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1. Finding Most-Shattering Minimum Vertex Cuts of Polylogarithmic Size in Near-Linear Time

   https://doi.org/10.4230/LIPIcs.ICALP.2024.87  

   Hua, Kevin ; Li, Daniel ; Park, Jaewoo ; Saranurak, Thatchaphol ( January 2024 , Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bringmann, Karl ; Grohe, Martin ; Puppis, Gabriele ; Svensson, Ola (Ed.)

   We show the first near-linear time randomized algorithms for listing all minimum vertex cuts of polylogarithmic size that separate the graph into at least three connected components (also known as shredders) and for finding the most shattering one, i.e., the one maximizing the number of connected components. Our algorithms break the quadratic time bound by Cheriyan and Thurimella (STOC'96) for both problems that has been unimproved for more than two decades. Our work also removes an important bottleneck to near-linear time algorithms for the vertex connectivity augmentation problem (Jordan '95) and finding an even-length directed cycle in a graph, a problem shown to be equivalent to many other fundamental problems (Vazirani and Yannakakis '90, Robertson et al. '99). Note that it is necessary to list only minimum vertex cuts that separate the graph into at least three components because there can be an exponential number of minimum vertex cuts in general. To obtain a near-linear time algorithm, we have extended techniques in local flow algorithms developed by Forster et al. (SODA'20) to list shredders on a local scale. We also exploit fast queries to a pairwise vertex connectivity oracle subject to vertex failures (Long and Saranurak FOCS'22, Kosinas ESA'23). This is the first application of using connectivity oracles subject to vertex failures to speed up a static graph algorithm. 

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2. Maximal k-Edge-Connected Subgraphs in Almost-Linear Time for Small k

   https://doi.org/10.4230/LIPIcs.ESA.2023.92  

   Saranurak, Thatchaphol ; Yuan, Wuwei ( January 2023 , Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

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

   We give the first almost-linear time algorithm for computing the maximal k-edge-connected subgraphs of an undirected unweighted graph for any constant k. More specifically, given an n-vertex m-edge graph G = (V,E) and a number k = log^o(1) n, we can deterministically compute in O(m+n^{1+o(1)}) time the unique vertex partition {V_1,… ,V_z} such that, for every i, V_i induces a k-edge-connected subgraph while every superset V'_i ⊃ V_{i} does not. Previous algorithms with linear time work only when k ≤ 2 [Tarjan SICOMP'72], otherwise they all require Ω(m+n√n) time even when k = 3 [Chechik et al. SODA'17; Forster et al. SODA'20]. Our algorithm also extends to the decremental graph setting; we can deterministically maintain the maximal k-edge-connected subgraphs of a graph undergoing edge deletions in m^{1+o(1)} total update time. Our key idea is a reduction to the dynamic algorithm supporting pairwise k-edge-connectivity queries [Jin and Sun FOCS'20]. 

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3. Minimum Cuts in Directed Graphs via Partial Sparsification

   https://doi.org/10.1109/FOCS52979.2021.00113  

   Cen, Ruoxu ; Li, Jason ; Nanongkai, Danupon ; Panigrahi, Debmalya ; Saranurak, Thatchaphol ; Quanrud, Kent ( February 2022 , 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022)

   We give an algorithm to find a minimum cut in an edge-weighted directed graph with n vertices and m edges in O ̃(n · max{m^{2/3}, n}) time. This improves on the 30 year old bound of O ̃(nm) obtained by Hao and Orlin for this problem. Using similar techniques, we also obtain O ̃ (n^2 /ε^2 )-time (1+ε)-approximation algorithms for both the minimum edge and minimum vertex cuts in directed graphs, for any fixed ε. Before our work, no (1+ε)-approximation algorithm better than the exact runtime of O ̃(nm) is known for either problem. Our algorithms follow a two-step template. In the first step, we employ a partial sparsification of the input graph to preserve a critical subset of cut values approximately. In the second step, we design algorithms to find the (edge/vertex) mincut among the preserved cuts from the first step. For edge mincut, we give a new reduction to O ̃ (min{n/m^{1/3} , √n}) calls of any maxflow subroutine, via packing arborescences in the sparsifier. For vertex mincut, we develop new local flow algorithms to identify small unbalanced cuts in the sparsified graph. 

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4. Can We Break Symmetry with o(m) Communication?

   https://doi.org/10.1145/3465084  

   Pai, Shreyas ; Pandurangan, Gopal ; Pemmaraju, Sriram ; Robinson, Peter ( July 2021 , PODC'21: Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing)

   null (Ed.)

