We study the problem of finding a maximal independent set (MIS) in the standard LOCAL model of distributed computing. Classical algorithms by Luby [JACM’86] and Alon, Babai, and Itai [JALG’86] find an MIS in O(logn) rounds in n-node graphs with high probability. Despite decades of research, the existence of any o(logn)-round algorithm for general graphs remains one of the major open problems in the field.

Abstract In this work we present the first constant-round algorithms for computing spanners and approximate All-Pairs Shortest Paths (APSP) in the distributed Congested Clique model. Specifically, we show the following results for undirected n -node graphs. For every integer $$k \ge 1$$ , O (1)-round algorithms for constructing O ( k )-spanners with $$O(n^{1+1/k})$$ edges in unweighted graphs, and O ( k )-spanners with $$O(n^{1+1/k} \log {n})$$ edges in weighted graphs. An O (1)-round algorithm for $$O(\log {n})$$ -approximation for APSP in unweighted graphs. An O (1)-round algorithm for $$O(\log ^2{n})$$ -approximation for APSP in weighted graphs. All our algorithms are randomized and succeed with high probability. Prior to our work, the fastest algorithms for computing O ( k )-spanners in this model require $${{\,\textrm{poly}\,}}(\log {k})$$ rounds [Parter, Yogev, DISC ’18] [Biswas, Dory, Ghaffari, Mitrovic, Nazari, SPAA ’21], and the fastest algorithms for approximate shortest paths require $${{\,\textrm{poly}\,}}(\log {\log {n}})$$ rounds [Dory, Parter, PODC ’20]. Our results extend to the closely related massively parallel computation (MPC) model with near-linear memory per machine, leading to the first O (1)-round algorithms for spanners and approximate shortest paths in this model as well.

Maximal Independent Set (MIS) and Maximal Matching (MM) play a vital role in distributed symmetry breaking. Despite decades of research, the complexity of both problems in the standard LOCAL model remains unresolved, and several gaps between the best-known upper and lower bounds persist. For n-node graphs with maximum degree $\Delta$, the best current upper bound for randomized algorithms is $O(\log \Delta+\operatorname{poly}(\log \log n))$, shown by Barenboim, Elkin, Pettie, and Schneider for MM [FOCS’12, JACM’16], and by Ghaffari for MIS [SODA’16]. On the other hand, the best-known lower bound for the two problems is $\Omega\left(\min \left\{\sqrt{\frac{\log n}{\log \log n}}, \frac{\log \Delta}{\log \log \Delta}\right\}\right)$, shown by Kuhn, Moscibroda, and Wattenhofer [PODC’04, JACM’16].In this work, we present an $\Omega(\min \{\log \Delta, \sqrt{\log n}\})$ lower bound for MM in $\Delta$-ary trees against randomized algorithms. By a folklore reduction, the same lower bound applies to MIS, albeit not in trees. As a function of n, this is the first advancement in our understanding of the randomized complexity of the two problems in more than two decades. As a function of $\Delta$, this shows that the current upper bounds are optimal for a wide range of $\Delta \in 2^{O(\sqrt{\log n})}$, answering an open question by Balliu, Brandt, Hirvonen, Olivetti, Rabie, and Suomela [FOCS’19, JACM’21].Moreover, our result implies a surprising and counterintuitive separation between MIS and MM in trees, as it was very recently shown that MIS in trees can be solved in $o(\sqrt{\log n})$ rounds. While MIS can be used to find an MM in general graphs, the reduction does not preserve the tree structure when applied to trees. Our separation shows that this is not an artifact of the reduction, but a fundamental difference between the two problems in trees. This also implies that MIS is strictly harder in general graphs compared to trees.Our main technical contribution is a novel technique in which we show that there is a self-reduction from a matching problem in r rounds to the same matching problem in r-1rounds (with slightly weaker probabilistic guarantees). Conceptually, this resembles the celebrated round elimination technique, which transforms an r-round algorithm for a problem $\Pi$ into an (r-1)round algorithm for a different problem $\Pi^{\prime}$. However, our proof differs significantly from the round elimination framework in several fundamental aspects. One of the key concepts we analyze in achieving our result is vertex survival probability, where we show that after $r \ll \min \{\log \Delta, \sqrt{\log n}\}$ rounds, any algorithm that finds a matching must leave two surviving unmatched nodes that are adjacent.

