matching} problems can be solved in time $\tilde{O}(m\sqrt{n})$, a bound that has resisted improvement despite decades of research. (Here $m$ and $n$ are the number of edges and vertices.) In this article we demonstrate that this "$m\sqrt{n}$ barrier" is extremely fragile, in the following sense. For any $ε>0$, we give an algorithm that computes a $(1-ε)$-approximate maximum weight matching in $O(mε^{-1}\logε^{-1})$ time, that is, optimal {\em linear time} for any fixed $ε$. Our algorithm is dramatically simpler than the best exact maximum …
Let G be an edge-weighted hypergraph on n vertices, m edges of size \le s, where the edges have real weights in an interval [1,W]. We show that if we can approximate a maximum weight matching in G within factor alpha in time T(n,m,W) then we can find a matching of weight at least (alpha-epsilon) times the maximum weight of a matching in G in time (epsilon^{-1})^{O(1)}max_{1\le q \le O(epsilon \frac {log {\frac n {epsilon}}} {log epsilon^{-1}})} max_{m_1+...m_q=m} sum_1^qT(min{n,sm_j},m_{j},(epsilon^{-1})^{O(epsilon^{-1})}). In particular, if we combine our result with the recent (1-ε)-approxim…
We consider geometric instances of the Maximum Weighted Matching Problem (MWMP) and the Maximum Traveling Salesman Problem (MTSP) with up to 3,000,000 vertices. Making use of a geometric duality relationship between MWMP, MTSP, and the Fermat-Weber-Problem (FWP), we develop a heuristic approach that yields in near-linear time solutions as well as upper bounds. Using various computational tools, we get solutions within considerably less than 1% of the optimum. An interesting feature of our approach is that, even though an FWP is hard to compute in theory and Edmonds' algorithm for maximum weigh…
In this paper we revisit the problem of constructing randomized composable coresets for bipartite matching. In this problem the input graph is randomly partitioned across $k$ players, each of which sends a single message to a coordinator, who then must output a good approximation to the maximum matching in the input graph. Assadi and Khanna gave the first such coreset, achieving a $1/9$-approximation by having every player send a maximum matching, i.e. at most $n/2$ words per player. The approximation factor was improved to $1/3$ by Bernstein et al. In this paper, we show that the matching ske…
We present a simple deterministic single-pass $(2+ε)$-approximation algorithm for the maximum weight matching problem in the semi-streaming model. This improves upon the currently best known approximation ratio of $(4+ε)$. Our algorithm uses $O(n\log^2 n)$ bits of space for constant values of $ε$. It relies on a variation of the local-ratio theorem, which may be of use for other algorithms in the semi-streaming model as well.
We focus on belief propagation for the assignment problem, also known as the maximum weight bipartite matching problem. We provide a constructive proof that the well-known upper bound on the number of iterations (Bayati, Shah, Sharma 2008) is tight up to a factor of four. Furthermore, we investigate the behavior of belief propagation when convergence is not required. We show that the number of iterations required for a sharp approximation consumes a large portion of the convergence time. Finally, we propose an "approximate belief propagation" algorithm for the assignment problem.
