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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. Hardness of approximation in p via short cycle removal: cycle detection, distance oracles, and beyond

   https://doi.org/10.1145/3519935.3520066  

   Abboud, Amir ; Bringmann, Karl ; Khoury, Seri ; Zamir, Or ( June 2022 , STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing)

   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. Triangle finding is at the base of many conditional lower bounds in P, mainly for distance computation problems, and the existence of many 4- or 5-cycles in a worst-case instance had been the obstacle towards resolving major open questions. Hardness of approximation: Are there distance oracles with m1+o(1) preprocessing time and mo(1) query time that achieve a constant approximation? Existing algorithms with such desirable time bounds only achieve super-constant approximation factors, while only 3− factors were conditionally ruled out (Pătraşcu, Roditty, and Thorup; FOCS 2012). We prove that no O(1) approximations are possible, assuming the 3-SUM or APSP conjectures. In particular, we prove that k-approximations require Ω(m1+1/ck) time, which is tight up to the constant c. The lower bound holds even for the offline version where we are given the queries in advance, and extends to other problems such as dynamic shortest paths. The 4-Cycle problem: An infamous open question in fine-grained complexity is to establish any surprising consequences from a subquadratic or even linear-time algorithm for detecting a 4-cycle in a graph. This is arguably one of the simplest problems without a near-linear time algorithm nor a conditional lower bound. We prove that Ω(m1.1194) time is needed for k-cycle detection for all k≥ 4, unless we can detect a triangle in √n-degree graphs in O(n2−δ) time; a breakthrough that is not known to follow even from optimal matrix multiplication algorithms. 

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3. Improving the Smoothed Complexity of FLIP for Max Cut Problems

   https://doi.org/10.1145/3454125  

   Bibak, Ali ; Carlson, Charles ; Chandrasekaran, Karthekeyan ( August 2021 , ACM Transactions on Algorithms)

   null (Ed.)

   Finding locally optimal solutions for MAX-CUT and MAX- k -CUT are well-known PLS-complete problems. An instinctive approach to finding such a locally optimum solution is the FLIP method. Even though FLIP requires exponential time in worst-case instances, it tends to terminate quickly in practical instances. To explain this discrepancy, the run-time of FLIP has been studied in the smoothed complexity framework. Etscheid and Röglin (ACM Transactions on Algorithms, 2017) showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres, and Wei (STOC, 2017) showed that the smoothed complexity of FLIP for max-cut in complete graphs is ( O Φ 5 n 15.1 ), where Φ is an upper bound on the random edge-weight density and Φ is the number of vertices in the input graph. While Angel, Bubeck, Peres, and Wei’s result showed the first polynomial smoothed complexity, they also conjectured that their run-time bound is far from optimal. In this work, we make substantial progress toward improving the run-time bound. We prove that the smoothed complexity of FLIP for max-cut in complete graphs is O (Φ n 7.83 ). Our results are based on a carefully chosen matrix whose rank captures the run-time of the method along with improved rank bounds for this matrix and an improved union bound based on this matrix. In addition, our techniques provide a general framework for analyzing FLIP in the smoothed framework. We illustrate this general framework by showing that the smoothed complexity of FLIP for MAX-3-CUT in complete graphs is polynomial and for MAX - k - CUT in arbitrary graphs is quasi-polynomial. We believe that our techniques should also be of interest toward showing smoothed polynomial complexity of FLIP for MAX - k - CUT in complete graphs for larger constants k . 

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4. Counting Simplices in Hypergraph Streams

   Chakrabarti, Amit ; Haris, Themistoklis ( October 2022 , Leibniz international proceedings in informatics)

   We consider the problem of space-efficiently estimating the number of simplices in a hypergraph stream. This is the most natural hypergraph generalization of the highly-studied problem of estimating the number of triangles in a graph stream. Our input is a k-uniform hypergraph H with n vertices and m hyperedges, each hyperedge being a k-sized subset of vertices. A k-simplex in H is a subhypergraph on k+1 vertices X such that all k+1 possible hyperedges among X exist in H. The goal is to process the hyperedges of H, which arrive in an arbitrary order as a data stream, and compute a good estimate of T_k(H), the number of k-simplices in H. We design a suite of algorithms for this problem. As with triangle-counting in graphs (which is the special case k = 2), sublinear space is achievable but only under a promise of the form T_k(H) ≥ T. Under such a promise, our algorithms use at most four passes and together imply a space bound of O(ε^{-2} log δ^{-1} polylog n ⋅ min{(m^{1+1/k})/T, m/(T^{2/(k+1)})}) for each fixed k ≥ 3, in order to guarantee an estimate within (1±ε)T_k(H) with probability ≥ 1-δ. We also give a simpler 1-pass algorithm that achieves O(ε^{-2} log δ^{-1} log n⋅ (m/T) (Δ_E + Δ_V^{1-1/k})) space, where Δ_E (respectively, Δ_V) denotes the maximum number of k-simplices that share a hyperedge (respectively, a vertex), which generalizes a previous result for the k = 2 case. We complement these algorithmic results with space lower bounds of the form Ω(ε^{-2}), Ω(m^{1+1/k}/T), Ω(m/T^{1-1/k}) and Ω(mΔ_V^{1/k}/T) for multi-pass algorithms and Ω(mΔ_E/T) for 1-pass algorithms, which show that some of the dependencies on parameters in our upper bounds are nearly tight. Our techniques extend and generalize several different ideas previously developed for triangle counting in graphs, using appropriate innovations to handle the more complicated combinatorics of hypergraphs. 

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5. Adversarially Robust Coloring for Graph Streams

   Chakrabarti, Amit ; Ghosh, Prantar ; Stoeckl, Manuel ( January 2022 , Leibniz international proceedings in informatics)

   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 breakthrough (Assadi, Chen, and Khanna; SODA 2019) shows that in the standard, non-robust, streaming setting, (Δ+1)-colorings can be obtained while using only Õ(n) space. Here, we prove that an adversarially robust algorithm running under a similar space bound must spend almost Ω(Δ²) colors and that robust O(Δ)-coloring requires a linear amount of space, namely Ω(nΔ). We in fact obtain a more general lower bound, trading off the space usage against the number of colors used. From a complexity-theoretic standpoint, these lower bounds provide (i) the first significant separation between adversarially robust algorithms and ordinary randomized algorithms for a natural problem on insertion-only streams and (ii) the first significant separation between randomized and deterministic coloring algorithms for graph streams, since deterministic streaming algorithms are automatically robust. We complement our lower bounds with a suite of positive results, giving adversarially robust coloring algorithms using sublinear space. In particular, we can maintain an O(Δ²)-coloring using Õ(n √Δ) space and an O(Δ³)-coloring using Õ(n) space. 

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