We study the dynamic correlation clustering problem with \emph{adaptive} edge label flips. In correlation clustering, we are given a $n$-vertex complete graph whose edges are labeled either $(+)$ or $(-)$, and the goal is to minimize the total number of $(+)$ edges between clusters and the number of $(-)$ edges within clusters. We consider the dynamic setting with adversarial robustness, in which the \emph{adaptive} adversary can flip the label of an edge based on the current output of the algorithm. Our main result is a randomized algorithm that always maintains an $O(1)$-approximation to the optimal correlation clustering with $O(\log^{2}{n})$ amortized update time. Prior to our work, no algorithm with $O(1)$-approximation and $\text{polylog}{(n)}$ update time for the adversarially robust setting was known. We further validate our theoretical results with experiments on synthetic and real-world datasets with competitive empirical performances. Our main technical ingredient is an algorithm that maintains \emph{sparse-dense decomposition} with $\text{polylog}{(n)}$ update time, which could be of independent interest.

Round complexity is an extensively studied metric of distributed algorithms. In contrast, our knowledge of the message complexity of distributed computing problems and its relationship (if any) with round complexity is still quite limited. To illustrate, for many fundamental distributed graph optimization problems such as (exact) diameter computation, All-Pairs Shortest Paths (APSP), Maximum Matching etc., while (near) round-optimal algorithms are known, message-optimal algorithms are hitherto unknown. More importantly, the existing round-optimal algorithms are not message-optimal. This raises two important questions: (1) Can we design message-optimal algorithms for these problems? (2) Can we give message-time tradeoffs for these problems in case the message-optimal algorithms are not round-optimal?

We connect three distinct lines of research that have recently explored extensions of the classical LOCAL model of distributed computing: A. distributed quantum computing and non-signaling distributions [e.g. STOC 2024], B. finitely-dependent processes [e.g. Forum Math. Pi 2016], and C. locality in online graph algorithms and dynamic graph algorithms [e.g. ICALP 2023]. We prove new results on the capabilities and limitations of all of these models of computing, for locally checkable labeling problems (LCLs). We show that all these settings can be sandwiched between the classical LOCAL model and what we call the randomized online-LOCAL model. Our work implies limitations on the quantum advantage in the distributed setting, and we also exhibit a new barrier for proving tighter bounds. Our main technical results are these: 1. All LCL problems solvable with locality $O(\log^\star n)$ in the classical deterministic LOCAL model admit a finitely-dependent distribution with locality $O(1)$. This answers an open question by Holroyd [2024], and also presents a new barrier for proving bounds on distributed quantum advantage using causality-based arguments. 2. In rooted trees, if we can solve an LCL problem with locality $o(\log \log \log n)$ in the randomized online-LOCAL model (or any of the weaker models, such as quantum-LOCAL), we can solve it with locality $O(\log^\star n)$ in the classical deterministic LOCAL model. One of many implications is that in rooted trees, $O(\log^\star n)$ locality in quantum-LOCAL is not stronger than $O(\log^\star n)$ locality in classical LOCAL.

The message complexity of a distributed algorithm is the total number of messages sent by all nodes over the course of the algorithm. This paper studies the message complexity of distributed algorithms for fundamental graph optimization problems. We focus on four classical graph optimization problems: Maximum Matching (MaxM), Minimum Vertex Cover (MVC), Minimum Dominating Set (MDS), and Maximum Independent Set (MaxIS). In the sequential setting, these problems are representative of a wide spectrum of hardness of approximation. While there has been some progress in understanding the round complexity of distributed algorithms (for both exact and approximate versions) for these problems, much less is known about their message complexity and its relation with the quality of approximation. We almost fully quantify the message complexity of distributed graph optimization by showing the following results: 1) Cubic regime: Our first main contribution is showing essentially cubic, i.e., Ω̃(n³) lower bounds (where n is the number of nodes in the graph) on the message complexity of distributed exact computation of Minimum Vertex Cover (MVC), Minimum Dominating Set (MDS), and Maximum Independent Set (MaxIS). Our lower bounds apply to any distributed algorithm that runs in polynomial number of rounds (a mild and necessary restriction). Our result is significant since, to the best of our knowledge, this are the first ω(m) (where m is the number of edges in the graph) message lower bound known for distributed computation of such classical graph optimization problems. Our bounds are essentially tight, as all these problems can be solved trivially using O(n³) messages in polynomial rounds. All these bounds hold in the standard CONGEST model of distributed computation in which messages are of O(log n) size. 2) Quadratic regime: In contrast, we show that if we allow approximate computation then Θ̃(n²) messages are both necessary and sufficient. Specifically, we show that Ω̃(n²) messages are required for constant-factor approximation algorithms for all four problems. For MaxM and MVC, these bounds hold for any constant-factor approximation, whereas for MDS and MaxIS they hold for any approximation factor better than some specific constants. These lower bounds hold even in the LOCAL model (in which messages can be arbitrarily large) and they even apply to algorithms that take arbitrarily many rounds. We show that our lower bounds are essentially tight, by showing that if we allow approximation to within an arbitrarily small constant factor, then all these problems can be solved using Õ(n²) messages even in the CONGEST model. 3) Linear regime: We complement the above lower bounds by showing distributed algorithms with Õ(n) message complexity that run in polylogarithmic rounds and give constant-factor approximations for all four problems on random graphs. These results imply that almost linear (in n) message complexity is achievable on almost all (connected) graphs of every edge density.