In this paper, we propose a depth-first search (DFS) algorithm for searching maximum matchings in general graphs. Unlike blossom shrinking algorithms, which store all possible alternative alternating paths in the super-vertices shrunk from blossoms, the newly proposed algorithm does not involve blossom shrinking. The basic idea is to deflect the alternating path when facing blossoms. The algorithm maintains detour information in an auxiliary stack to minimize the redundant data structures. A benefit of our technique is to avoid spending time on shrinking and expanding blossoms. This DFS algori…
We initiate the study of the Diverse Pair of (Maximum/ Perfect) Matchings problems which given a graph $G$ and an integer $k$, ask whether $G$ has two (maximum/perfect) matchings whose symmetric difference is at least $k$. Diverse Pair of Matchings (asking for two not necessarily maximum or perfect matchings) is NP-complete on general graphs if $k$ is part of the input, and we consider two restricted variants. First, we show that on bipartite graphs, the problem is polynomial-time solvable, and second we show that Diverse Pair of Maximum Matchings is FPT parameterized by $k$. We round off the …
We consider the communication complexity of finding an approximate maximum matching in a graph in a multi-party message-passing communication model. The maximum matching problem is one of the most fundamental graph combinatorial problems, with a variety of applications. The input to the problem is a graph $G$ that has $n$ vertices and the set of edges partitioned over $k$ sites, and an approximation ratio parameter $α$. The output is required to be a matching in $G$ that has to be reported by one of the sites, whose size is at least factor $α$ of the size of a maximum matching in $G$. We show …
In this paper, we study the problem of finding a maximum matching in the semi-streaming model when edges arrive in a random order. In the semi-streaming model, an algorithm receives a stream of edges and it is allowed to have a memory of $\tilde{O}(n)$ where $n$ is the number of vertices in the graph. A recent inspiring work by Assadi et al. shows that there exists a streaming algorithm with the approximation ratio of $\frac{2}{3}$ that uses $\tilde{O}(n^{1.5})$ memory. However, the memory of their algorithm is much larger than the memory constraint of the semi-streaming algorithms. In this wo…
We present a randomized algorithm to maintain a maximal matching without 3 length augmenting paths in the fully dynamic setting. Consequently, we maintain a $3/2$ approximate maximum cardinality matching. Our algorithm takes expected amortized $O(\sqrt{n})$ time where $n$ is the number of vertices in the graph when the update sequence is generated by an oblivious adversary. Over any sequence of $t$ edge insertions and deletions presented by an oblivious adversary, the total update time of our algorithm is $O(t\sqrt{n})$ in expectation and $O(t\sqrt{n} + n \log n)$ with high probability. To the…
We consider the problem of estimating the weight of a maximum weighted matching of a weighted graph $G(V,E)$ whose edges are revealed in a streaming fashion. We develop a reduction from the maximum weighted matching problem to the maximum cardinality matching problem that only doubles the approximation factor of a streaming algorithm developed for the maximum cardinality matching problem. Our results hold for the insertion-only and the dynamic (i.e, insertion and deletion) edge-arrival streaming models. The previous best-known reduction is due to Bury and Schwiegelshohn (ESA 2015) who develop …
In this paper, we study linear programming based approaches to the maximum matching problem in the semi-streaming model. The semi-streaming model has gained attention as a model for processing massive graphs as the importance of such graphs has increased. This is a model where edges are streamed-in in an adversarial order and we are allowed a space proportional to the number of vertices in a graph. In recent years, there has been several new results in this semi-streaming model. However broad techniques such as linear programming have not been adapted to this model. We present several techniqu…
We consider the problem of approximating a maximum weighted matching, when the edges of an underlying weighted graph $G(V,E)$ are revealed in a streaming fashion. We analyze a variant of the previously best-known $(4+ε)$-approximation algorithm due to Crouch and Stubbs (APPROX, 2014), and prove their conjecture that it achieves a tight approximation factor of $3.5+ε$. The algorithm splits the stream into substreams on which it runs a greedy maximum matching algorithm. At the end of the stream, the selected edges are given as input to an optimal maximum weighted matching algorithm. To analyze t…
Finding a maximum-cardinality or maximum-weight matching in (edge-weighted) undirected graphs is among the most prominent problems of algorithmic graph theory. For $n$-vertex and $m$-edge graphs, the best known algorithms run in $\widetilde{O}(m\sqrt{n})$ time. We build on recent theoretical work focusing on linear-time data reduction rules for finding maximum-cardinality matchings and complement the theoretical results by presenting and analyzing (thereby employing the kernelization methodology of parameterized complexity analysis) new (near-)linear-time data reduction rules for both the unwe…
In the Student / Project Allocation problem (SPA) we seek to assign students to individual or group projects offered by lecturers. Students provide a list of projects they find acceptable in order of preference. Each student can be assigned to at most one project and there are constraints on the maximum number of students that can be assigned to each project and lecturer. We seek matchings of students to projects that are optimal with respect to profile, which is a vector whose rth component indicates how many students have their rth-choice project. We present an efficient algorithm for findin…