The main goal in distributed symmetry-breaking is to understand the locality of problems: the radius of the neighborhood that a node must explore to determine its part of a global solution. In this work, we study the locality of matching problems in the family of regular graphs, which is one of the main benchmarks for establishing lower bounds on the locality of symmetry-breaking problems, as well as for obtaining classification results. Our main results are summarized as follows: 1) Approximate matching: We develop randomized algorithms to show that (1 + ε)-approximate matching in regular graphs is truly local, i.e., the locality depends only on ε and is independent of all other graph parameters. Furthermore, as long as the degree Δ is not very small (namely, as long as Δ ≥ poly(1/ε)), this dependence is only logarithmic in 1/ε. This stands in sharp contrast to maximal matching in regular graphs which requires some dependence on the number of nodes n or the degree Δ. 2) Maximal matching: Our techniques further allow us to establish a strong separation between the node-averaged complexity and worst-case complexity of maximal matching in regular graphs, by showing that the former is only O(1). Central to our main technical contribution is a novel martingale-based analysis for the ≈ 40-year-old algorithm by Luby. In particular, our analysis shows that applying one round of Luby’s algorithm on the line graph of a Δ-regular graph results in an almost Δ/2-regular graph.

A central goal of cryptography is Secure Multi-party Computation (MPC), where n parties desire to compute a function of their joint inputs without letting any party learn about the inputs of its peers. Unfortunately, it is well-known that MPC guaranteeing output delivery to every party is infeasible when a majority of the parties are malicious. In fact, parties operating over a point-to-point network (i.e., without access to a broadcast channel) cannot even reach an agreement on the output when more than one third of the parties are malicious (Lamport, Shostak, and Pease, JACM 1980).

This is a survey of the exciting recent progress made in understanding the complexity of distributed subgraph finding problems. It overviews the results and techniques for assorted variants of subgraph finding problems in various models of distributed computing, and states intriguing open questions. This version contains some updates over the ICALP 2021 version, and I will try to keep updating it as additional progress is made.

We present a new technique for efficiently removing almost all short cycles in a graph without unintentionally removing its triangles. Consequently, triangle finding problems do not become easy even in almost k-cycle free graphs, for any constant k≥ 4.

In this work we present the first constant-round algorithms for computing spanners and approximate All-Pairs Shortest Paths (APSP) in the distributed CONGESTED CLIQUE model. Specifically, we show the following results for undirected n-node graphs. ulFor every integer k ≥ 1, O(1)-round algorithms for constructing O(k)-spanners with O(n1+1/k) edges in unweighted graphs, and O(k)-spanners with O(n1+1/k log n) edges in weighted graphs. An O(1)-round algorithm for O(log n)-approximation for APSP in unweighted graphs. An O(1)-round algorithm for O(log2n)-approximation for APSP in weighted graphs. All our algorithms are randomized and succeed with high probability. Prior to our work, the fastest algorithms for computing O(k)-spanners in this model require poly(log k) rounds [Parter, Yogev, DISC '18] [Biswas et al., SPAA '21], and the fastest algorithms for approximate shortest paths require poly(log log n) rounds [Dory, Parter, PODC '20]. Our results extend to the closely related massively parallel computation (MPC) model with near-linear memory per machine, leading to the first O(1)-round algorithms for spanners and approximate shortest paths in this model as well.

This article 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-shortest-paths 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.

By far the most fruitful technique for showing lower bounds for the CONGEST model is reductions to two-party communication complexity. This technique has yielded nearly tight results for various fundamental problems such as distance computations, minimum spanning tree, minimum vertex cover, and more.

