The maximal matching problem has received considerable attention in the self-stabilizing community. Previous work has given different self-stabilizing algorithms that solves the problem for both the adversarial and fair distributed daemon, the sequential adversarial daemon, as well as the synchronous daemon. In the following we present a single self-stabilizing algorithm for this problem that unites all of these algorithms in that it stabilizes in the same number of moves as the previous best algorithms for the sequential adversarial, the distributed fair, and the synchronous daemon. In addition, the algorithm improves the previous best moves complexities for the distributed adversarial daemon from O(n^2) and O(delta m) to O(m) where n is the number of processes, m is thenumber of edges, and delta is the maximum degree in the graph.
We investigate the distributed complexity of maximal matching and maximal independent set (MIS) in hypergraphs in the LOCAL model. A maximal matching of a hypergraph $H=(V_H,E_H)$ is a maximal disjoint set $M\subseteq E_H$ of hyperedges and an MIS $S\subseteq V_H$ is a maximal set of nodes such that no hyperedge is fully contained in $S$. Both problems can be solved by a simple sequential greedy algorithm, which can be implemented naively in $O(Δr + \log^* n)$ rounds, where $Δ$ is the maximum degree, $r$ is the rank, and $n$ is the number of nodes. We show that for maximal matching, this naive…
Enumerating matchings is a classical problem in the field of enumeration algorithms. There are polynomial-delay enumeration algorithms for several settings, such as enumerating perfect matchings, maximal matchings, and (weighted) matchings in specific orders. In this paper, we present polynomial-delay enumeration algorithms for maximal matchings with cardinality at least given threshold $t$. Our algorithm enumerates all such matchings in $O(nm)$ delay with exponential space, where $n$ and $m$ are the number of vertices and edges of an input graph, respectively. We also present a polynomial-del…
Let G = (A U P, E) be a bipartite graph where A denotes a set of agents, P denotes a set of posts and ranks on the edges denote preferences of the agents over posts. A matching M in G is rank-maximal if it matches the maximum number of applicants to their top-rank post, subject to this, the maximum number of applicants to their second rank post and so on. In this paper, we develop a switching graph characterization of rank-maximal matchings, which is a useful tool that encodes all rank-maximal matchings in an instance. The characterization leads to simple and efficient algorithms for several i…
We consider networks of small, autonomous devices that communicate with each other wirelessly. Minimizing energy usage is an important consideration in designing algorithms for such networks, as battery life is a crucial and limited resource. Working in a model where both sending and listening for messages deplete energy, we consider the problem of finding a maximal matching of the nodes in a radio network of arbitrary and unknown topology. We present a distributed randomized algorithm that produces, with high probability, a maximal matching. The maximum energy cost per node is $O(\log^2 n)$, …
An edge-weighted graph $G=(V,E)$ is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in some interesting game theory problems, such as network bargaining games and cooperative matching games, because they characterize instances which admit stable outcomes. Motivated by this, in the last few years many researchers have investigated the algorithmic problem of turning a given graph into a stable one, via edge- and vertex-removal operations. However, all the algorithmic results developed in the li…
In this paper we study the problem of fully dynamic maximal matching with lookahead. In a fully dynamic $n$-vertex graph setting, we have to handle updates (insertions and removals of edges), and answer queries regarding the current graph, preferably with a better time bound than that when running the trivial deterministic algorithm with worst-case time of $O(m)$ (where $m$ is the all-time maximum number of the edges) and recompute the matching from scratch each time a query arrives. We show that a maximal matching can be maintained in an (undirected) general graph with a deterministic amortiz…
The greedy sequential algorithm for maximal independent set (MIS) loops over the vertices in arbitrary order adding a vertex to the resulting set if and only if no previous neighboring vertex has been added. In this loop, as in many sequential loops, each iterate will only depend directly on a subset of the previous iterates (i.e. knowing that any one of a vertices neighbors is in the MIS or knowing that it has no previous neighbors is sufficient to decide its fate). This leads to a dependence structure among the iterates. If this structure is shallow then running the iterates in parallel whil…