Classic symmetry-breaking problems on graphs have gained a lot of attention in models of modern parallel computation. The Adaptive Massively Parallel Computation (AMPC) is a model that captures the central challenges in data center computations. Chang et al. [PODC'2019] gave an extremely fast, constant time, algorithm for the (Δ+1)-coloring problem, where Δ is the maximum degree of an input graph of n nodes. The algorithm works in the most restrictive low-space setting, where each machine has nδ local space for a constant 0 < δ < 1.

Grouping together similar elements in datasets is a common task in data mining and machine learning. In this paper, we study streaming and parallel algorithms for correlation clustering, where each pair of elements is labeled either similar or dissimilar. The task is to partition the elements and the objective is to minimize disagreements, that is, the number of dissimilar elements grouped together and similar elements that get separated.

We present a deterministic algorithm for solving a wide range of dynamic programming problems in trees in O(log D) rounds in the massively parallel computation model (MPC), with O(nδ) words of local memory per machine, for any given constant 0 < δ < 1. Here D is the diameter of the tree and n is the number of nodes---we emphasize that our running time is independent of n.

In this work, we present a constant-round algorithm for the $2$-ruling set problem in the Congested Clique model. As a direct consequence, we obtain a constant round algorithm in the MPC model with linear space-per-machine and optimal total space. Our results improve on the $O(\log \log \log n)$-round algorithm by [HPS, DISC'14] and the $O(\log \log Δ)$-round algorithm by [GGKMR, PODC'18]. Our techniques can also be applied to the semi-streaming model to obtain an $O(1)$-pass algorithm. Our main technical contribution is a novel sampling procedure that returns a small subgraph such that almost all nodes in the input graph are adjacent to the sampled subgraph. An MIS on the sampled subgraph provides a $2$-ruling set for a large fraction of the input graph. As a technical challenge, we must handle the remaining part of the graph, which might still be relatively large. We overcome this challenge by showing useful structural properties of the remaining graph and show that running our process twice yields a $2$-ruling set of the original input graph with high probability.

We show the first conditionally optimal deterministic algorithm for $3$-coloring forests in the low-space massively parallel computation (MPC) model. Our algorithm runs in $O(\log \log n)$ rounds and uses optimal global space. The best previous algorithm requires $4$ colors [Ghaffari, Grunau, Jin, DISC'20] and is randomized, while our algorithm are inherently deterministic. Our main technical contribution is an $O(\log \log n)$-round algorithm to compute a partition of the forest into $O(\log n)$ ordered layers such that every node has at most two neighbors in the same or higher layers. Similar decompositions are often used in the area and we believe that this result is of independent interest. Our results also immediately yield conditionally optimal deterministic algorithms for maximal independent set and maximal matching for forests, matching the state of the art [Giliberti, Fischer, Grunau, SPAA'23]. In contrast to their solution, our algorithms are not based on derandomization, and are arguably simpler.

In this paper we present a deterministic O(log log n)-round algorithm for the 2-ruling set problem in the Massively Parallel Computation model with Õ(n) memory; this algorithm also runs in O(log log n) rounds in the Congested Clique model. This is exponentially faster than the fastest known deterministic 2-ruling set algorithm for these models, which is simply the O(log Δ)-round deterministic Maximal Independent Set algorithm due to Czumaj, Davies, and Parter (SPAA 2020). Our result is obtained by derandomizing the 2-ruling set algorithm of Kothapalli and Pemmaraju (FSTTCS 2012).

