We consider the classic maximal and maximum independent set problems in three models of graph streams:
In the edge-arrival model we see a stream of edges which collectively define a graph, this model has been well-studied for a variety of problems. We first show that the space complexity for a one-pass streaming algorithm to find a maximal independent set is quadratic (i.e. we must store all edges). We further show that the problem does not become much easier if we only require approximate maximality.
In the "explicit" vertex stream model, the input stream is a sequence of vertices making up the graph, where every vertex arrives along with its incident edges that connect to previously arrived vertices. Various graph problems require substantially less space to solve in this setting than for edge-arrival streams. We show that every one-pass $c$-approximation algorithm for maximum independent set (MIS) on explicit vertex streams requires space $Ω(\frac{n^2}{c^7})$, where $n$ is the number of vertices of the input graph, and it is already known that space $\tildeΘ(\frac{n^2}{c^2})$ is necessary and sufficient in the edge arrival model (Halldórsson et al. 2012). The MIS problem is thus not significantly easier to solve under the explicit vertex arrival order assumption. Our result is proved via a reduction to a new multi-party communication problem closely related to pointer jumping.
In the "implicit" vertex stream model, the input stream consists of a sequence of objects, one per vertex. The algorithm is equipped with a function that can map a pair of objects to the presence or absence of an edge, thus defining the graph. This model captures, for example, geometric intersection graphs such as unit disc graphs. Our final set of results consists of several improved upper and lower bounds for ball intersection graphs, in both explicit and implicit streams.
We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the…
In this paper we present improved bounds for approximating maximum matchings in bipartite graphs in the streaming model. First, we consider the question of how well maximum matching can be approximated in a single pass over the input using $\tilde O(n)$ space, where $n$ is the number of vertices in the input graph. Two natural variants of this problem have been considered in the literature: (1) the edge arrival setting, where edges arrive in the stream and (2) the vertex arrival setting, where vertices on one side of the graph arrive in the stream together with all their incident edges. The la…
We provide prophet inequality algorithms for online weighted matching in general (non-bipartite) graphs, under two well-studied arrival models, namely edge arrival and vertex arrival. The weight of each edge is drawn independently from an a-priori known probability distribution. Under edge arrival, the weight of each edge is revealed upon arrival, and the algorithm decides whether to include it in the matching or not. Under vertex arrival, the weights of all edges from the newly arriving vertex to all previously arrived vertices are revealed, and the algorithm decides which of these edges, if …
We provide online algorithms for secretary matching in general weighted graphs, under the well-studied models of vertex and edge arrivals. In both models, edges are associated with arbitrary weights that are unknown from the outset, and are revealed online. Under vertex arrival, vertices arrive online in a uniformly random order; upon the arrival of a vertex $v$, the weights of edges from $v$ to all previously arriving vertices are revealed, and the algorithm decides which of these edges, if any, to include in the matching. Under edge arrival, edges arrive online in a uniformly random order; u…
The online matching problem was introduced by Karp, Vazirani and Vazirani nearly three decades ago. In that seminal work, they studied this problem in bipartite graphs with vertices arriving only on one side, and presented optimal deterministic and randomized algorithms for this setting. In comparison, more general arrival models, such as edge arrivals and general vertex arrivals, have proven more challenging and positive results are known only for various relaxations of the problem. In particular, even the basic question of whether randomization allows one to beat the trivially-optimal determ…
The bipartite matching problem in the online and streaming settings has received a lot of attention recently. The classical vertex arrival setting, for which the celebrated Karp, Vazirani and Vazirani (KVV) algorithm achieves a $1-1/e$ approximation, is rather well understood: the $1-1/e$ approximation is optimal in both the online and semi-streaming setting, where the algorithm is constrained to use $n\cdot \log^{O(1)} n$ space. The more challenging the edge arrival model has seen significant progress recently in the online algorithms literature. For the strictly online model (no preemption) …
We study a graph coloring problem that is otherwise easy but becomes quite non-trivial in the one-pass streaming model. In contrast to previous graph coloring problems in streaming that try to find an assignment of colors to vertices, our main work is on estimating the number of conflicting or monochromatic edges given a coloring function that is streaming along with the graph; we call the problem {\sc Conflict-Est}. The coloring function on a vertex can be read or accessed only when the vertex is revealed in the stream. If we need the color on a vertex that has streamed past, then that color,…