We present a simple randomized reduction from fully-dynamic integral matching algorithms to fully-dynamic "approximately-maximal" fractional matching algorithms. Applying this reduction to the recent fractional matching algorithm of Bhattacharya, Henzinger, and Nanongkai (SODA 2017), we obtain a novel result for the integral problem. Specifically, our main result is a randomized fully-dynamic $(2+ε)$-approximate integral matching algorithm with small polylog worst-case update time. For the $(2+ε)$-approximation regime only a \emph{fractional} fully-dynamic $(2+ε)$-matching algorithm with worst…
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 induced matching in a graph is a set of edges whose endpoints induce a $1$-regular subgraph. It is known that any $n$-vertex graph has at most $10^{n/5} \approx 1.5849^n$ maximal induced matchings, and this bound is best possible. We prove that any $n$-vertex triangle-free graph has at most $3^{n/3} \approx 1.4423^n$ maximal induced matchings, and this bound is attained by any disjoint union of copies of the complete bipartite graph $K_{3,3}$. Our result implies that all maximal induced matchings in an $n$-vertex triangle-free graph can be listed in time $O(1.4423^n)$, yielding the fastest …
The success of modern parallel paradigms such as MapReduce, Hadoop, or Spark, has attracted a significant attention to the Massively Parallel Computation (MPC) model over the past few years, especially on graph problems. In this work, we consider symmetry breaking problems of maximal independent set (MIS) and maximal matching (MM), which are among the most intensively studied problems in distributed/parallel computing, in MPC. These problems are known to admit efficient MPC algorithms if the space per machine is near-linear in $n$, the number of vertices in the graph. This space requirement ho…
We present an algorithm for maintaining maximal matching in a graph under addition and deletion of edges. Our data structure is randomized that takes O(log n) expected amortized time for each edge update where n is the number of vertices in the graph. While there is a trivial O(n) algorithm for edge update, the previous best known result for this problem for a graph with n vertices and m edges is O({(n+ m)}^{0.7072})which is sub-linear only for a sparse graph. For the related problem of maximum matching, Onak and Rubinfield designed a randomized data structure that achieves O(log^2 n) amortize…
In the distributed backup-placement problem each node of a network has to select one neighbor, such that the maximum number of nodes that make the same selection is minimized. This is a natural relaxation of the perfect matching problem, in which each node is selected just by one neighbor. Previous (approximate) solutions for backup placement are non-trivial, even for simple graph topologies, such as dense graphs. In this paper we devise an algorithm for dense graph topologies, including unit disk graphs, unit ball graphs, line graphs, graphs with bounded diversity, and many more. Our algorith…
We present $O(\log\log n)$ round scalable Massively Parallel Computation algorithms for maximal independent set and maximal matching, in trees and more generally graphs of bounded arboricity, as well as for constant coloring trees. Following the standards, by a scalable MPC algorithm, we mean that these algorithms can work on machines that have capacity/memory as small as $n^δ$ for any positive constant $δ<1$. Our results improve over the $O(\log^2\log n)$ round algorithms of Behnezhad et al. [PODC'19]. Moreover, our matching algorithm is presumably optimal as its bound matches an $Ω(\log\log …
We study a combinatorial problem called Minimum Maximal Matching, where we are asked to find in a general graph the smallest that can not be extended. We show that this problem is hard to approximate with a constant smaller than 2, assuming the Unique Games Conjecture. As a corollary we show, that Minimum Maximal Matching in bipartite graphs is hard to approximate with constant smaller than $\frac{4}{3}$, with the same assumption. With a stronger variant of the Unique Games Conjecture --- that is Small Set Expansion Hypothesis --- we are able to improve the hardness result up to the factor of …
A Faster Deterministic Algorithm for Fully Dynamic Maximal Matching
Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；Courant Institute School of Mathematics, Computing, and Data Science, New York University（纽约大学库朗数学科学与技术学院）
本文提出了一种新的确定性框架，用于完全动态最大匹配的验证与维护，实现了 amortized update time 为 n^{1/2+o(1)} 的高效算法。
