In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an O(n 1−2/ω ) round matrix multiplication algorithm, where ω < 2.3728639 is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -triangle and 4-cycle counting in O(n 0.158 ) rounds, improving upon the O(n 1/3 ) algorithm of Dolev et al. [DISC 2012], -a (1 + o(1))-approximation of all-pairs shortest paths in O(n 0.158 ) rounds, improving upon theÕ(n 1/2 )-round (2 + o(1))-approximation algorithm of Nanongkai [STOC 2014], and computing the girth in O(n 0.158 ) rounds, which is the first non-trivial solution in this model.In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.

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We present a simple deterministic single-pass (2 + )-approximation algorithm for the maximum weight matching problem in the semi-streaming model. This improves upon the currently best known approximation ratio of (3.5 + ).Our algorithm uses O(n log 2 n) space for constant values of . It relies on a variation of the local-ratio theorem, which may be of independent interest in the semi-streaming model.

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In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an O(n 1−2/ω ) round matrix multiplication algorithm, where ω < 2.3728639 is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.

This paper studies lower bounds for fundamental optimization problems in the congest model. We show that solving problems exactly in this model can be a hard task, by providing Ω(n 2 ) lower bounds for cornerstone problems, such as minimum dominating set (MDS), Hamiltonian path, Steiner tree and max-cut. These are almost tight, since all of these problems can be solved optimally in O(n 2 ) rounds. Moreover, we show that even in bounded-degree graphs and even in simple graphs with maximum degree 5 and logarithmic diameter, it holds that various tasks, such as finding a maximum independent set (MaxIS) or a minimum vertex cover, are still difficult, requiring a near-tight number ofΩ(n) rounds.Furthermore, we show that in some cases even approximations are difficult, by providing añ Ω(n 2 ) lower bound for a (7/8 + )-approximation for MaxIS, and a nearly-linear lower bound for an O(log n)-approximation for the k-MDS problem for any constant k ≥ 2, as well as for several variants of the Steiner tree problem.Our lower bounds are based on a rich variety of constructions that leverage novel observations, and reductions among problems that are specialized for the congest model. However, for several additional approximation problems, as well as for exact computation of some central problems in P , such as maximum matching and max flow, we show that such constructions cannot be designed, by which we exemplify some limitations of this framework.Lower bounds for exact computation. We show that in many cases, solving problems exactly in the congest model is hard, by providing many newΩ(n 2 ) lower bounds for fundamental optimization problems, such as MDS, max-cut, Hamiltonian path, Steiner tree and minimum 2edge-connected spanning subgraph (2-ECSS). Such results were previously known only for the minimum vertex cover (MVC), MaxIS and minimum chromatic number problems [10]. Our results are inspired by [10], but combine many new technical ingredients. In particular, one of the key components in our lower bounds are reductions between problems. After having a lower bound for MDS, a cleverly designed reduction allows us to build a new lower bound construction for Hamiltonian path. These constructions serve as a basis for our constructions for the Steiner tree and minimum 2-ECSS. We emphasize that we cannot use directly known reductions from the sequential setting, but rather we must create reductions that can be applied efficiently on lower bound constructions.To demonstrate the challenge, we now give more details about the general framework. We use the well-known framework of reductions from 2-party communication complexity, as originated in [44] and used in many additional works, e.g., [1, 9,14,17,18,47]. In communication complexity, two players, Alice and Bob, receive private input strings and their goal is to solve some problem related to their inputs, for example, decide whether their inputs are disjoint, by communicating the minimum number of bits possible. To show a lower bound for the congest model, the highlevel idea is t...

This paper proves strong lower bounds for distributed computing in the congest model, by presenting the bit-gadget: a new technique for constructing graphs with small cuts.The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can be obtained when using dense graphs.Second, small cuts are essential for proving super-linear lower bounds, none of which were known prior to this work. In fact, they allow us to show near-quadratic lower bounds for several problems, such as exact minimum vertex cover or maximum independent set, as well as for coloring a graph with its chromatic number. Such strong lower bounds are not limited to NP-hard problems, as given by two simple graph problems in P which are shown to require a quadratic and near-quadratic number of rounds. All of the above are optimal up to logarithmic factors. In addition, in this context, the complexity of the all-pairs-shortestpaths problem is discussed.Finally, it is shown that graph constructions for congest lower bounds translate to lower bounds for the semi-streaming model, despite being very different in its nature.

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