In this paper, we study the non-bipartite maximum matching problem in the semi-streaming model. The maximum matching problem in the semi-streaming model has received a significant amount of attention lately. While the problem has been somewhat well solved for bipartite graphs, the known algorithms for non-bipartite graphs use $2^{\frac1ε}$ passes or $n^{\frac1ε}$ time to compute a $(1-ε)$ approximation. In this paper we provide the first FPTAS (polynomial in $n,\frac1ε$) for the problem which is efficient in both the running time and the number of passes. We also show that we can estimate the …
In this paper we consider graph algorithms in models of computation where the space usage (random accessible storage, in addition to the read only input) is sublinear in the number of edges $m$ and the access to input data is constrained. These questions arises in many natural settings, and in particular in the analysis of MapReduce or similar algorithms that model constrained parallelism with sublinear central processing. In SPAA 2011, Lattanzi etal. provided a $O(1)$ approximation of maximum matching using $O(p)$ rounds of iterative filtering via mapreduce and $O(n^{1+1/p})$ space of central…
Let c denote a non-negative constant. Suppose that we are given an edge-weighted bipartite graph G=(V,E) with its 2-layered drawing and a family X of intersecting edge pairs. We consider the problem of finding a maximum weighted matching M* such that each edge in M* intersects with at most c other edges in M*, and that all edge crossings in M* are contained in X. In the present paper, we propose polynomial-time algorithms for the cases of c=1 and 2. The time complexities of the algorithms are O((k+m)log n+n) for c=1 and O(k^3+k^2n+m(m+log n)) for c=2, respectively, where n=|V|, m=|E| and k=|X|…
This paper addresses the problem of maximizing the expected size of a matching in the case of unreliable vertices and/or edges. The assumption is that upon failure, remaining vertices that have not been matched may be subject to a new assignment. This process may be repeated a given number of times, and the objective is to end with the overall maximum number of matched vertices. The origin of this problem is in kidney exchange programs, going on in several countries, where a vertex is an incompatible patient-donor pair; the objective is to match these pairs so as to maximize the number of serv…
We study the maximum induced matching problem on a graph g. Induced matchings correspond to independent sets in L2(g), the square of the line graph of g. The problem is NP-complete on bipartite graphs. In this work, we show that for a number of graph families with forbidden vertex orderings, almost all forbidden patterns on three vertices are preserved when taking the square of the line graph. These orderings can be computed in linear time in the size of the input graph. In particular, given a graph class G characterized by a vertex ordering, and a graph g = (V, E) in G with a corresponding ve…
We study the problem of matching agents who arrive at a marketplace over time and leave after d time periods. Agents can only be matched while they are present in the marketplace. Each pair of agents can yield a different match value, and the planner's goal is to maximize the total value over a finite time horizon. First we study the case in which vertices arrive in an adversarial order. We provide a randomized 0.25-competitive algorithm building on a result by Feldman et al. (2009) and Lehman et al. (2006). We extend the model to the case in which departure times are drawn independently from …
This paper presents an algorithm for estimating the weight of a maximum weighted matching by augmenting any estimation routine for the size of an unweighted matching. The algorithm is implementable in any streaming model including dynamic graph streams. We also give the first constant estimation for the maximum matching size in a dynamic graph stream for planar graphs (or any graph with bounded arboricity) using $\tilde{O}(n^{4/5})$ space which also extends to weighted matching. Using previous results by Kapralov, Khanna, and Sudan (2014) we obtain a $\mathrm{polylog}(n)$ approximation for gen…
We consider the sensitivity of algorithms for the maximum matching problem against edge and vertex modifications. Algorithms with low sensitivity are desirable because they are robust to edge failure or attack. In this work, we show a randomized $(1-ε)$-approximation algorithm with worst-case sensitivity $O_ε(1)$, which substantially improves upon the $(1-ε)$-approximation algorithm of Varma and Yoshida (arXiv 2020) that obtains average sensitivity $n^{O(1/(1+ε^2))}$ sensitivity algorithm, and show a deterministic $1/2$-approximation algorithm with sensitivity $\exp(O(\log^*n))$ for bounded-de…
We consider the unweighted bipartite maximum matching problem in the one-pass turnstile streaming model where the input stream consists of edge insertions and deletions. In the insertion-only model, a one-pass $2$-approximation streaming algorithm can be easily obtained with space $O(n \log n)$, where $n$ denotes the number of vertices of the input graph. We show that no such result is possible if edge deletions are allowed, even if space $O(n^{3/2-δ})$ is granted, for every $δ> 0$. Specifically, for every $0 \le ε\le 1$, we show that in the one-pass turnstile streaming model, in order to comp…
Let G be a bipartite graph with positive integer weights on the edges and without isolated nodes. Let n, N and W be the node count, the largest edge weight and the total weight of G. Let k(x,y) be log(x)/log(x^2/y). We present a new decomposition theorem for maximum weight bipartite matchings and use it to design an O(sqrt(n)W/k(n,W/N))-time algorithm for computing a maximum weight matching of G. This algorithm bridges a long-standing gap between the best known time complexity of computing a maximum weight matching and that of computing a maximum cardinality matching. Given G and a maximum wei…
The approach mapping from a matching of bipartite graphs to digraphs has been successfully used for forcing set problem, in this paper, it is extended to uniquely restricted matching problem. We show to determine a uniquely restricted matching in a bipartite graph is equivalent to recognition a acyclic digraph. Based on these results, it proves that determine the bipartite graphs with all maximum matching are uniquely restricted is polynomial time. This answers an open question of Levit and Mandrescu(Discrete Applied Mathematics 132(2004) 163-164).