Motivated by display advertising on the internet, the online stochastic matching problem is proposed by Feldman, Mehta, Mirrokni, and Muthukrishnan (FOCS 2009). Consider a stochastic bipartite graph with offline vertices on one side and with i.i.d. online vertices on the other side. The algorithm knows the offline vertices and the distribution of the online vertices in advance. Upon the arrival of each online vertex, its type is realized and the algorithm immediately and irrevocably decides how to match it. In the vertex-weighted version of the problem, each offline vertex is associated with a…
Online bipartite matching with one-sided arrival and its variants have been extensively studied since the seminal work of Karp, Vazirani, and Vazirani (STOC 1990). Motivated by real-life applications with dynamic market structures, e.g. ride-sharing, two generalizations of the classical one-sided arrival model are proposed to allow non-bipartite graphs and to allow all vertices to arrive online. Namely, online matching with general vertex arrival is introduced by Wang and Wong (ICALP 2015), and fully online matching is introduced by Huang et al. (JACM 2020). In this paper, we study the fractio…
We present a streaming algorithm for the vertex connectivity problem in dynamic streams with a (nearly) optimal space bound: for any $n$-vertex graph $G$ and any integer $k \geq 1$, our algorithm with high probability outputs whether or not $G$ is $k$-vertex-connected in a single pass using $\widetilde{O}(k n)$ space. Our upper bound matches the known $Ω(k n)$ lower bound for this problem even in insertion-only streams -- which we extend to multi-pass algorithms in this paper -- and closes one of the last remaining gaps in our understanding of dynamic versus insertion-only streams. Our result …
Maximum coverage and minimum set cover problems --collectively called coverage problems-- have been studied extensively in streaming models. However, previous research not only achieve sub-optimal approximation factors and space complexities, but also study a restricted set arrival model which makes an explicit or implicit assumption on oracle access to the sets, ignoring the complexity of reading and storing the whole set at once. In this paper, we address the above shortcomings, and present algorithms with improved approximation factor and improved space complexity, and prove that our result…
Most prior work on online matching problems has been with the flexibility of keeping some vertices unmatched. We study three related online matching problems with the constraint of matching every vertex, i.e., with no rejections. We adopt a model in which vertices arrive in uniformly random order and the non-negative edge-weights are arbitrary. For the capacitated online bipartite matching problem, in which the vertices of one side of the graph are offline and those of the other side arrive online, we give a 4.62-competitive algorithm when the capacity of each offline vertex is 2. For the onli…
We consider the prophet inequality problem for (not necessarily bipartite) matching problems with independent edge values, under both edge arrivals and vertex arrivals. We show constant-factor prophet inequalities for the case where the online algorithm has only limited access to the value distributions through samples. First, we give a $16$-approximate prophet inequality for matching in general graphs under edge arrivals that uses only a single sample from each value distribution as prior information. Then, for bipartite matching and (one-sided) vertex arrivals, we show an improved bound of $…
As graphs continue to grow in size, we seek ways to effectively process such data at scale. The model of streaming graph processing, in which a compact summary is maintained as each edge insertion/deletion is observed, is an attractive one. However, few results are known for optimization problems over such dynamic graph streams. In this paper, we introduce a new approach to handling graph streams, by instead seeking solutions for the parameterized versions of these problems where we are given a parameter $k$ and the objective is to decide whether there is a solution bounded by $k$. By combinin…
In this paper, we initiate the study of the vertex coloring problem of a graph in the semi streaming model. In this model, the input graph is defined by a stream of edges, arriving in adversarial order and any algorithm must process the edges in the order of arrival using space linear (up to polylogarithmic factors) in the number of vertices of the graph. In the offline settings, there is a simple greedy algorithm for $(Δ+1)$-vertex coloring of a graph with maximum degree $Δ$. We design a one pass randomized streaming algorithm for $(1+\varepsilon)Δ$-vertex coloring problem for any constant $\…
Nearly three decades ago, Bar-Noy, Motwani and Naor showed that no online edge-coloring algorithm can edge color a graph optimally. Indeed, their work, titled "the greedy algorithm is optimal for on-line edge coloring", shows that the competitive ratio of $2$ of the naïve greedy algorithm is best possible online. However, their lower bound required bounded-degree graphs, of maximum degree $Δ= O(\log n)$, which prompted them to conjecture that better bounds are possible for higher-degree graphs. While progress has been made towards resolving this conjecture for restricted inputs and arrivals or…
We consider directed graph algorithms in a streaming setting, focusing on problems concerning orderings of the vertices. This includes such fundamental problems as topological sorting and acyclicity testing. We also study the related problems of finding a minimum feedback arc set (edges whose removal yields an acyclic graph), and finding a sink vertex. We are interested in both adversarially-ordered and randomly-ordered streams. For arbitrary input graphs with edges ordered adversarially, we show that most of these problems have high space complexity, precluding sublinear-space solutions. Some…
