For over a decade now we have been witnessing the success of {\em massive parallel computation} (MPC) frameworks, such as MapReduce, Hadoop, Dryad, or Spark. One of the reasons for their success is the fact that these frameworks are able to accurately capture the nature of large-scale computation. In particular, compared to the classic distributed algorithms or PRAM models, these frameworks allow for much more local computation. The fundamental question that arises in this context is though: can we leverage this additional power to obtain even faster parallel algorithms?
A prominent example here is the {\em maximum matching} problem---one of the most classic graph problems. It is well known that in the PRAM model one can compute a 2-approximate maximum matching in $O(\log{n})$ rounds. However, the exact complexity of this problem in the MPC framework is still far from understood. Lattanzi et al. showed that if each machine has $n^{1+Ω(1)}$ memory, this problem can also be solved $2$-approximately in a constant number of rounds. These techniques, as well as the approaches developed in the follow up work, seem though to get stuck in a fundamental way at roughly $O(\log{n})$ rounds once we enter the near-linear memory regime. It is thus entirely possible that in this regime, which captures in particular the case of sparse graph computations, the best MPC round complexity matches what one can already get in the PRAM model, without the need to take advantage of the extra local computation power.
In this paper, we finally refute that perplexing possibility. That is, we break the above $O(\log n)$ round complexity bound even in the case of {\em slightly sublinear} memory per machine. In fact, our improvement here is {\em almost exponential}: we are able to deliver a $(2+ε)$-approximation to maximum matching, for any fixed constant $ε>0$, in $O((\log \log n)^2)$ rounds.
Recently, Czumaj et.al. (arXiv 2017) presented a parallel (almost) $2$-approximation algorithm for the maximum matching problem in only $O({(\log\log{n})^2})$ rounds of the massive parallel computation (MPC) framework, when the memory per machine is $O(n)$. The main approach in their work is a way of compressing $O(\log{n})$ rounds of a distributed algorithm for maximum matching into only $O({(\log\log{n})^2})$ MPC rounds. In this note, we present a similar algorithm for the closely related problem of approximating the minimum vertex cover in the MPC framework. We show that one can achieve an …
The Massive Parallel Computation (MPC) model is a theoretical framework for popular parallel and distributed platforms such as MapReduce, Hadoop, or Spark. We consider the task of computing a large matching or small vertex cover in this model when the space per machine is $n^δ$ for $δ\in (0,1)$, where $n$ is the number of vertices in the input graph. A direct simulation of classic PRAM and distributed algorithms from the 1980s results in algorithms that require at least a logarithmic number of MPC rounds. We give the first algorithm that breaks this logarithmic barrier and runs in $\tilde O(\s…
We present $O(\log\log n)$-round algorithms in the Massively Parallel Computation (MPC) model, with $\tilde{O}(n)$ memory per machine, that compute a maximal independent set, a $1+ε$ approximation of maximum matching, and a $2+ε$ approximation of minimum vertex cover, for any $n$-vertex graph and any constant $ε>0$. These improve the state of the art as follows: - Our MIS algorithm leads to a simple $O(\log\log Δ)$-round MIS algorithm in the Congested Clique model of distributed computing, which improves on the $\tilde{O}(\sqrt{\log Δ})$-round algorithm of Ghaffari [PODC'17]. - Our $O(\log\log…
For numerous online bipartite matching problems, such as edge-weighted matching and matching under two-sided vertex arrivals, the state-of-the-art fractional algorithms outperform their randomized integral counterparts. This gap is surprising, given that the bipartite fractional matching polytope is integral, and so lossless rounding is possible. This gap was explained by Devanur et al.~(SODA'13), who showed that \emph{online} lossless rounding is impossible. Despite the above, we initiate the study of lossless online rounding for online bipartite matching problems. Our key observation is that…
Graphics Processing Units allow for running massively parallel applications offloading the CPU from computationally intensive resources, however GPUs have a limited amount of memory. In this paper a trie compression algorithm for massively parallel pattern matching is presented demonstrating 85% less space requirements than the original highly efficient parallel failure-less aho-corasick, whilst demonstrating over 22 Gbps throughput. The algorithm presented takes advantage of compressed row storage matrices as well as shared and texture memory on the GPU.
