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Register a new userDistributed Weighted Matching via Randomized Composable Coresets
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Added by
Sepehr Assadi
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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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.
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A common approach for designing scalable algorithms for massive data sets is to distribute the computation across, say $k$, machines and process the data using limited communication between them. A particularly appealing framework here is the simultaneous communication model whereby each machine constructs a small representative summary of its own data and one obtains an approximate/exact solution from the union of the representative summaries. If the representative summaries needed for a problem are small, then this results in a communication-efficient and round-optimal protocol. While many fundamental graph problems admit efficient solutions in this model, two prominent problems are notably absent from the list of successes, namely, the maximum matching problem and the minimum vertex cover problem. Indeed, it was shown recently that for both these problems, even achieving a polylog$(n)$ approximation requires essentially sending the entire input graph from each machine. The main insight of our work is that the intractability of matching and vertex cover in the simultaneous communication model is inherently connected to an adversarial partitioning of the underlying graph across machines. We show that when the underlying graph is randomly partitioned across machines, both these problems admit randomized composable coresets of size $widetilde{O}(n)$ that yield an $widetilde{O}(1)$-approximate solution. This results in an $widetilde{O}(1)$-approximation simultaneous protocol for these problems with $widetilde{O}(nk)$ total communication when the input is randomly partitioned across $k$ machines. We further prove the optimality of our results. Finally, by a standard application of composable coresets, our results also imply MapReduce algorithms with the same approximation guarantee in one or two rounds of communication
Studying distributed computing through the lens of algebraic topology has been the source of many significant breakthroughs during the last two decades, especially in the design of lower bounds or impossibility results for deterministic algorithms. This paper aims at studying randomized synchronous distributed computing through the lens of algebraic topology. We do so by studying the wide class of (input-free) symmetry-breaking tasks, e.g., leader election, in synchronous fault-free anonymous systems. We show that it is possible to redefine solvability of a task locally, i.e., for each simplex of the protocol complex individually, without requiring any global consistency. However, this approach has a drawback: it eliminates the topological aspect of the computation, since a single facet has a trivial topological structure. To overcome this issue, we introduce a projection $pi$ of both protocol and output complexes, where every simplex $sigma$ is mapped to a complex $pi(sigma)$; the later has a rich structure that replaces the structure we lost by considering one single facet at a time. To show the significance and applicability of our topological approach, we derive necessary and sufficient conditions for solving leader election in synchronous fault-free anonymous shared-memory and message-passing models. In both models, we consider scenarios in which there might be correlations between the random values provided to the nodes. In particular, different parties might have access to the same randomness source so their randomness is not independent but equal. Interestingly, we find that solvability of leader election relates to the number of parties that possess correlated randomness, either directly or via their greatest common divisor, depending on the specific communication model.
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.
Diameter, radius and eccentricities are fundamental graph parameters, which are extensively studied in various computational settings. Typically, computing approximate answers can be much more efficient compared with computing exact solutions. In this paper, we give a near complete characterization of the trade-offs between approximation ratios and round complexity of distributed algorithms for approximating these parameters, with a focus on the weighted and directed variants. Furthermore, we study emph{bi-chromatic} variants of these parameters defined on a graph whose vertices are colored either red or blue, and one focuses only on distances for pairs of vertices that are colored differently. Motivated by applications in computational geometry, bi-chromatic diameter, radius and eccentricities have been recently studied in the sequential setting [Backurs et al. STOC18, Dalirrooyfard et al. ICALP19]. We provide the first distributed upper and lower bounds for such problems. Our technical contributions include introducing the notion of emph{approximate pseudo-center}, which extends the emph{pseudo-centers} of [Choudhary and Gold SODA20], and presenting an efficient distributed algorithm for computing approximate pseudo-centers. On the lower bound side, our constructions introduce the usage of new functions into the framework of reductions from 2-party communication complexity to distributed algorithms.
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