We optimally resolve the space complexity for the problem of finding an $α$-approximate minimum vertex cover ($α$MVC) in dynamic graph streams. We give a randomised algorithm for $α$MVC which uses $O(n^2/α^2)$ bits of space matching Dark and Konrad's lower bound [CCC 2020] up to constant factors. By computing a random greedy matching, we identify `easy' instances of the problem which can trivially be solved by returning the entire vertex set. The remaining `hard' instances, then have sparse induced subgraphs which we exploit to get our space savings and solve $α$MVC. Achieving this type of opt…
In the online bipartite matching with reassignments problem, an algorithm is initially given only one side of the vertex set of a bipartite graph; the vertices on the other side are revealed to the algorithm one by one, along with its incident edges. The algorithm is required to maintain a matching in the current graph, where the algorithm revises the matching after each vertex arrival by reassigning vertices. Bernstein, Holm, and Rotenberg showed that an online algorithm can maintain a matching of maximum cardinality by performing amortized $O(\log^2 n)$ reassignments per arrival. In this pap…
Streaming is a model where an input graph is provided one edge at a time, instead of being able to inspect it at will. In this work, we take a parameterized approach by assuming a vertex cover of the graph is given, building on work of Bishnu et al. [COCOON 2020]. We show the further potency of combining this parameter with the Adjacency List streaming model to obtain results for vertex deletion problems. This includes kernels, parameterized algorithms, and lower bounds for the problems of Pi-free Deletion, H-free Deletion, and the more specific forms of Cluster Vertex Deletion and Odd Cycle T…
Motivated by the properties of unending real-world cybersecurity streams, we present a new graph streaming model: XStream. We maintain a streaming graph and its connected components at single-edge granularity. In cybersecurity graph applications, input streams typically consist of edge insertions; individual deletions are not explicit. Analysts maintain as much history as possible and will trigger customized bulk deletions when necessary Despite a variety of dynamic graph processing systems and some canonical literature on theoretical sliding-window graph streaming, XStream is the first model …
In the W-streaming model, an algorithm is given $O(n \mathrm{polylog} n)$ space and must process a large graph of up to $O(n^2)$ edges. In this short note we give two algorithms for edge colouring under the W-streaming model. For edge colouring in W-streaming, a colour for every edge must be determined by the time all the edges are streamed. Our first algorithm uses $Δ+ o(Δ)$ colours in $O(n \log^2 n)$ space when the edges arrive according to a uniformly random permutation. The second algorithm uses $(1 + o(1))Δ^2 / s$ colours in $\tilde{O}(n s)$ space when edges arrival adversarially.
A rotor walk in a directed graph can be thought of as a deterministic version of a Markov Chain, where a pebble moves from vertex to vertex following a simple rule until a terminal vertex, or sink, is reached. The ARRIVAL problem, as defined by Dohrau and al., consists in determining which sink will be reached. While the walk itself can take an exponential number of steps, this problem belongs to the complexity class NP$\cap$co-NP without being known to be in P. Several variants have been studied where we add one or two players to the model, defining deterministic analogs of stochastic models …
In this paper, we study max-weight stochastic matchings on online bipartite graphs under both vertex and edge arrivals. We focus on designing polynomial time approximation algorithms with respect to the online benchmark, which was first considered by Papadimitriou, Pollner, Saberi, and Wajc [EC'21]. In the vertex arrival version of the problem, the goal is to find an approximate max-weight matching of a given bipartite graph when the vertices in one part of the graph arrive online in a fixed order with independent chances of failure. Whenever a vertex arrives we should decide, irrevocably, whe…
In this paper, we give simple optimal lower bounds on the one-way two-party communication complexity of approximate Maximum Matching and Minimum Vertex Cover with deletions. In our model, Alice holds a set of edges and sends a single message to Bob. Bob holds a set of edge deletions, which form a subset of Alice's edges, and needs to report a large matching or a small vertex cover in the graph spanned by the edges that are not deleted. Our results imply optimal space lower bounds for insertion-deletion streaming algorithms for Maximum Matching and Minimum Vertex Cover. Previously, Assadi et al…
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…
In this paper we study the mincut problem in the online setting. We consider two distinct models: A) competitive analysis and B) regret analysis. In the competitive setting we consider the vertex arrival model; whenever a new vertex arrives it's neighborhood with respect to the set of known vertices is revealed. An online algorithm must make an irrevocable decision to determine the side of the cut that the vertex must belong to in order to minimize the size of the final cut. Various models are considered. 1) For classical and advice models we give tight bounds on the competitive ratio of deter…
The study of parameterized streaming complexity on graph problems was initiated by Fafianie et al. (MFCS'14) and Chitnis et al. (SODA'15 and SODA'16). Simply put, the main goal is to design streaming algorithms for parameterized problems such that $O\left(f(k)\log^{O(1)}n\right)$ space is enough, where $f$ is an arbitrary computable function depending only on the parameter $k$. However, in the past few years, very few positive results have been established. Most of the graph problems that do have streaming algorithms of the above nature are ones where localized checking is required, like Verte…