We present improved deterministic distributed algorithms for a number of well-studied matching problems, which are simpler, faster, more accurate, and/or more general than their known counterparts. The common denominator of these results is a deterministic distributed rounding method for certain linear programs, which is the first such rounding method, to our knowledge. A sampling of our end results is as follows. -- An $O(\log^2 Δ\cdot \log n)$-round deterministic distributed algorithm for computing a maximal matching, in $n$-node graphs with maximum degree $Δ$. This is the first improvement …
We present a simple distributed $Δ$-approximation algorithm for maximum weight independent set (MaxIS) in the $\mathsf{CONGEST}$ model which completes in $O(\texttt{MIS}(G)\cdot \log W)$ rounds, where $Δ$ is the maximum degree, $\texttt{MIS}(G)$ is the number of rounds needed to compute a maximal independent set (MIS) on $G$, and $W$ is the maximum weight of a node. %Whether our algorithm is randomized or deterministic depends on the \texttt{MIS} algorithm used as a black-box. Plugging in the best known algorithm for MIS gives a randomized solution in $O(\log n \log W)$ rounds, where $n$ is th…
We present a framework for deterministically rounding a dynamic fractional matching. Applying our framework in a black-box manner on top of existing fractional matching algorithms, we derive the following new results: (1) The first deterministic algorithm for maintaining a $(2-δ)$-approximate maximum matching in a fully dynamic bipartite graph, in arbitrarily small polynomial update time. (2) The first deterministic algorithm for maintaining a $(1+δ)$-approximate maximum matching in a decremental bipartite graph, in polylogarithmic update time. (3) The first deterministic algorithm for maintai…
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…
$ \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 new dynamic matching sparsification scheme. From this scheme we derive a framework for dynamically rounding fractional matchings against \emph{adaptive adversaries}. Plugging in known dynamic fractional matching algorithms into our framework, we obtain numerous randomized dynamic matching algorithms which work against adaptive adversaries (the first such algorithms, as all previous randomized algorithms for this problem assumed an \emph{oblivious} adversary). In particular, for any constant $ε>0$, our framework yields $(2+ε)$-approximate algorithms with constant update time or pol…
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…
We develop a new approach for distributed distance computation in planar graphs that is based on a variant of the metric compression problem recently introduced by Abboud et al. [SODA'18]. One of our key technical contributions is in providing a compression scheme that encodes all $S \times T$ distances using $\widetilde{O}(|S|\cdot poly(D)+|T|)$ bits for unweighted graphs with diameter $D$. This significantly improves the state of the art of $\widetilde{O}(|S|\cdot 2^{D}+|T| \cdot D)$ bits. We also consider an approximate version of the problem for \emph{weighted} graphs, where the goal is to…
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 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 this paper, we reduce the maximum weighted matching problem to the largest cardinality matching in {\bf CONGEST}. The paper presents two technical contributions. The first of them is a simple $poly(\log n, \frac{1}{\varepsilon}, t, \ln w_t)$-round {\bf CONGEST} algorithm for reducing the maximum weighted matching problem to the largest cardinality matching problem. This is achieved under the assumption that all vertices know all edge-weights $\{w_1,....,w_t\}$ (in particular, they know $t$, the number of different edge-weights), though a particular vertex may not know the weight of a partic…
This paper presents an $O(\log\log \bar{d})$ round massively parallel algorithm for $1+ε$ approximation of maximum weighted $b$-matchings, using near-linear memory per machine. Here $\bar{d}$ denotes the average degree in the graph and $ε$ is an arbitrarily small positive constant. Recall that $b$-matching is the natural and well-studied generalization of the matching problem where different vertices are allowed to have multiple (and differing number of) incident edges in the matching. Concretely, each vertex $v$ is given a positive integer budget $b_v$ and it can have up to $b_v$ incident edg…
$ $We study the $d$-Uniform Hypergraph Matching ($d$-UHM) problem: given an $n$-vertex hypergraph $G$ where every hyperedge is of size $d$, find a maximum cardinality set of disjoint hyperedges. For $d\geq3$, the problem of finding the maximum matching is NP-complete, and was one of Karp's 21 $\mathcal{NP}$-complete problems. In this paper we are interested in the problem of finding matchings in hypergraphs in the massively parallel computation (MPC) model that is a common abstraction of MapReduce-style computation. In this model, we present the first three parallel algorithms for $d$-Uniform …
We develop a general deterministic distributed method for locally rounding fractional solutions of graph problems for which the analysis can be broken down into analyzing pairs of vertices. Roughly speaking, the method can transform fractional/probabilistic label assignments of the vertices into integral/deterministic label assignments for the vertices, while approximately preserving a potential function that is a linear combination of functions, each of which depends on at most two vertices (subject to some conditions usually satisfied in pairwise analyses). The method unifies and significant…
As massive graphs become more prevalent, there is a rapidly growing need for scalable algorithms that solve classical graph problems, such as maximum matching and minimum vertex cover, on large datasets. For massive inputs, several different computational models have been introduced, including the streaming model, the distributed communication model, and the massively parallel computation (MPC) model that is a common abstraction of MapReduce-style computation. In each model, algorithms are analyzed in terms of resources such as space used or rounds of communication needed, in addition to the m…
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…
