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 $O(\log{n})$ approximation to minimum vertex cover in only $O(\log\log{n})$ MPC rounds when the memory per machine is $O(n)$. Our algorithm for vertex cover is similar to the maximum matching algorithm of Czumaj et.al. but avoids many of the intricacies in their approach and as a result admits a considerably simpler analysis (at a cost of a worse approximation guarantee). We obtain this result by modifying a previous parallel algorithm by Khanna and the author (SPAA 2017) for vertex cover that allowed for compressing $O(\log{n})$ rounds of a distributed algorithm into constant MPC rounds when the memory allowed per machine is $O(n\sqrt{n})$.
We present a simple deterministic distributed $(2+ε)$-approximation algorithm for minimum weight vertex cover, which completes in $O(\logΔ/ε\log\logΔ)$ rounds, where $Δ$ is the maximum degree in the graph, for any $ε>0$ which is at most $O(1)$. For a constant $ε$, this implies a constant approximation in $O(\logΔ/\log\logΔ)$ rounds, which contradicts the lower bound of [KMW10].
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…
We present a deterministic distributed $2$-approximation algorithm for the Minimum Weight Vertex Cover problem in the CONGEST model whose round complexity is $O(\log n \log Δ/ \log^2 \log Δ)$. This improves over the currently best known deterministic 2-approximation implied by [KVY94]. Our solution generalizes the $(2+ε)$-approximation algorithm of [BCS17], improving the dependency on $ε^{-1}$ from linear to logarithmic. In addition, for every $ε=(\log Δ)^{-c}$, where $c\geq 1$ is a constant, our algorithm computes a $(2+ε)$-approximation in $O(\log Δ/ \log \log Δ)$~rounds (which is asymptotic…
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 a simple deterministic distributed algorithm that computes a $(Δ+1)$-vertex coloring in $O(\log^2 Δ\cdot \log n)$ rounds. The algorithm can be implemented with $O(\log n)$-bit messages. The algorithm can also be extended to the more general $(degree+1)$-list coloring problem. Obtaining a polylogarithmic-time deterministic algorithm for $(Δ+1)$-vertex coloring had remained a central open question in the area of distributed graph algorithms since the 1980s, until a recent network decomposition algorithm of Rozhoň and Ghaffari [STOC'20]. The current state of the art is based on an impr…
We present ${\rm poly\log\log n}$-round randomized distributed algorithms to compute vertex splittings, a partition of the vertices of a graph into $k$ parts such that a node of degree $d(u)$ has $\approx d(u)/k$ neighbors in each part. Our techniques can be seen as the first progress towards general ${\rm poly\log\log n}$-round algorithms for the Lovász Local Lemma. As the main application of our result, we obtain a randomized ${\rm poly\log\log n}$-round CONGEST algorithm for $(1+ε)Δ$-edge coloring $n$-node graphs of sufficiently large constant maximum degree $Δ$, for any $ε>0$. Further, our…
In this work, we provide the first practical evaluation of the structural rounding framework for approximation algorithms. Structural rounding works by first editing to a well-structured class, efficiently solving the edited instance, and "lifting" the partial solution to recover an approximation on the input. We focus on the well-studied Vertex Cover problem, and edit to the class of bipartite graphs (where Vertex Cover has an exact polynomial time algorithm). In addition to the naive lifting strategy for Vertex Cover described by Demaine et al., we introduce a suite of new lifting strategies…
Many modern parallel systems, such as MapReduce, Hadoop and Spark, can be modeled well by the MPC model. The MPC model captures well coarse-grained computation on large data --- data is distributed to processors, each of which has a sublinear (in the input data) amount of memory and we alternate between rounds of computation and rounds of communication, where each machine can communicate an amount of data as large as the size of its memory. This model is stronger than the classical PRAM model, and it is an intriguing question to design algorithms whose running time is smaller than in the PRAM …
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 he…
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…
We provide a simple and novel algorithmic design technique, for which we call iterative partial rounding, that gives a tight rounding-based approximation for vertex cover with hard capacities (VC-HC). In particular, we obtain an $f$-approximation for VC-HC on hypergraphs, improving over a previous results of Cheung et al (SODA 2014) to the tight extent. This also closes the gap of approximation since it was posted by Chuzhoy and Naor in (FOCS 2002). We believe that our rounding technique is of independence interests when hard constraints are considered. Our main technical tool for establishing…