Given a bipartite graph, where the two sets of vertices are applicants and posts and ranks on the edges represent preferences of applicants over posts, a {\em rank-maximal} matching is one in which the maximum number of applicants is matched to their rank one posts and subject to this condition, the maximum number of applicants is matched to their rank two posts, and so on. We study the dynamic version of the problem in which a new applicant or post may be added to the graph and we would like to maintain a rank-maximal matching. We show that after the arrival of one vertex, we are always able …
In this paper, we consider the problem of computing an optimal matching in a bipartite graph where elements of one side of the bipartition specify preferences over the other side, and one or both sides can have capacities and classifications. The input instance is a bipartite graph G=(A U P,E), where A is a set of applicants, P is a set of posts, and each applicant ranks its neighbors in an order of preference, possibly involving ties. Moreover, each vertex v in A U P has a quota q(v) denoting the maximum number of partners it can have in any allocation of applicants to posts - referred to as …
We consider the problem of matching applicants to posts where applicants have preferences over posts. Thus the input to our problem is a bipartite graph G = (A U P,E), where A denotes a set of applicants, P is a set of posts, and there are ranks on edges which denote the preferences of applicants over posts. A matching M in G is called rank-maximal if it matches the maximum number of applicants to their rank 1 posts, subject to this the maximum number of applicants to their rank 2 posts, and so on. We consider this problem in a dynamic setting, where vertices and edges can be added and deleted…
In this paper we analyze the expected time complexity of the auction algorithm for the matching problem on random bipartite graphs. We prove that the expected time complexity of the auction algorithm for bipartite matching is $O\left(\frac{N\log^2(N)}{\log\left(Np\right)}\right)$ on sequential machines. This is equivalent to other augmenting path algorithms such as the HK algorithm. Furthermore, we show that the algorithm can be implemented on parallel machines with $O(\log(N))$ processors and shared memory with an expected time complexity of $O(N\log(N))$.
We consider manipulation strategies for the rank-maximal matching problem. In the rank-maximal matching problem we are given a bipartite graph $G = (A \cup P, E)$ such that $A$ denotes a set of applicants and $P$ a set of posts. Each applicant $a \in A$ has a preference list over the set of his neighbours in $G$, possibly involving ties. Preference lists are represented by ranks on the edges - an edge $(a,p)$ has rank $i$, denoted as $rank(a,p)=i$, if post $p$ belongs to one of $a$'s $i$-th choices. A rank-maximal matching is one in which the maximum number of applicants is matched to their ra…
We present a new algorithm which detects the maximal possible number of matched disjoint pairs satisfying a given caliper when a bipartite matching is done with respect to a scalar index (e.g., propensity score), and constructs a corresponding matching. Variable width calipers are compatible with the technique, provided that the width of the caliper is a Lipschitz function of the index. If the observations are ordered with respect to the index then the matching needs $O(N)$ operations, where $N$ is the total number of subjects to be matched. The case of 1-to-$n$ matching is also considered. We…
A maximal matching can be maintained in fully dynamic (supporting both addition and deletion of edges) $n$-vertex graphs using a trivial deterministic algorithm with a worst-case update time of O(n). No deterministic algorithm that outperforms the na\"ıve O(n) one was reported up to this date. The only progress in this direction is due to Ivković and Lloyd \cite{IL93}, who in 1993 devised a deterministic algorithm with an \emph{amortized} update time of $O((n+m)^{\sqrt{2}/2})$, where $m$ is the number of edges. In this paper we show the first deterministic fully dynamic algorithm that outperfo…
Despite significant research efforts, the state-of-the-art algorithm for maintaining an approximate matching in fully dynamic graphs has a polynomial {worst-case} update time, even for very poor approximation guarantees. In a recent breakthrough, Bhattacharya, Henzinger and Nanongkai showed how to maintain a constant approximation to the minimum vertex cover, and thus also a constant-factor estimate of the maximum matching size, with polylogarithmic worst-case update time. Later (in SODA'17 Proc.) they improved the approximation factor all the way to $2+ε$. Nevertheless, the longstanding funda…