The clustering problem, in its many variants, has numerous applications in operations research and computer science (e.g., in applications in bioinformatics, image processing, social network analysis, etc.). As sizes of data sets have grown rapidly, researchers have focused on designing algorithms for clustering problems in models of computation suited for large-scale computation such as MapReduce, Pregel, and streaming models. The k-machine model (Klauck et al., SODA 2015) is a simple, message-passing model for large-scale distributed graph processing. This paper considers three of the most prominent examples of clustering problems: the uncapacitated facility location problem, the p-median problem, and the p-center problem and presents O (1)-factor approximation algorithms for these problems running in Õ (n/k) rounds in the k -machine model. These algorithms are optimal upto polylogarithmic factors because this paper also shows Ω (n/k) lower bounds for obtaining poly(n)-factor approximation algorithms for these problems. These are the first results for clustering problems in the k -machine model.

This paper studies the problem of detecting asymptomatic cases in a temporal contact network in which multiple outbreaks have occurred. For many infections, asymptomatic cases present a major obstacle to obtaining a precise understanding of infection-spread. We show that the key to detecting asymptomatic cases well, is taking into account both individual risk as well as the likelihood of disease-flow along edges. Most related research has ignored the interplay between these dual aspects influencing disease-spread. We take both aspects into account by formulating the asymptomatic case detection problem as a Directed Prize-Collecting Steiner Tree (DIRECTED PCST) problem. We present an approximation-preserving reduction from this problem to the Directed Steiner Tree problem and use this reduction to obtain scalable algorithms for the DIRECTED PCST problem. Using these algorithms, we solve instances with more than 1.5M edges obtained from both synthetic and actual fine-grained hospital data. On synthetic data, we demonstrate that our detection methods significantly outperform various baselines (with a gain of $3.6 \times$). As an application of our methods, we use a measure of exposure to detected asymptomatic Clostridioides difficile (C. diff) infection (CDI) cases as an additional feature for the important task of predicting symptomatic CDI cases. In this application, our method outperforms all baselines, including those that don’t use asymptomatic CDI cases as a feature and those that use other methods for detecting asymptomatic CDI cases. We also demonstrate that the solutions returned by our approach are clinically meaningful by presenting a case study.

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?

Motivated by recent progress on symmetry breaking problems such as maximal independent set (MIS) and maximal matching in the low-memory Massively Parallel Computation (MPC) model (e.g., Behnezhad et al.~PODC 2019; Ghaffari-Uitto SODA 2019), we investigate the complexity of ruling set problems in this model. The MPC model has become very popular as a model for large-scale distributed computing and it comes with the constraint that the memory-per-machine is strongly sublinear in the input size. For graph problems, extremely fast MPC algorithms have been designed assuming $\tildeΩ(n)$ memory-per-machine, where $n$ is the number of nodes in the graph (e.g., the $O(\log\log n)$ MIS algorithm of Ghaffari et al., PODC 2018). However, it has proven much more difficult to design fast MPC algorithms for graph problems in the low-memory MPC model, where the memory-per-machine is restricted to being strongly sublinear in the number of nodes, i.e., $O(n^\eps)$ for $0 < \eps < 1$. In this paper, we present an algorithm for the 2-ruling set problem, running in $\tilde{O}(\log^{1/6} Δ)$ rounds whp, in the low-memory MPC model. We then extend this result to $β$-ruling sets for any integer $β> 1$. Specifically, we show that a $β$-ruling set can be computed in the low-memory MPC model with $O(n^\eps)$ memory-per-machine in $\tilde{O}(β\cdot \log^{1/(2^{β+1}-2)} Δ)$ rounds, whp. From this it immediately follows that a $β$-ruling set for $β= Ω(\log\log\log Δ)$-ruling set can be computed in in just $O(β\log\log n)$ rounds whp. The above results assume a total memory of $\tilde{O}(m + n^{1+\eps})$. We also present algorithms for $β$-ruling sets in the low-memory MPC model assuming that the total memory over all machines is restricted to $\tilde{O}(m)$.