   We study the communication cost (or message complexity) of fundamental distributed symmetry breaking problems, namely, coloring and MIS. While significant progress has been made in understanding and improving the running time of such problems, much less is known about the message complexity of these problems. In fact, all known algorithms need at least Ω(m) communication for these problems, where m is the number of edges in the graph. We addressthe following question in this paper: can we solve problems such as coloring and MIS using sublinear, i.e., o(m) communication, and if sounder what conditions? In a classical result, Awerbuch, Goldreich, Peleg, and Vainish [JACM 1990] showed that fundamental global problems such asbroadcast and spanning tree construction require at least o(m) messages in the KT-1 Congest model (i.e., Congest model in which nodes have initial knowledge of the neighbors' ID's) when algorithms are restricted to be comparison-based (i.e., algorithms inwhich node ID's can only be compared). Thirty five years after this result, King, Kutten, and Thorup [PODC 2015] showed that onecan solve the above problems using Õ(n) messages (n is the number of nodes in the graph) in Õ(n) rounds in the KT-1 Congest model if non-comparison-based algorithms are permitted. An important implication of this result is that one can use the synchronous nature of the KT-1 Congest model, using silence to convey information,and solve any graph problem using non-comparison-based algorithms with Õ(n) messages, but this takes an exponential number of rounds. In the asynchronous model, even this is not possible. In contrast, much less is known about the message complexity of local symmetry breaking problems such as coloring and MIS. Our paper fills this gap by presenting the following results. Lower bounds: In the KT-1 CONGEST model, we show that any comparison-based algorithm, even a randomized Monte Carlo algorithm with constant success probability, requires Ω(n 2) messages in the worst case to solve either (△ + 1)-coloring or MIS, regardless of the number of rounds. We also show that Ω(n) is a lower bound on the number ofmessages for any (△ + 1)-coloring or MIS algorithm, even non-comparison-based, and even with nodes having initial knowledge of up to a constant radius. Upper bounds: In the KT-1 CONGEST model, we present the following randomized non-comparison-based algorithms for coloring that, with high probability, use o(m) messages and run in polynomially many rounds.(a) A (△ + 1)-coloring algorithm that uses Õ(n1.5) messages, while running in Õ(D + √ n) rounds, where D is the graph diameter. Our result also implies an asynchronous algorithm for (△ + 1)-coloring with the same message bound but running in Õ(n) rounds. (b) For any constantε > 0, a (1+ε)△-coloring algorithm that uses Õ(n/ε 2 ) messages, while running in Õ(n) rounds. If we increase our input knowledge slightly to radius 2, i.e.,in the KT-2 CONGEST model, we obtain:(c) A randomized comparison-based MIS algorithm that uses Õ(n 1.5) messages. while running in Õ( √n) rounds. While our lower bound results can be viewed as counterparts to the classical result of Awerbuch, Goldreich, Peleg, and Vainish [JACM 90], but for local problems, our algorithms are the first-known algorithms for coloring and MIS that take o(m) messages and run in polynomially many rounds. 

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5. New hardness results for planar graph problems in p and an algorithm for sparsest cut

   https://doi.org/10.1145/3357713.3384310  

   Abboud, Amir ; Cohen-Addad, Vincent ; Klein, Philip N. ( June 2020 , Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing)

   null (Ed.)

   The Sparsest Cut is a fundamental optimization problem that have been extensively studied. For planar inputs the problem is in P and can be solved in Õ(n 3 ) time if all vertex weights are 1. Despite a significant amount of effort, the best algorithms date back to the early 90’s and can only achieve O(log n)-approximation in Õ(n) time or 3.5-approximation in Õ(n 2 ) time [Rao, STOC92]. Our main result is an Ω(n 2−ε ) lower bound for Sparsest Cut even in planar graphs with unit vertex weights, under the (min, +)-Convolution conjecture, showing that approxima- tions are inevitable in the near-linear time regime. To complement the lower bound, we provide a 3.3-approximation in near-linear time, improving upon the 25-year old result of Rao in both time and accuracy. We also show that our lower bound is not far from optimal by observing an exact algorithm with running time Õ(n 5/2 ) improving upon the Õ(n 3 ) algorithm of Park and Phillips [STOC93]. Our lower bound accomplishes a repeatedly raised challenge by being the first fine-grained lower bound for a natural planar graph problem in P. Building on our construction we prove near-quadratic lower bounds under SETH for variants of the closest pair problem in planar graphs, and use them to show that the popular Average-Linkage procedure for Hierarchical Clustering cannot be simulated in truly subquadratic time. At the core of our constructions is a diamond-like gadget that also settles the complexity of Diameter in distributed planar networks. We prove an Ω(n/ log n) lower bound on the number of communication rounds required to compute the weighted diameter of a network in the CONGET model, even when the underlying graph is planar and all nodes are D = 4 hops away from each other. This is the first poly(n) lower bound in the planar-distributed setting, and it complements the recent poly(D, log n) upper bounds of Li and Parter [STOC 2019] for (exact) unweighted diameter and for (1 + ε) approximate weighted diameter. 

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