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1. On the Distributed Complexity of Large-Scale Graph Computations

   https://doi.org/10.1145/3460900  

   Pandurangan, Gopal; Robinson, Peter; Scquizzato, Michele (June 2021, ACM Transactions on Parallel Computing)

   null (Ed.)

   Motivated by the increasing need to understand the distributed algorithmic foundations of large-scale graph computations, we study some fundamental graph problems in a message-passing model for distributed computing where k ≥ 2 machines jointly perform computations on graphs with n nodes (typically, n >> k). The input graph is assumed to be initially randomly partitioned among the k machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal is to minimize the number of communication rounds of the computation. Our main contribution is the General Lower Bound Theorem , a theorem that can be used to show non-trivial lower bounds on the round complexity of distributed large-scale data computations. This result is established via an information-theoretic approach that relates the round complexity to the minimal amount of information required by machines to solve the problem. Our approach is generic, and this theorem can be used in a “cookbook” fashion to show distributed lower bounds for several problems, including non-graph problems. We present two applications by showing (almost) tight lower bounds on the round complexity of two fundamental graph problems, namely, PageRank computation and triangle enumeration . These applications show that our approach can yield lower bounds for problems where the application of communication complexity techniques seems not obvious or gives weak bounds, including and especially under a stochastic partition of the input. We then present distributed algorithms for PageRank and triangle enumeration with a round complexity that (almost) matches the respective lower bounds; these algorithms exhibit a round complexity that scales superlinearly in k , improving significantly over previous results [Klauck et al., SODA 2015]. Specifically, we show the following results: PageRank: We show a lower bound of Ὼ(n/k 2 ) rounds and present a distributed algorithm that computes an approximation of the PageRank of all the nodes of a graph in Õ(n/k 2 ) rounds. Triangle enumeration: We show that there exist graphs with m edges where any distributed algorithm requires Ὼ(m/k 5/3 ) rounds. This result also implies the first non-trivial lower bound of Ὼ(n 1/3 ) rounds for the congested clique model, which is tight up to logarithmic factors. We then present a distributed algorithm that enumerates all the triangles of a graph in Õ(m/k 5/3 + n/k 4/3 ) rounds. 

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2. Streaming Maximal Matching with Bounded Deletions

   https://doi.org/10.4230/lipics.icalp.2025.106  

   Khanna, Sanjeev; Konrad, Christian; Dark, Jacques (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Censor-Hillel, Keren; Grandoni, Fabrizio; Ouaknine, Joel; Puppis, Gabriele (Ed.)

   Abstract: We initiate the study of the Maximal Matching problem in bounded-deletion graph streams. In this setting, a graph G is revealed as an arbitrary sequence of edge insertions and deletions, where the number of insertions is unrestricted but the number of deletions is guaranteed to be at most K, for some given parameter K. The single-pass streaming space complexity of this problem is known to be Θ(n²) when K is unrestricted, where n is the number of vertices of the input graph. In this work, we present new randomized and deterministic algorithms and matching lower bound results that together give a tight understanding (up to poly-log factors) of how the space complexity of Maximal Matching evolves as a function of the parameter K: The randomized space complexity of this problem is Θ̃(n ⋅ √K), while the deterministic space complexity is Θ̃(n ⋅ K). We further show that if we relax the maximal matching requirement to an α-approximation to Maximum Matching, for any constant α > 2, then the space complexity for both, deterministic and randomized algorithms, strikingly changes to Θ̃(n + K). A key conceptual contribution of our work that underlies all our algorithmic results is the introduction of the hierarchical maximal matching data structure, which computes a hierarchy of L maximal matchings on the substream of edge insertions, for an integer L. This deterministic data structure allows recovering a Maximal Matching even in the presence of up to L-1 edge deletions, which immediately yields an optimal deterministic algorithm with space Õ(n ⋅ K). To reduce the space to Õ(n ⋅ √K), we compute only √K levels of our hierarchical matching data structure and utilize a randomized linear sketch, i.e., our matching repair data structure, to repair any damage due to edge deletions. Using our repair data structure, we show that the level that is least affected by deletions can be repaired back to be globally maximal. The repair data structure is computed independently of the hierarchical maximal matching data structure and stores information for vertices at different scales with a gradually smaller set of vertices storing more and more information about their incident edges. The repair process then makes progress either by rematching a vertex to a previously unmatched vertex, or by strategically matching it to another matched vertex whose current mate is in a better position to find a new mate in that we have stored more information about its incident edges. Our lower bound result for randomized algorithms is obtained by establishing a lower bound for a generalization of the well-known Augmented-Index problem in the one-way two-party communication setting that we refer to as Embedded-Augmented-Index, and then showing that an instance of Embedded-Augmented-Index reduces to computing a maximal matching in bounded-deletion streams. To obtain our lower bound for deterministic algorithms, we utilize a compression argument to show that a deterministic algorithm with space o(n ⋅ K) would yield a scheme to compress a suitable class of graphs below the information-theoretic threshold. 

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3. A Deterministic Algorithm for Balanced Cut with Applications to Dynamic Connectivity, Flows, and Beyond

   https://doi.org/10.1109/FOCS46700.2020.00111  

   Chuzhoy, Julia; Gao, Yu; Li, Jason; Nanongkai, Danupon; Peng, Richard; Saranurak, Thatchaphol (November 2020, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020)

   null (Ed.)