We present improved algorithms for approximating maximum-weight independent set (MaxIS) in the CONGEST model. Given an input graph, let n and Δ be the number of nodes and maximum degree, respectively, and let MIS(n, Δ) be the running time of finding a maximal independent set (MIS) in the CONGEST model. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a Δ-approximation for MaxIS in O(MIS(n, Δ) log W) rounds, where W is the maximum weight of a node in the graph, which can be as high as poly(n). Whether their algorithm is deterministic or randomized depends on the MIS algorithm that is used as a black-box. Our results:

We study the problem of approximating the diameter D of an unweighted and undirected n-node graph in the congest model. Through a connection to extremal combinatorics, we show that a (6/11 + ε)-approximation requires Ω(n^{1/6}/log n) rounds, a (4/7 + ε)-approximation requires Ω(n^{1/4}/log n) rounds, and a (3/5 + ε)-approximation requires Ω(n^{1/3}/log n) rounds. These lower bounds are robust in the sense that they hold even against algorithms that are allowed to return an additional small additive error. Prior to our work, only lower bounds for (2/3 + ε)-approximation were known [Frischknecht et al. SODA 2012, Abboud et al. DISC 2016]. Furthermore, we prove that distinguishing graphs of diameter 3 from graphs of diameter 5 requires Ω(n/log n) rounds. This stands in sharp contrast to previous work: while there is an algorithm that returns an estimate ⌊ 2/3D ⌋ ≤ D̃ ≤ D in Õ(√n+D) rounds [Holzer et al. DISC 2014], our lower bound implies that any algorithm for returning an estimate 2/3D ≤ D̃ ≤ D requires ̃Ω(n) rounds.

We present improved results for approximating maximum-weight independent set (MaxIS) in the CONGEST and LOCAL models of distributed computing. Given an input graph, let n and Δ be the number of nodes and maximum degree, respectively, and let MIS(n,Δ) be the running time of finding a maximal independent set (MIS) in the CONGEST model. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a Δ-approximation for MaxIS in O(MIS(n,Δ)log W) rounds, where W is the maximum weight of a node in the graph, which can be as large as poly (n). Whether their algorithm is deterministic or randomized that succeeds with high probability depends on the MIS algorithm that is used as a black-box. Our results: 1) A deterministic O(MIS(n,Δ)/ε)-round algorithm that finds a (1+ε)Δ-approximation for MaxIS in the CONGEST model. 2) A randomized (poly(log log n)/ε)-round algorithm that finds, with high probability, a (1+ε)Δ-approximation for MaxIS in the CONGEST model. That is, by sacrificing only a tiny fraction of the approximation guarantee, we achieve an exponential speed-up in the running time over the previous best known result. 3) A randomized O(log n⋅ poly(log log n)/ε)-round algorithm that finds, with high probability, a 8(1+ε)α-approximation for MaxIS in the CONGEST model, where α is the arboricity of the graph. For graphs of arboricity α < Δ/(8(1+ε)), this result improves upon the previous best known result in both the approximation factor and the running time. One may wonder whether it is possible to approximate MaxIS with high probability in fewer than poly(log log n) rounds. Interestingly, a folklore randomized ranking algorithm by Boppana implies a single round algorithm that gives an expected Δ-approximation in the CONGEST model. However, it is unclear how to convert this algorithm to one that succeeds with high probability without sacrificing a large number of rounds. For unweighted graphs of maximum degree Δ ≤ n/log n, we show a new analysis of the randomized ranking algorithm, which we combine with the local-ratio technique, to provide a O(1/ε)-round algorithm in the CONGEST model that, with high probability, finds an independent set of size at least n/((1+ε)(Δ+1)). This result cannot be extended to very high degree graphs, as we show a lower bound of Ω(log^*n) rounds for any randomized algorithm that with probability at least 1-1/log n finds an independent set of size Ω(n/Δ). This lower bound holds even for the LOCAL model. The hard instances that we use to prove our lower bound are graphs of maximum degree Δ = Ω(n/log^*n).

We present the first super-linear lower bounds for natural graph problems in the CONGEST model, answering a long-standing open question. Specifically, we show that any exact computation of a minimum vertex cover or a maximum independent set requires a near-quadratic number of rounds in the CONGEST model, as well as any algorithm for computing the chromatic number of the graph. We further show that such strong lower bounds are not limited to NP-hard problems, by showing two simple graph problems in P which require a quadratic and near-quadratic number of rounds. Finally, we address the problem of computing an exact solution to weighted all-pairs-shortest-paths (APSP), which arguably may be considered as a candidate for having a super-linear lower bound. We show a simple linear lower bound for this problem, which implies a separation between the weighted and unweighted cases, since the latter is known to have a sub-linear complexity. We also formally prove that the standard Alice-Bob framework is incapable of providing a super-linear lower bound for exact weighted APSP, whose complexity remains an intriguing open question.