Any graph with maximum degree ∆ admits a proper vertex coloring with ∆ + 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:

We study the problem of finding an approximate maximum matching in two closely related computational models, namely, the dynamic graph streaming model and the simultaneous multi-party communication model. In the dynamic graph streaming model, the input graph is revealed as a stream of edge insertions and deletions, and the goal is to design a small space algorithm to approximate the maximum matching. In the simultaneous model, the input graph is partitioned across k players, and the goal is to design a protocol where the k players simultaneously send a small-size message to a coordinator, and the coordinator computes an approximate matching. Dynamic graph streams. We resolve the space complexity of single-pass turnstile streaming algorithms for approximating matchings by showing that for any > 0, Θ(n 2−3) space is both sufficient and necessary (up to polylogarithmic factors) to compute an n-approximate matching; here n denotes the number of vertices in the input graph. The simultaneous communication model. Our results for dynamic graph streams also resolve the (per-player) simultaneous communication complexity for approximating matchings in the edge partition model. For the vertex partition model, we design new randomized and deterministic protocols for k players to achieve an α-approximation. Specifically, for α ≥ √ k, we provide a randomized protocol with total communication of O(nk/α 2) and a deterministic protocol with total communication of O(nk/α). Both these bounds are tight. Our work generalizes the results established by Dobzinski et al. (STOC 2014) for the special case of k = n. Finally, for the case of α = o(√ k), we establish a new lower bound on the simultaneous communication complexity which is super-linear in n.

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 more traditional approximation ratio.In this paper, we give a single unified approach that yields better approximation algorithms for matching and vertex cover in all these models. The highlights include:• The first one pass, significantly-better-than-2-approximation for matching in random arrival streams that uses subquadratic space, namely a (1.5 + ε)-approximation streaming algorithm that uses O(n 1.5 ) space for constant ε > 0. • The first 2-round, better-than-2-approximation for matching in the MPC model that uses subquadratic space per machine, namely a (1.5 + ε)-approximation algorithm with O( √ mn + n) memory per machine for constant ε > 0.By building on our unified approach, we further develop parallel algorithms in the MPC model that give a (1 + ǫ)-approximation to matching and an O(1)-approximation to vertex cover in only O(log log n) MPC rounds and O(n/polylog(n)) memory per machine. These results settle multiple open questions posed in the recent paper of Czumaj et al. [STOC 2018].We obtain our results by a novel combination of two previously disjoint set of techniques, namely randomized composable coresets and edge degree constrained subgraphs (EDCS). We significantly extend the power of these techniques and prove several new structural results. For example, we show that an EDCS is a sparse certificate for large matchings and small vertex covers that is quite robust to sampling and composition. * sassadi@cis.upenn.edu.As massive graphs become more prevalent, there is a rapidly growing need for scalable algorithms that solve classical graph problems on large datasets. When dealing with massive data, the entire input graph is orders of magnitude larger than the amount of storage on one processor and hence any algorithm needs to explicitly address this issue. For massive inputs, several different computational models have been introduced, each focusing on certain additional resources needed to solve largescale problems. Some examples include the streaming model, the distributed communication model, and the massively parallel computation (MPC) model that is a common abstraction of MapReducestyle computation (see Section 2 for a definition of MPC). The target resources in these models are the number of rounds of communication and the local storage on each machine.Given the variety of relevant models, there has been a lot of attention on designing general algorithmic techniques that can be applicable across a wide ran...

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 (requiring essentially no interaction between the machines) protocol. Some well-known examples of techniques for creating summaries include sampling, linear sketching, and composable coresets. These techniques have been successfully used to design communication efficient solutions for many fundamental graph problems. However, 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 modest approximation factor of polylog(n) requires using representative summaries of size Ω(n 2 ) i.e. essentially no better summary exists than each machine simply sending its entire input graph.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 O(n) that yield an O(1)-approximate solution 1 . In other words, a small subgraph of the input graph at each machine can be identified as its representative summary and the final answer then is obtained by simply running any maximum matching or minimum vertex cover algorithm on these combined subgraphs. This results in an O(1)-approximation simultaneous protocol for these problems with O(nk) total communication when the input is randomly partitioned across k machines. We also prove our results are optimal in a very strong sense: we not only rule out existence of smaller randomized composable coresets for these problems but in fact show that our O(nk) bound for total communication is optimal for any simultaneous communication protocol (i.e. not only for randomized coresets) for these two problems. 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, improving the previous best known round complexity for these problems.