This paper gives poly-logarithmic-round, distributed D-approximation algorithms for covering problems with submodular cost and monotone covering constraints (Submodular-cost Covering). The approximation ratio D is the maximum number of variables in any constraint. Special cases include Covering Mixed Integer Linear Programs (CMIP), and Weighted Vertex Cover (with D=2). Via duality, the paper also gives poly-logarithmic-round, distributed D-approximation algorithms for Fractional Packing linear programs (where D is the maximum number of constraints in which any variable occurs), and for Max Wei…
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 consider the following matching-based routing problem. Initially, each vertex $v$ of a connected graph $G$ is occupied by a pebble which has a unique destination $π(v)$. In each round the pebbles across the edges of a selected matching in $G$ are swapped, and the goal is to route each pebble to its destination vertex in as few rounds as possible. We show that if $G$ is a sufficiently strong $d$-regular spectral expander then any permutation $π$ can be achieved in $O(\log n)$ rounds. This is optimal for constant $d$ and resolves a problem of Alon, Chung, and Graham [SIAM J. Discrete Math., 7…
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…
We design a generic method for reducing the task of finding weighted matchings to that of finding short augmenting paths in unweighted graphs. This method enables us to provide efficient implementations for approximating weighted matchings in the streaming model and in the massively parallel computation (MPC) model. In the context of streaming with random edge arrivals, our techniques yield a $(1/2+c)$-approximation algorithm thus breaking the natural barrier of $1/2$. For multi-pass streaming and the MPC model, we show that any algorithm computing a $(1-δ)$-approximate unweighted matching in …
We give efficient distributed algorithms for the minimum vertex cover problem in bipartite graphs in the CONGEST model. From Kőnig's theorem, it is well known that in bipartite graphs the size of a minimum vertex cover is equal to the size of a maximum matching. We first show that together with an existing $O(n\log n)$-round algorithm for computing a maximum matching, the constructive proof of Kőnig's theorem directly leads to a deterministic $O(n\log n)$-round CONGEST algorithm for computing a minimum vertex cover. We then show that by adapting the construction, we can also convert an \emph{a…
Iterative rounding has enjoyed tremendous success in elegantly resolving open questions regarding the approximability of problems dominated by covering constraints. Although iterative rounding methods have been applied to packing problems, no single method has emerged that matches the effectiveness and simplicity afforded by the covering case. We offer a simple iterative packing technique that retains features of Jain's seminal approach, including the property that the magnitude of the fractional value of the element rounded during each iteration has a direct impact on the approximation guaran…
Graphs can be used to represent a wide variety of data belonging to different domains. Graphs can capture the relationship among data in an efficient way, and have been widely used. In recent times, with the advent of Big Data, there has been a need to store and compute on large data sets efficiently. However, considering the size of the data sets in question, finding optimal methods to store and process the data has been a challenge. Therefore, in this paper, we study different graph compression techniques and propose novel algorithms to do the same. Specifically, given a graph G = (V, E), wh…
We study dynamic graph algorithms in the Massively Parallel Computation model, which was inspired by practical data processing systems. Our goal is to provide algorithms that can efficiently handle large batches of edge insertions and deletions. We show algorithms that require fewer rounds to update a solution to problems such as Minimum Spanning Forest, 2-Edge Connected Components, and Maximal Matching than would be required by their static counterparts to compute it from scratch. They work in the most restrictive memory regime, in which local memory per machine is strongly sublinear in the n…
We introduce a `concrete complexity' model for studying algorithms for matching in bipartite graphs. The model is based on the "demand query" model used for combinatorial auctions. Most (but not all) known algorithms for bipartite matching seem to be translatable into this model including exact, approximate, sequential, parallel, and online ones. A perfect matching in a bipartite graph can be found in this model with O(n^{3/2}) demand queries (in a bipartite graph with n vertices on each side) and our main open problem is to either improve the upper bound or prove a lower bound. An improved up…
Relaxation and rounding approaches became a standard and extremely versatile tool for constrained submodular function maximization. One of the most common rounding techniques in this context are contention resolution schemes. Such schemes round a fractional point by first rounding each coordinate independently, and then dropping some elements to reach a feasible set. Also the second step, where elements are dropped, is typically randomized. This leads to an additional source of randomization within the procedure, which can complicate the analysis. We suggest a different, polyhedral viewpoint t…
This paper studies the online correlated selection (OCS) problem. It was introduced by Fahrbach, Huang, Tao, and Zadimoghaddam (2020) to obtain the first edge-weighted online bipartite matching algorithm that breaks the $0.5$ barrier. Suppose that we receive a pair of elements in each round and immediately select one of them. Can we select with negative correlation to be more effective than independent random selections? Our contributions are threefold. For semi-OCS, which considers the probability that an element remains unselected after appearing in $k$ rounds, we give an optimal algorithm t…
When LP is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings
考虑一个随机图模型，其中每个可能的边$e$以一定概率$p_e$独立存在。给定这些概率，我们希望在随机生成的图中构建一个大/重匹配。然而，我们发现边是否存在的唯一方法是查询它，如果边确实存在于图中，我们被迫将其添加到匹配中。此外，允许查询每个顶点$i$最多$t_i$次。我们应该如何自适应地查询边以最大化匹配的期望权重？我们认为，在这个一般的框架匹配问题（其中一些出现在肾脏交换和网上约会，和其他出现在建模的在线广告），我们给LP舍入为基础的常数因子近似算法这些问题。我们的主要结果如下：我们给出了一般图的加权随机匹配的4近似，和二部图的3近似。这回答了[Chen et al ICALP 09]的一个开放性问题。结合LP舍入算法和自然贪心算法，我们给出了一般图上无权随机匹配的改进的3.46近似。我们介绍了一个泛化的随机在线匹配问题[Feldman等人FOCS 09]，也模型的偏好不确定性和超时的买家，并给出了一个常数因子近似算法。
