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Register a new userDistributed Maximum Matching Verification in CONGEST
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Added by
Mohamad Ahmadi
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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 $G$ and $ell$ is the length of a shortest augmenting path. We hope that our algorithm constitutes a significant step towards developing a CONGEST algorithm to compute a maximum matching in time $tilde{O}(s^*)$, where $s^*$ is the size of a maximum matching.
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We present simple deterministic algorithms for subgraph finding and enumeration in the broadcast CONGEST model of distributed computation: -- For any constant $k$, detecting $k$-paths and trees on $k$ nodes can be done in $O(1)$ rounds. -- For any constant $k$, detecting $k$-cycles and pseudotrees on $k$ nodes can be done in $O(n)$ rounds. -- On $d$-degenerate graphs, cliques and $4$-cycles can be enumerated in $O(d + log n)$ rounds, and $5$-cycles in $O(d^2 + log n)$ rounds. In many cases, these bounds are tight up to logarithmic factors. Moreover, we show that the algorithms for $d$-degenerate graphs can be improved to optimal complexity $O(d/log n)$ and $O(d^2/log n)$, respectively, in the supported CONGEST model, which can be seen as an intermediate model between CONGEST and the congested clique.
We give efficient randomized and deterministic distributed algorithms for computing a distance-$2$ vertex coloring of a graph $G$ in the CONGEST model. In particular, if $Delta$ is the maximum degree of $G$, we show that there is a randomized CONGEST model algorithm to compute a distance-$2$ coloring of $G$ with $Delta^2+1$ colors in $O(logDeltacdotlog n)$ rounds. Further if the number of colors is slightly increased to $(1+epsilon)Delta^2$ for some $epsilon>1/{rm polylog}(n)$, we show that it is even possible to compute a distance-$2$ coloring deterministically in polylog$(n)$ time in the CONGEST model. Finally, we give a $O(Delta^2 + log^* n)$-round deterministic CONGEST algorithm to compute distance-$2$ coloring with $Delta^2+1$ colors.
We present improved results for approximating maximum-weight independent set ($MaxIS$) in the CONGEST and LOCAL models of distributed computing. Given an input graph, let $n$ and $Delta$ be the number of nodes and maximum degree, respectively, and let $MIS(n,Delta)$ be the the running time of finding a emph{maximal} independent set ($MIS$) in the CONGEST model. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a $Delta$-approximation for $MaxIS$ in $O(MIS(n,Delta)log W)$ rounds, where $W$ is the maximum weight of a node in the graph, which can be as high as $poly (n)$. Whether their algorithm is deterministic or randomized depends on the $MIS$ algorithm that is used as a black-box. Our main result in this work is a randomized $(poly(loglog n)/epsilon)$-round algorithm that finds, with high probability, a $(1+epsilon)Delta$-approximation for $MaxIS$ in the CONGEST model. That is, by sacrificing only a tiny fraction of the approximation guarantee, we achieve an emph{exponential} speed-up in the running time over the previous best known result. Due to a lower bound of $Omega(sqrt{log n/log log n})$ that was given by Kuhn, Moscibroda and Wattenhofer [JACM, 2016] on the number of rounds for any (possibly randomized) algorithm that finds a maximal independent set (even in the LOCAL model) this result implies that finding a $(1+epsilon)Delta$-approximation for $MaxIS$ is exponentially easier than $MIS$.
Maximum weight matching is one of the most fundamental combinatorial optimization problems with a wide range of applications in data mining and bioinformatics. Developing distributed weighted matching algorithms is challenging due to the sequential nature of efficient algorithms for this problem. In this paper, we develop a simple distributed algorithm for the problem on general graphs with approximation guarantee of $2+varepsilon$ that (nearly) matches that of the sequential greedy algorithm. A key advantage of this algorithm is that it can be easily implemented in only two rounds of computation in modern parallel computation frameworks such as MapReduce. We also demonstrate the efficiency of our algorithm in practice on various graphs (some with half a trillion edges) by achieving objective values always close to what is achievable in the centralized setting.
Distributed vertex coloring is one of the classic problems and probably also the most widely studied problems in the area of distributed graph algorithms. We present a new randomized distributed vertex coloring algorithm for the standard CONGEST model, where the network is modeled as an $n$-node graph $G$, and where the nodes of $G$ operate in synchronous communication rounds in which they can exchange $O(log n)$-bit messages over all the edges of $G$. For graphs with maximum degree $Delta$, we show that the $(Delta+1)$-list coloring problem (and therefore also the standard $(Delta+1)$-coloring problem) can be solved in $O(log^5log n)$ rounds. Previously such a result was only known for the significantly more powerful LOCAL model, where in each round, neighboring nodes can exchange messages of arbitrary size. The best previous $(Delta+1)$-coloring algorithm in the CONGEST model had a running time of $O(logDelta + log^6log n)$ rounds. As a function of $n$ alone, the best previous algorithm therefore had a round complexity of $O(log n)$, which is a bound that can also be achieved by a na{i}ve folklore algorithm. For large maximum degree $Delta$, our algorithm hence is an exponential improvement over the previous state of the art.
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