The question of the existence of a polynomial kernelization of the Vertex Cover Above LP problem has been a longstanding, notorious open problem in Parameterized Complexity. Five years ago, the breakthrough work by Kratsch and Wahlstrom on representative sets has finally answered this question in the affirmative [FOCS 2012]. In this paper, we present an alternative, algebraic compression of the Vertex Cover Above LP problem into the Rank Vertex Cover problem. Here, the input consists of a graph G, a parameter k, and a bijection between V (G) and the set of columns of a representation of a matr…
By processing all minimal cutsets of a graph G, and by using novel wildcards, all spanning trees of G can be compactly encoded. Surprisingly, a 1986 algorithm of Winter seems to achieve (Conjecture 2) exactly the same compression, although in totally unrelated ways! Although Winter's algorithm is faster, our method adapts to relevant variations of the task, e.g. to the enumeration of only those spanning trees that obey prescribed degree conditions
The round-trip distance function on a geographic network (such as a road network, flight network, or utility distribution grid) defines the "distance" from a single vertex to a pair of vertices as the minimum length tour visiting all three vertices and ending at the starting vertex. Given a geographic network and a subset of its vertices called "sites" (for example a road network with a list of grocery stores), a two-site round-trip Voronoi diagram labels each vertex in the network with the pair of sites that minimizes the round-trip distance from that vertex. Alternatively, given a geographic…
We present time-efficient distributed algorithms for decomposing graphs with large edge or vertex connectivity into multiple spanning or dominating trees, respectively. As their primary applications, these decompositions allow us to achieve information flow with size close to the connectivity by parallelizing it along the trees. More specifically, our distributed decomposition algorithms are as follows: (I) A decomposition of each undirected graph with vertex-connectivity $k$ into (fractionally) vertex-disjoint weighted dominating trees with total weight $Ω(\frac{k}{\log n})$, in $\widetilde{O…
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…
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…
We present an algorithm for distributed networks to efficiently find a small vertex cut in the CONGEST model. Given a positive integer $κ$, our algorithm can, with high probability, either find $κ$ vertices whose removal disconnects the network or return that such $κ$ vertices do not exist. Our algorithm takes $κ^3\cdot \tilde{O}(D+\sqrt{n})$ rounds, where $n$ is the number of vertices in the network and $D$ denotes the network's diameter. This implies $\tilde{O}(D+\sqrt{n})$ round complexity whenever $κ=\text{polylog}(n)$. Prior to our result, a bound of $\tilde{O}(D)$ is known only when $κ=1…
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 $Δ$, we show that the $(Δ+1)$-list coloring problem (and therefore also the standard $(Δ+1)$-coloring problem) …
We (nearly) settle the time complexity for computing vertex fault-tolerant (VFT) spanners with optimal sparsity (up to polylogarithmic factors). VFT spanners are sparse subgraphs that preserve distance information, up to a small multiplicative stretch, in the presence of vertex failures. These structures were introduced by [Chechik et al., STOC 2009] and have received a lot of attention since then. We provide algorithms for computing nearly optimal $f$-VFT spanners for any $n$-vertex $m$-edge graph, with near optimal running time in several computational models: - A randomized sequential algor…
In this paper we give a f-approximation algorithm for the minimum unweighted Vertex Cover problem with Hard Capacity constraints (VCHC) on f-hypergraphs. This problem generalizes standard vertex cover for which the best known approximation ratio is also f and cannot be improved assuming the unique game conjecture. Our result is therefore essentially the best possible. This improves over the previous 2.155 (for f=2) and 2f-approximation algorithms by Cheung, Goemans and Wong (CGW). At the heart of our approach is to apply iterative rounding to the problem with ideas coming from several previous…
$ \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 give deterministic distributed $(1+ε)$-approximation algorithms for Minimum Vertex Coloring and Maximum Independent Set on chordal graphs in the LOCAL model. Our coloring algorithm runs in $O(\frac{1}ε \log n)$ rounds, and our independent set algorithm has a runtime of $O(\frac{1}ε\log(\frac{1}ε)\log^* n)$ rounds. For coloring, existing lower bounds imply that the dependencies on $\frac{1}ε$ and $\log n$ are best possible. For independent set, we prove that $O(\frac{1}ε)$ rounds are necessary. Both our algorithms make use of a tree decomposition of the input chordal graph. They iteratively …