   We consider the classical Minimum Balanced Cut problem: given a graph $$G$$, compute a partition of its vertices into two subsets of roughly equal volume, while minimizing the number of edges connecting the subsets. We present the first {\em deterministic, almost-linear time} approximation algorithm for this problem. Specifically, our algorithm, given an $$n$$-vertex $$m$$-edge graph $$G$$ and any parameter $$1\leq r\leq O(\log n)$$, computes a $$(\log m)^{r^2}$$-approximation for Minimum Balanced Cut on $$G$$, in time $$O\left ( m^{1+O(1/r)+o(1)}\cdot (\log m)^{O(r^2)}\right )$$. In particular, we obtain a $$(\log m)^{1/\epsilon}$$-approximation in time $$m^{1+O(1/\sqrt{\epsilon})}$$ for any constant $$\epsilon$$, and a $$(\log m)^{f(m)}$$-approximation in time $$m^{1+o(1)}$$, for any slowly growing function $$m$$. We obtain deterministic algorithms with similar guarantees for the Sparsest Cut and the Lowest-Conductance Cut problems. Our algorithm for the Minimum Balanced Cut problem in fact provides a stronger guarantee: it either returns a balanced cut whose value is close to a given target value, or it certifies that such a cut does not exist by exhibiting a large subgraph of $$G$$ that has high conductance. We use this algorithm to obtain deterministic algorithms for dynamic connectivity and minimum spanning forest, whose worst-case update time on an $$n$$-vertex graph is $$n^{o(1)}$$, thus resolving a major open problem in the area of dynamic graph algorithms. Our work also implies deterministic algorithms for a host of additional problems, whose time complexities match, up to subpolynomial in $$n$$ factors, those of known randomized algorithms. The implications include almost-linear time deterministic algorithms for solving Laplacian systems and for approximating maximum flows in undirected graphs. 

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4. Faster Detours in Undirected Graphs

   https://doi.org/10.4230/LIPIcs.ESA.2023.7  

   Akmal, Shyan; Vassilevska_Williams, Virginia; Williams, Ryan; Xu, Zixuan (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Gørtz, Inge Li; Farach-Colton, Martin; Puglisi, Simon J; Herman, Grzegorz (Ed.)

   The k-Detour problem is a basic path-finding problem: given a graph G on n vertices, with specified nodes s and t, and a positive integer k, the goal is to determine if G has an st-path of length exactly dist(s,t) + k, where dist(s,t) is the length of a shortest path from s to t. The k-Detour problem is NP-hard when k is part of the input, so researchers have sought efficient parameterized algorithms for this task, running in f(k)poly(n) time, for f(⋅) as slow-growing as possible. We present faster algorithms for k-Detour in undirected graphs, running in 1.853^k poly(n) randomized and 4.082^kpoly(n) deterministic time. The previous fastest algorithms for this problem took 2.746^k poly(n) randomized and 6.523^k poly(n) deterministic time [Bezáková-Curticapean-Dell-Fomin, ICALP 2017]. Our algorithms use the fact that detecting a path of a given length in an undirected graph is easier if we are promised that the path belongs to what we call a "bipartitioned" subgraph, where the nodes are split into two parts and the path must satisfy constraints on those parts. Previously, this idea was used to obtain the fastest known algorithm for finding paths of length k in undirected graphs [Björklund-Husfeldt-Kaski-Koivisto, JCSS 2017], intuitively by looking for paths of length k in randomly bipartitioned subgraphs. Our algorithms for k-Detour stem from a new application of this idea, which does not involve choosing the bipartitioned subgraphs randomly. Our work has direct implications for the k-Longest Detour problem, another related path-finding problem. In this problem, we are given the same input as in k-Detour, but are now tasked with determining if G has an st-path of length at least dist(s,t)+k. Our results for k-Detour imply that we can solve k-Longest Detour in 3.432^k poly(n) randomized and 16.661^k poly(n) deterministic time. The previous fastest algorithms for this problem took 7.539^k poly(n) randomized and 42.549^k poly(n) deterministic time [Fomin et al., STACS 2022]. 

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5. Deterministic Lower Bounds for k-Edge Connectivity in the Distributed Sketching Model

   https://doi.org/10.1145/3801897  

   Robinson, Peter; Tan, Ming Ming (May 2026, Proceedings of the ACM on Management of Data)

   We study thek-edge connectivity problem on undirected graphs in the distributed sketching model, where we havennodes and a referee. Each node sends a single message to the referee based on its 1-hop neighborhood in the graph, and the referee must decide whether the graph isk-edge connected by taking into account the received messages. We present the first lower bound for deciding a graph connectivity problem in this model with a deterministic algorithm. Concretely, we show that the worst case message length is Ω(k) bits fork-edge connectivity, for any super-constantk=O(√n). Previously, only a lower bound of Ω(log³n) bits was known for (1-edge) connectivity, due to Yu (SODA 2021). In fact, our result is the first super-polylogarithmic lower bound for a connectivity decision problem in the distributed graph sketching model. To obtain our result, we introduce a new lower bound graph construction, as well as a new communication complexity problem that we call Overlap. As this problem does not appear to be amenable to reductions to existing hard problems such as set disjointness or indexing due to correlations between the inputs of the three players, we leverage results from cross-intersecting set families to prove the hardness of Overlap for deterministic algorithms in the 3-party model with simultaneous messages. Finally, we obtain the sought lower bound for decidingk-edge connectivity via a novel simulation argument that, in contrast to previous works, does not introduce any probability of error and thus works for deterministic algorithms. 

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