We study the maximum cardinality matching problem in a standard distributed setting, where the nodes $V$ of a given $n$-node network graph $G=(V,E)$ communicate over the edges $E$ in synchronous rounds. More specifically, we consider the distributed CONGEST model, where in each round, each node of $G$ can send an $O(\log n)$-bit message to each of its neighbors. We show that for every graph $G$ and a matching $M$ of $G$, there is a randomized CONGEST algorithm to verify $M$ being a maximum matching of $G$ in time $O(|M|)$ and disprove it in time $O(D + \ell)$, where $D$ is the diameter of $G$ …
In a graph where vertices have preferences over their neighbors, a matching is called popular if it does not lose a head-to-head election against any other matching when the vertices vote between the matchings. Popular matchings can be seen as an intermediate category between stable matchings and maximum-size matchings. In this paper, we aim to maximize the utility of a matching that is popular but admits only a few blocking edges. For general graphs already finding a popular matching with at most one blocking edge is NP-complete. For bipartite instances, we study the problem of finding a maxi…
The Maximum Induced Matching problem asks to find the maximum $k$ such that, given a graph $G=(V,E)$, can we find a subset of vertices $S$ of size $k$ for which every vertices $v$ in the induced graph $G[S]$ has exactly degree $1$. In this paper, we design an exact algorithm running in $O(1.2630^n)$ time and polynomial space to solve the Maximum Induced Matching problem for graphs where each vertex has degree at most 3. Prior work solved the problem by finding the Maximum Independent Set using polynomial space in the line graph $L(G^2)$; this method uses $O(1.3139^n)$ time.
We design quantum algorithms for maximum matching. Working in the query model, in both adjacency matrix and adjacency list settings, we improve on the best known algorithms for general graphs, matching previously obtained results for bipartite graphs. In particular, for a graph with $n$ nodes and $m$ edges, our algorithm makes $O(n^{7/4})$ queries in the matrix model and $O(n^{3/4}(m+n)^{1/2})$ queries in the list model. Our approach combines Gabow's classical maximum matching algorithm [Gabow, Fundamenta Informaticae, '17] with the guessing tree method of Beigi and Taghavi [Beigi and Taghavi,…
We reduce the problem of finding an augmenting path in a general graph to a reachability problem in a directed bipartite graph. A slight modification of depth-first search leads to an algorithm for finding such paths. Although this setting is equivalent to the traditional terminology of blossoms due to Edmonds, there are some advantages. Mainly, this point of view enables the description of algorithms for the solution of matching problems without explicit analysis of blossoms, nested blossoms, and so on. Exemplary, we describe an efficient realization of the Hopcroft-Karp approach for the comp…
We prove a lower bound on the space complexity of two-pass semi-streaming algorithms that approximate the maximum matching problem. The lower bound is parameterized by the density of Ruzsa-Szemeredi graphs: * Any two-pass semi-streaming algorithm for maximum matching has approximation ratio at least $(1- Ω(\frac{\log{RS(n)}}{\log{n}}))$, where $RS(n)$ denotes the maximum number of induced matchings of size $Θ(n)$ in any $n$-vertex graph, i.e., the largest density of a Ruzsa-Szemeredi graph. Currently, it is known that $n^{Ω(1/\!\log\log{n})} \leq RS(n) \leq \frac{n}{2^{O(\log^*{\!(n)})}}$ and …
Maximum bipartite matching is a fundamental algorithmic problem which can be solved in polynomial time. We consider a natural variant in which there is a separation constraint: the vertices on one side lie on a path or a grid, and two vertices that are close to each other are not allowed to be matched simultaneously. We show that the problem is hard to approximate even for paths, and provide constant-factor approximation algorithms for both paths and grids.