Baswana, Gupta and Sen [FOCS'11] showed that fully dynamic maximal matching can be maintained in general graphs with logarithmic amortized update time. More specifically, starting from an empty graph on $n$ fixed vertices, they devised a randomized algorithm for maintaining maximal matching over any sequence of $t$ edge insertions and deletions with a total runtime of $O(t \log n)$ in expectation and $O(t \log n + n \log^2 n)$ with high probability. Whether or not this runtime bound can be improved towards $O(t)$ has remained an important open problem. Despite significant research efforts, thi…
$ \renewcommand{\tilde}{\widetilde} $We present an $\tilde{O}(\log^2 n)$ round deterministic distributed algorithm for the maximal independent set problem. By known reductions, this round complexity extends also to maximal matching, $Δ+1$ vertex coloring, and $2Δ-1$ edge coloring. These four problems are among the most central problems in distributed graph algorithms and have been studied extensively for the past four decades. This improved round complexity comes closer to the $\tildeΩ(\log n)$ lower bound of maximal independent set and maximal matching [Balliu et al. FOCS '19]. The previous b…
We present a deterministic distributed algorithm that computes a $(2Δ-1)$-edge-coloring, or even list-edge-coloring, in any $n$-node graph with maximum degree $Δ$, in $O(\log^7 Δ\log n)$ rounds. This answers one of the long-standing open questions of \emph{distributed graph algorithms} from the late 1980s, which asked for a polylogarithmic-time algorithm. See, e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and Elkin. The previous best round complexities were $2^{O(\sqrt{\log n})}$ by Panconesi and Srinivasan [STOC'92] and $\tilde{O}(\sqrtΔ) + O(\log^* n)$ by Fraigniau…
In recent years, significant advances have been made in the design and analysis of fully dynamic maximal matching algorithms. However, these theoretical results have received very little attention from the practical perspective. Few of the algorithms are implemented and tested on real datasets, and their practical potential is far from understood. In this paper, we attempt to bridge the gap between theory and practice that is currently observed for the fully dynamic maximal matching problem. We engineer several algorithms and empirically study those algorithms on an extensive set of dynamic in…
Maximal independent set (MIS), maximal matching (MM), and $(Δ+1)$-coloring in graphs of maximum degree $Δ$ are among the most prominent algorithmic graph theory problems. They are all solvable by a simple linear-time greedy algorithm and up until very recently this constituted the state-of-the-art. In SODA 2019, Assadi, Chen, and Khanna gave a randomized algorithm for $(Δ+1)$-coloring that runs in $\widetilde{O}(n\sqrt{n})$ time, which even for moderately dense graphs is sublinear in the input size. The work of Assadi et al. however contained a spoiler for MIS and MM: neither problems provably…
Many dynamic graph algorithms have an amortized update time, rather than a stronger worst-case guarantee. But amortized data structures are not suitable for real-time systems, where each individual operation has to be executed quickly. For this reason, there exist many recent randomized results that aim to provide a guarantee stronger than amortized expected. The strongest possible guarantee for a randomized algorithm is that it is always correct (Las Vegas), and has high-probability worst-case update time, which gives a bound on the time for each individual operation that holds with high prob…
In the (fully) dynamic set cover problem, we have a collection of $m$ sets from a universe of size $n$ that undergo element insertions and deletions; the goal is to maintain an approximate set cover of the universe after each update. We give an $O(f^2)$ update time algorithm for this problem that achieves an $f$-approximation, where $f$ is the maximum number of sets that an element belongs to; under the unique games conjecture, this approximation is best possible for any fixed $f$. This is the first algorithm for dynamic set cover with approximation ratio that {exactly} matches $f$ (as opposed…
By prior work, there is a distributed algorithm that finds a maximal fractional matching (maximal edge packing) in $O(Δ)$ rounds, where $Δ$ is the maximum degree of the graph. We show that this is optimal: there is no distributed algorithm that finds a maximal fractional matching in $o(Δ)$ rounds. Our work gives the first linear-in-$Δ$ lower bound for a natural graph problem in the standard model of distributed computing---prior lower bounds for a wide range of graph problems have been at best logarithmic in $Δ$.
