In this paper, we present a new simple degree-based estimator for the size of maximum matching in bounded arboricity graphs. When the arboricity of the graph is bounded by $α$, the estimator gives a $α+2$ factor approximation of the matching size. For planar graphs, we show the estimator does better and returns a $3.5$ approximation of the matching size.
Using this estimator, we get new results for approximating the matching size of planar graphs in the streaming and distributed models of computation. In particular, in the vertex-arrival streams, we get a randomized $O(\frac{\sqrt{n}}{ε^2}\log n)$ space algorithm for approximating the matching size within $(3.5+ε)$ factor in a planar graph on $n$ vertices. Similarly, we get a simultaneous protocol in the vertex-partition model for approximating the matching size within $(3.5+ε)$ factor using $O(\frac{n^{2/3}}{ε^2}\log n)$ communication from each player.
In comparison with the previous estimators, the estimator in this paper does not need to know the arboricity of the input graph and improves the approximation factor for the case of planar graphs.
We present a data stream algorithm for estimating the size of the maximum matching of a low arboricity graph. Recall that a graph has arboricity $α$ if its edges can be partitioned into at most $α$ forests and that a planar graph has arboricity $α=3$. Estimating the size of the maximum matching in such graphs has been a focus of recent data stream research. A surprising result on this problem was recently proved by Cormode et al. They designed an ingenious algorithm that returned a $(22.5α+6)(1+ε)$ approximation using a single pass over the edges of the graph (ordered arbitrarily) and $O(ε^{-2…
Estimating the size of the maximum matching is a canonical problem in graph algorithms, and one that has attracted extensive study over a range of different computational models. We present improved streaming algorithms for approximating the size of maximum matching with sparse (bounded arboricity) graphs. * Insert-Only Streams: We present a one-pass algorithm that takes O(c log^2 n) space and approximates the size of the maximum matching in graphs with arboricity c within a factor of O(c). This improves significantly on the state-of-the-art O~(cn^{2/3})-space streaming algorithms. * Dynamic S…
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…
Given query access to an undirected graph $G$, we consider the problem of computing a $(1\pmε)$-approximation of the number of $k$-cliques in $G$. The standard query model for general graphs allows for degree queries, neighbor queries, and pair queries. Let $n$ be the number of vertices, $m$ be the number of edges, and $n_k$ be the number of $k$-cliques. Previous work by Eden, Ron and Seshadhri (STOC 2018) gives an $O^*(\frac{n}{n^{1/k}_k} + \frac{m^{k/2}}{n_k})$-time algorithm for this problem (we use $O^*(\cdot)$ to suppress $\poly(\log n, 1/ε, k^k)$ dependencies). Moreover, this bound is ne…
We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the out-degree of each vertex is bounded. On one hand, we show how to orient the edges such that the out-degree of each vertex is proportional to the arboricity $α$ of the graph, in a worst-case update time of $O(\log^3 n \log α)$. On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off, namely the improved worst case update time of $O(\log ^2 n \log α)$ for the problem of maintaining an edge-orientation with at most $O(α+ \log n)$ out-edges…
In graph sparsification, the goal has almost always been of {global} nature: compress a graph into a smaller subgraph ({sparsifier}) that maintains certain features of the original graph. Algorithms can then run on the sparsifier, which in many cases leads to improvements in the overall runtime and memory. This paper studies sparsifiers that have bounded (maximum) degree, and are thus {locally} sparse, aiming to improve local measures of runtime and memory. To improve those local measures, it is important to be able to compute such sparsifiers {locally}. We initiate the study of local algorith…
Identifying clusters of similar elements in a set is a common task in data analysis. With the immense growth of data and physical limitations on single processor speed, it is necessary to find efficient parallel algorithms for clustering tasks. In this paper, we study the problem of correlation clustering in bounded arboricity graphs with respect to the Massively Parallel Computation (MPC) model. More specifically, we are given a complete graph where the edges are either positive or negative, indicating whether pairs of vertices are similar or dissimilar. The task is to partition the vertices …
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 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 …
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 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 study the problem of estimating the maximum matching size in graphs whose edges are revealed in a streaming manner. We consider both insertion-only streams and dynamic streams and present new upper and lower bound results for both models. On the upper bound front, we show that an $α$-approximate estimate of the matching size can be computed in dynamic streams using $\widetilde{O}({n^2/α^4})$ space, and in insertion-only streams using $\widetilde{O}(n/α^2)$-space. On the lower bound front, we prove that any $α$-approximation algorithm for estimating matching size in dynamic graph streams req…
Kernelization via Sampling with Applications to Dynamic Graph Streams
在本文中，我们提出了一个简单但功能强大的子图采样原语，适用于各种计算模型，包括动态图流（其中输入图是由一系列的边/超边插入和删除定义）和分布式系统，如MapReduce。在动态图流的情况下，我们使用这个原语来证明以下结果：-匹配：首先，存在一个$\tilde{O}(k^2)$空间算法，返回一个确切的最大匹配的假设基数是最多$k$。最好的以前的算法使用$\tilde{O}(kn)$空间，其中$n$是在图中的顶点数，我们证明了我们的结果是最佳的对数因子。我们的算法有$\tilde{O}(1)$更新时间。其次，存在一个$\tilde{O}(n^2/α^3)$空间算法，它返回一个$α$ -近似的匹配任意大小。（Assadi et al.（2015）表明这是最佳的，并且独立和同时建立了相同的上限。我们推广这两个结果加权匹配。第三，存在一个$\tilde{O}(n^{4/5})$空间算法，返回一个常数近似图有界荫度。--顶点覆盖和命中集：存在解决最小命中集问题的$\tilde{O}(k^d)$空间算法，其中$d$是输入集的基数，而$k$是最小命中集大小的上限。我们证明这是最佳的对数因子。我们的算法有$\tilde{O}(1)$更新时间。情况$d=2$对应于最小顶点覆盖。最后，我们考虑一个更大的家庭参数化问题（包括$b$匹配，不相交的路径，顶点着色等），我们的子图采样原语产生快速，小空间的动态图…
In their breakthrough ICALP'15 paper, Bernstein and Stein presented an algorithm for maintaining a $(3/2+ε)$-approximate maximum matching in fully dynamic {\em bipartite} graphs with a {\em worst-case} update time of $O_ε(m^{1/4})$; we use the $O_ε$ notation to suppress the $ε$-dependence. Their main technical contribution was in presenting a new type of bounded-degree subgraph, which they named an {\em edge degree constrained subgraph (EDCS)}, which contains a large matching -- of size that is smaller than the maximum matching size of the entire graph by at most a factor of $3/2+ε$. They demo…
We analyze the behavior of the Barvinok estimator of the hafnian of even dimension, symmetric matrices with nonnegative entries. We introduce a condition under which the Barvinok estimator achieves subexponential errors, and show that this condition is almost optimal. Using that hafnians count the number of perfect matchings in graphs, we conclude that Barvinok's estimator gives a polynomial-time algorithm for the approximate (up to subexponential errors) evaluation of the number of perfect matchings.