Majority dynamics on the binomial Erdős-Rényi graph $\mathsf{G}(n,p)$ with $p=λ/\sqrt{n}$ is studied. In this process, each vertex has a state in $\{0,1\}$ and at each round, every vertex adopts the state of the majority of its neighbors, retaining its state in the case of a tie. It was conjectured by Benjamini et al. and proved by Fountoulakis et al. that this process reaches unanimity with high probability in at most four rounds. By adding some extra randomness and allowing the underlying graph to be drawn anew in each communication round, we improve on their result and prove that this proce…
We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank $f$. This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by $f$. The approximation factor of our algorithm is $(f+ε)$. Our algorithm runs in the CONGEST model and requires $O(\logΔ/ \log\logΔ)$ rounds, for constants $ε\in(0,1]$ and $f\in N^+$. This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. For constant va…
The notion of vertex sparsification is introduced in \cite{M}, where it was shown that for any graph $G = (V, E)$ and a subset of $k$ terminals $K \subset V$, there is a polynomial time algorithm to construct a graph $H = (K, E_H)$ on just the terminal set so that simultaneously for all cuts $(A, K-A)$, the value of the minimum cut in $G$ separating $A$ from $K -A$ is approximately the same as the value of the corresponding cut in $H$. We give the first super-constant lower bounds for how well a cut-sparsifier $H$ can simultaneously approximate all minimum cuts in $G$. We prove a lower bound o…
We describe an elegant O*(2^k) algorithm for the disjoint compression problem for Odd Cycle Transversal based on a reduction to Above Guarantee Vertex Cover. We believe that this algorithm refines the understanding of the Odd Cycle Transversal algorithm by Reed, Smith and Vetta.
In this paper, we study the problem of fast constructions of source-wise round-trip spanners in weighted directed graphs. For a source vertex set $S\subseteq V$ in a graph $G(V,E)$, an $S$-sourcewise round-trip spanner of $G$ of stretch $k$ is a subgraph $H$ of $G$ such that for every pair of vertices $u,v\in S\times V$, their round-trip distance in $H$ is at most $k$ times of their round-trip distance in $G$. We show that for a graph $G(V,E)$ with $n$ vertices and $m$ edges, an $s$-sized source vertex set $S\subseteq V$ and an integer $k>1$, there exists an algorithm that in time $O(ms^{1/k}\…
We present a massively parallel algorithm, with near-linear memory per machine, that computes a $(2+\varepsilon)$-approximation of minimum-weight vertex cover in $O(\log\log d)$ rounds, where $d$ is the average degree of the input graph. Our result fills the key remaining gap in the state-of-the-art MPC algorithms for vertex cover and matching problems; two classic optimization problems, which are duals of each other. Concretely, a recent line of work---by Czumaj et al. [STOC'18], Ghaffari et al. [PODC'18], Assadi et al. [SODA'19], and Gamlath et al. [PODC'19]---provides $O(\log\log n)$ time a…
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 f…
In this paper, we present improved algorithms for the $(Δ+1)$ (vertex) coloring problem in the Congested-Clique model of distributed computing. In this model, the input is a graph on $n$ nodes, initially each node knows only its incident edges, and per round each two nodes can exchange $O(\log n)$ bits of information. Our key result is a randomized $(Δ+1)$ vertex coloring algorithm that works in $O(\log\log Δ\cdot \log^* Δ)$-rounds. This is achieved by combining the recent breakthrough result of [Chang-Li-Pettie, STOC'18] in the \local\ model and a degree reduction technique. We also get the f…
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…
Self-stabilization for non-masking fault-tolerant distributed system has received considerable research interest over the last decade. In this paper, we propose a self-stabilizing algorithm for 2-edge-connectivity and 2-vertex-connectivity of an asynchronous distributed computer network. It is based on a self-stabilizing depth-first search, and is not a composite algorithm in the sense that it is not composed of a number of self-stabilizing algorithms that run concurrently. The time and space complexities of the algorithm are the same as those of the underlying self-stabilizing depth-first sea…
Numerous approaches study the vulnerability of networks against social contagion. Graph burning studies how fast a contagion, modeled as a set of fires, spreads in a graph. The burning process takes place in synchronous, discrete rounds. In each round, a fire breaks out at a vertex, and the fire spreads to all vertices that are adjacent to a burning vertex. The selection of vertices where fires start defines a schedule that indicates the number of rounds required to burn all vertices. Given a graph, the objective of an algorithm is to find a schedule that minimizes the number of rounds to burn…