Temporal graphs are used to abstractly model real-life networks that are inherently dynamic in nature. Given a static underlying graph $G=(V,E)$, a temporal graph on $G$ is a sequence of snapshots $G_t$, one for each time step $t\geq 1$. In this paper we study stochastic temporal graphs, i.e. stochastic processes $\mathcal{G}$ whose random variables are the snapshots of a temporal graph on $G$. A natural feature observed in various real-life scenarios is a memory effect in the appearance probabilities of particular edges; i.e. the probability an edge $e\in E$ appears at time step $t$ depends o…
We introduce a fully online model of maximum cardinality matching in which all vertices arrive online. On the arrival of a vertex, its incident edges to previously-arrived vertices are revealed. Each vertex has a deadline that is after all its neighbors' arrivals. If a vertex remains unmatched until its deadline, the algorithm must then irrevocably either match it to an unmatched neighbor, or leave it unmatched. The model generalizes the existing one-sided online model and is motivated by applications including ride-sharing platforms, real-estate agency, etc. We show that the Ranking algorithm…
In this paper we present a simple but powerful subgraph sampling primitive that is applicable in a variety of computational models including dynamic graph streams (where the input graph is defined by a sequence of edge/hyperedge insertions and deletions) and distributed systems such as MapReduce. In the case of dynamic graph streams, we use this primitive to prove the following results: -- Matching: First, there exists an $\tilde{O}(k^2)$ space algorithm that returns an exact maximum matching on the assumption the cardinality is at most $k$. The best previous algorithm used $\tilde{O}(kn)$ spa…
A streaming algorithm is considered to be adversarially robust if it provides correct outputs with high probability even when the stream updates are chosen by an adversary who may observe and react to the past outputs of the algorithm. We grow the burgeoning body of work on such algorithms in a new direction by studying robust algorithms for the problem of maintaining a valid vertex coloring of an $n$-vertex graph given as a stream of edges. Following standard practice, we focus on graphs with maximum degree at most $Δ$ and aim for colorings using a small number $f(Δ)$ of colors. A recent brea…
We study the maximum matching problem in the random-order semi-streaming setting. In this problem, the edges of an arbitrary $n$-vertex graph $G=(V, E)$ arrive in a stream one by one and in a random order. The goal is to have a single pass over the stream, use $n \cdot poly(\log n)$ space, and output a large matching of $G$. We prove that for an absolute constant $ε_0 > 0$, one can find a $(2/3 + ε_0)$-approximate maximum matching of $G$ using $O(n \log n)$ space with high probability. This breaks the natural boundary of $2/3$ for this problem prevalent in the prior work and resolves an open p…
Given a labeled graph, the frequent-subgraph mining (FSM) problem asks to find all the $k$-vertex subgraphs that appear with frequency greater than a given threshold. FSM has numerous applications ranging from biology to network science, as it provides a compact summary of the characteristics of the graph. However, the task is challenging, even more so for evolving graphs due to the streaming nature of the input and the exponential time complexity of the problem. In this paper, we initiate the study of the approximate FSM problem in both incremental and fully-dynamic streaming settings, where …
Given a stream $\mathcal{S}$ of insertions and deletions of edges of an underlying graph $G$ (with fixed vertex set $V$ where $n=|V|$ is the number of vertices of $G$), we propose a dynamic algorithm that maintains a maximal independent set (MIS) of $G$ (at any time $t$ of the stream $\mathcal{S}$) with amortized update time $O(\log^3 n)$.
In this paper, we consider the online vertex-weighted bipartite matching problem in the random arrival model. We consider the generalization of the RANKING algorithm for this problem introduced by Huang, Tang, Wu, and Zhang (TALG 2019), who show that their algorithm has a competitive ratio of 0.6534. We show that assumptions in their analysis can be weakened, allowing us to replace their derivation of a crucial function $g$ on the unit square with a linear program that computes the values of a best possible $g$ under these assumptions on a discretized unit square. We show that the discretizati…
Subgraph counting is a fundamental primitive in graph processing, with applications in social network analysis (e.g., estimating the clustering coefficient of a graph), database processing and other areas. The space complexity of subgraph counting has been studied extensively in the literature, but many natural settings are still not well understood. In this paper we revisit the subgraph (and hypergraph) counting problem in the sketching model, where the algorithm's state as it processes a stream of updates to the graph is a linear function of the stream. This model has recently received a lot…
We study the following vertex-weighted online bipartite matching problem: $G(U, V, E)$ is a bipartite graph. The vertices in $U$ have weights and are known ahead of time, while the vertices in $V$ arrive online in an arbitrary order and have to be matched upon arrival. The goal is to maximize the sum of weights of the matched vertices in $U$. When all the weights are equal, this reduces to the classic \emph{online bipartite matching} problem for which Karp, Vazirani and Vazirani gave an optimal $\left(1-\frac{1}{e}\right)$-competitive algorithm in their seminal work~\cite{KVV90}. Our main resu…