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, which only contain edge insertions, and dynamic streams that allow both insertions and deletions of the edges, and present new upper and lower bound results for both cases.On the upper bound front, we show that an α-approximate estimate of the matching size can be computed in dynamic streams using O(n 2 /α 4 ) space, and in insertiononly streams using O(n/α 2 )-space. These bounds respectively shave off a factor of α from the space necessary to compute an α-approximate matching (as opposed to only size), thus proving a non-trivial separation between approximate estimation and approximate computation of matchings in data streams.On the lower bound front, we prove that any α-approximation algorithm for estimating matching size in dynamic graph streams requires Ω( √ n/α 2.5 ) bits of space, even if the underlying graph is both sparse and has arboricity bounded by O(α). We further improve our lower bound to Ω(n/α 2 ) in the case of dense graphs. These results establish the first non-trivial streaming lower bounds for superconstant approximation of matching size.Furthermore, we present the first super-linear space lower bound for computing a (1 + ε)-approximation of matching size even in insertion-only streams. In particular, we prove that a (1 + ε)-approximation to matching size requires RS(n) · n 1−O(ε) space; here, RS(n) denotes the maximum number of edge-disjoint induced matchings of size Θ(n) in an n-vertex graph. It is a major open problem with far-reaching implications to determine the value of RS(n), and current results leave open the possibility that RS(n) may be as large as n/ log n. Moreover, using the best known lower bounds for RS(n), our result already rules out any O(n · poly(log n/ε))-space algorithm for (1 + ε)-approximation of matchings. We also show how to avoid the dependency on the parameter RS(n) in proving lower bound for dynamic streams and present a near-optimal lower bound of n 2−O(ε) for (1 + ε)-approximation in this model. Using a well-known connection between matching size and matrix rank, all our lower bounds also hold for the problem of estimating matrix rank. In particular our results imply a near-optimal n 2−O(ε) bit lower bound for (1 + ε)-approximation of matrix ranks for dense matrices in dynamic streams, answering an open question of Li and Woodruff (STOC 2016).

Motivated by an application in kidney exchange, we study the following stochastic matching problem: we are given a graph G(V, E) (not necessarily bipartite), where each edge in E is realized with some constant probability p > 0 and the goal is to find a maximum matching in the realized graph. An algorithm in this setting is allowed to make queries to edges in E in order to determine whether or not they are realized. We design an adaptive algorithm for this problem that, for any graph G, computes a (1 − ε)-approximate maximum matching in the realized graph Gp with high probability, while making O log (1/εp) εp queries per vertex, where the edges to query are chosen adaptively in O log (1/εp) εp rounds. We further present a non-adaptive algorithm that makes O log (1/εp) εp queries per vertex and computes a (1 2 − ε)-approximate maximum matching in Gp with high probability. Both our adaptive and non-adaptive algorithms achieve the same approximation factor as the previous best algorithms of Blum et al. (EC 2015) for this problem, while requiring exponentially smaller number of per-vertex queries (and rounds of adaptive queries for the adaptive algorithm). Our results settle an open problem raised by Blum et al. by achieving only a polynomial dependency on both ε and p. Moreover, the approximation guarantee of our algorithms is instance-wise as opposed to only being competitive in expectation as is the case for Blum et al.. This is of particular relevance to the key application of stochastic matching in kidney exchange. We obtain our results via two main techniques, namely matching-covers and vertex sparsification that may be of independent interest.

A maximal independent set (MIS) can be maintained in an evolving m-edge graph by simply recomputing it from scratch in O(m) time after each update. But can it be maintained in time sublinear in m in fully dynamic graphs?We answer this fundamental open question in the affirmative. We present a deterministic algorithm with amortized update time O(min{∆, m 3/4 }), where ∆ is a fixed bound on the maximum degree in the graph and m is the (dynamically changing) number of edges.We further present a distributed implementation of our algorithm with O(min{∆, m 3/4 }) amortized message complexity, and O(1) amortized round complexity and adjustment complexity (the number of vertices that change their output after each update). This strengthens a similar result by Censor-Hillel, Haramaty, and Karnin (PODC'16) that required an assumption of a non-adaptive oblivious adversary.

A longstanding open problem in Algorithmic Mechanism Design is to design computationally-efficient truthful mechanisms for (approximately) maximizing welfare in combinatorial auctions with submodular bidders. The first such mechanism was obtained by Dobzinski, Nisan, and Schapira [STOC'06] who gave an O (log 2 m )-approximation where m is the number of items. This problem has been studied extensively since, culminating in an O ([EQUATION])-approximation mechanism by Dobzinski [STOC'16]. We present a computationally-efficient truthful mechanism with approximation ratio that improves upon the state-of-the-art by almost an exponential factor. In particular, our mechanism achieves an O ((log log m ) 3 )-approximation in expectation, uses only O ( n ) demand queries, and has universal truthfulness guarantee.