Applications that require Internet access to remote 3D datasets are often limited by the storage costs of 3D models. Several compression methods are available to address these limits for objects represented by triangle meshes. Many CAD and VRML models, however, are represented as quadrilateral meshes or mixed triangle/quadrilateral meshes, and these models may also require compression. We present an algorithm for encoding the connectivity of such quadrilateral meshes, and we demonstrate that by preserving and exploiting the original quad structure, our approach achieves encodings 30 - 80% smal…
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…
Any graph with maximum degree $Δ$ admits a proper vertex coloring with $Δ+ 1$ colors that can be found via a simple sequential greedy algorithm in linear time and space. But can one find such a coloring via a sublinear algorithm? We answer this fundamental question in the affirmative for several canonical classes of sublinear algorithms including graph streaming, sublinear time, and massively parallel computation (MPC) algorithms. In particular, we design: * A single-pass semi-streaming algorithm in dynamic streams using $\tilde{O}(n)$ space. The only known semi-streaming algorithm prior to ou…
We consider the problem of finding a maximal independent set (MIS) in the shared blackboard communication model with vertex-partitioned inputs. There are $n$ players corresponding to vertices of an undirected graph, and each player sees the edges incident on its vertex -- this way, each edge is known by both its endpoints and is thus shared by two players. The players communicate in simultaneous rounds by posting their messages on a shared blackboard visible to all players, with the goal of computing an MIS of the graph. While the MIS problem is well studied in other distributed models, and wh…
In order to manage massive graphs in practice, it is often necessary to resort to graph compression, which aims at reducing the memory used when storing and processing the graph. Efficient compression methods have been proposed in the literature, especially for web graphs. In most cases, they are combined with a vertex reordering pre-processing step which significantly improves the compression rate. However, these techniques are not as efficient when considering other kinds of graphs. In this paper, we focus on the class of bipartite graphs and adapt the vertex reordering phase to their specif…
In this paper we study fractional coloring from the angle of distributed computing. Fractional coloring is the linear relaxation of the classical notion of coloring, and has many applications, in particular in scheduling. It was proved by Hasemann, Hirvonen, Rybicki and Suomela (2016) that for every real $α>1$ and integer $Δ$, a fractional coloring of total weight at most $α(Δ+1)$ can be obtained deterministically in a single round in graphs of maximum degree $Δ$, in the LOCAL model of computation. However, a major issue of this result is that the output of each vertex has unbounded size. Here…
We consider the adversarial combinatorial multi-armed bandit (CMAB) problem, whose decision set can be exponentially large with respect to the number of given arms. To avoid dealing with such large decision sets directly, we propose an algorithm performed on a zero-suppressed binary decision diagram (ZDD), which is a compressed representation of the decision set. The proposed algorithm achieves either $O(T^{2/3})$ regret with high probability or $O(\sqrt{T})$ expected regret as the any-time guarantee, where $T$ is the number of past rounds. Typically, our algorithm works efficiently for CMAB p…
In 1994, Thomassen proved that every planar graph is 5-list-colorable. In 1995, Thomassen proved that every planar graph of girth at least five is 3-list-colorable. His proofs naturally lead to quadratic-time algorithms to find such colorings. Here, we provide the first such linear-time algorithms to find such colorings. For a fixed surface S, Thomassen showed in 1997 that there exists a linear-time algorithm to decide if a graph embedded in S is 5-colorable and similarly in 2003 if a graph of girth at least five embedded in S is 3-colorable. Using the theory of hyperbolic families, the author…
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…
The classical NP-complete problem Vertex Cover requires us to determine whether a graph contains at most $k$ vertices that cover all edges. In spite of its intractability, the problem can be solved in FPT time for parameter $k$ by various techniques. In this paper, we present half a dozen new and simple FPT algorithms for Vertex Cover with respect to parameter $k$. For this purpose, we explore structural properties of vertex covers and use these properties to obtain FPT algorithms by iterative compression, colour coding, and indirect certificating methods. In particular, we show that every gra…