Multi-pass streaming algorithm for Maximum Matching have been studied since more than 15 years and various algorithmic results are known today, including $2$-pass streaming algorithms that break the $1/2$-approximation barrier, and $(1-ε)$-approximation streaming algorithms that run in $O(\text{poly} \frac{1}ε)$ passes in bipartite graphs and in $O( (\frac{1}ε)^{\frac{1}ε})$ or $O(\text{poly} (\frac{1}ε) \cdot \log n)$ passes in general graphs, where $n$ is the number of vertices of the input graph. However, proving impossibility results for such algorithms has so far been elusive, and, for ex…
The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale parallel computation frameworks and has recently gained a lot of importance, especially in the context of classic graph problems. Unsatisfactorily, all current $\text{poly} (\log \log n)$-round MPC algorithms seem to get fundamentally stuck at the linear-memory barrier: their efficiency crucially relies on each machine having space at least linear in the number $n$ of nodes. As this might not only be prohibitively large, but also allows for easy if not trivial solutions for sparse graphs, w…
We study the problem of finding "fair" stable matchings in the Stable Marriage problem with Incomplete lists (SMI). In particular, we seek stable matchings that are optimal with respect to profile, which is a vector that indicates the number of agents who have their first-, second-, third-choice partner, etc. In a rank maximal stable matching, the maximum number of agents have their first-choice partner, and subject to this, the maximum number of agents have their second-choice partner, etc., whilst in a generous stable matching $M$, the minimum number of agents have their $d$th-choice partner…
The Quantum Alternating Operator Ansatz (QAOA+) framework has recently gained attention due to its ability to solve discrete optimization problems on noisy intermediate-scale quantum (NISQ) devices in a manner that is amenable to derivation of worst-case guarantees. We design a technique in this framework to tackle a few problems over maximal matchings in graphs. Even though maximum matching is polynomial-time solvable, most counting and sampling versions are #P-hard. We design a few algorithms that generates superpositions over matchings allowing us to sample from them. In particular, we get …
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 consider mining dense substructures (maximal cliques) from an uncertain graph, which is a probability distribution on a set of deterministic graphs. For parameter 0 < α < 1, we present a precise definition of an α-maximal clique in an uncertain graph. We present matching upper and lower bounds on the number of α-maximal cliques possible within an uncertain graph. We present an algorithm to enumerate α-maximal cliques in an uncertain graph whose worst-case runtime is near-optimal, and an experimental evaluation showing the practical utility of the algorithm.
In 1976, Knuth asked if the stable marriage problem (SMP) can be generalized to marriages consisting of 3 genders. In 1988, Alkan showed that the natural generalization of SMP to 3 genders ($3$GSM) need not admit a stable marriage. Three years later, Ng and Hirschberg proved that it is NP-complete to determine if given preferences admit a stable marriage. They further prove an analogous result for the $3$ person stable assignment ($3$PSA) problem. In light of Ng and Hirschberg's NP-hardness result for $3$GSM and $3$PSA, we initiate the study of approximate versions of these problems. In partic…