We prove three new lower bounds for graph connectivity in the $1$-bit broadcast congested clique model, BCC$(1)$. First, in the KT-$0$ version of BCC$(1)$, in which nodes are aware of neighbors only through port numbers, we show an $Ω(\log n)$ round lower bound for CONNECTIVITY even for constant-error randomized Monte Carlo algorithms. The deterministic version of this result can be obtained via the well-known "edge-crossing" argument, but, the randomized version of this result requires establishing new combinatorial results regarding the indistinguishability graph induced by inputs. In our second result, we show that the $Ω(\log n)$ lower bound result extends to the KT-$1$ version of the BCC$(1)$ model, in which nodes are aware of IDs of all neighbors, though our proof works only for deterministic algorithms. Since nodes know IDs of their neighbors in the KT-$1$ model, it is no longer possible to play "edge-crossing" tricks; instead we present a reduction from the 2-party communication complexity problem PARTITION in which Alice and Bob are give two set partitions on $[n]$ and are required to determine if the join of these two set partitions equals the trivial one-part set partition. While our KT-$1$ CONNECTIVITY lower bound holds only for deterministic algorithms, in our third result we extend this $Ω(\log n)$ KT-1 lower bound to constant-error Monte Carlo algorithms for the closely related CONNECTED COMPONENTS problem. We use information-theoretic techniques to obtain this result. All our results hold for the seemingly easy special case of CONNECTIVITY in which an algorithm has to distinguish an instance with one cycle from an instance with multiple cycles. Our results showcase three rather different lower bound techniques and lay the groundwork for further improvements in lower bounds for CONNECTIVITY in the BCC$(1)$ model.

We investigate graph problems in the following setting: we are given a graph G and we are required to solve a problem on G2. While we focus mostly on exploring this theme in the distributed CONGEST model, we also show new results and surprising connections to the centralized model of computation. In the CONGEST model, it is natural to expect that problems on G2 would be quite difficult to solve efficiently on G, due to congestion. However, we show that the picture is both more complicated and more interesting.

In this paper, we consider the colorful k-center problem, which is a generalization of the well-known k-center problem. Here, we are given red and blue points in a metric space, and a coverage requirement for each color. The goal is to find the smallest radius rho, such that with k balls of radius rho, the desired number of points of each color can be covered. We obtain a constant approximation for this problem in the Euclidean plane. We obtain this result by combining a "pseudo-approximation" algorithm that works in any metric space, and an approximation algorithm that works for a special class of instances in the plane. The latter algorithm uses a novel connection to a certain matching problem in graphs.

We prove three new lower bounds for graph connectivity in the 1-bit broadcast congested clique model, BCC(1). First, in the KT-0 version of BCC(1), in which nodes are aware of neighbors only through port numbers, we show an Ømega(log n) round lower bound for CONNECTIVITY even for constant-error randomized Monte Carlo algorithms. The deterministic version of this result can be obtained via the well-known "edge-crossing" argument, but, the randomized version of this result requires establishing new combinatorial results regarding the indistinguishability graph induced by inputs. In our second result, we show that the Ømega(log n) lower bound result extends to the KT-1 version of the BCC(1) model, in which nodes are aware of IDs of all neighbors, though our proof works only for deterministic algorithms. Since nodes know IDs of their neighbors in the KT-1 model, it is no longer possible to play "edge-crossing" tricks; instead we present a reduction from the 2-party communication complexity problem PARTITION in which Alice and Bob are give two set partitions on [n] and are required to determine if the join of these two set partitions equals the trivial one-part set partition. While our KT-1 CONNECTIVITY lower bound holds only for deterministic algorithms, in our third result we extend this Ømega(log n) KT-1 lower bound to constant-error Monte Carlo algorithms for the closely related CONNECTED COMPONENTS problem. We use information-theoretic techniques to obtain this result. All our results hold for the seemingly easy special case of CONNECTIVITY in which an algorithm has to distinguish an instance with one cycle from an instance with multiple cycles. Our results showcase three rather different lower bound techniques and lay the groundwork for further improvements in lower bounds for CONNECTIVITY in the BCC(1) model.

This paper presents fast, distributed, $O(1)$-approximation algorithms for metric facility location problems with outliers in the Congested Clique model, Massively Parallel Computation (MPC) model, and in the $k$-machine model. The paper considers Robust Facility Location and Facility Location with Penalties, two versions of the facility location problem with outliers proposed by Charikar et al. (SODA 2001). The paper also considers two alternatives for specifying the input: the input metric can be provided explicitly (as an $n \times n$ matrix distributed among the machines) or implicitly as the shortest path metric of a given edge-weighted graph. The results in the paper are: - Implicit metric: For both problems, $O(1)$-approximation algorithms running in $O(\mbox{poly}(\log n))$ rounds in the Congested Clique and the MPC model and $O(1)$-approximation algorithms running in $\tilde{O}(n/k)$ rounds in the $k$-machine model. - Explicit metric: For both problems, $O(1)$-approximation algorithms running in $O(\log\log\log n)$ rounds in the Congested Clique and the MPC model and $O(1)$-approximation algorithms running in $\tilde{O}(n/k)$ rounds in the $k$-machine model. Our main contribution is to show the existence of Mettu-Plaxton-style $O(1)$-approximation algorithms for both Facility Location with outlier problems. As shown in our previous work (Berns et al., ICALP 2012, Bandyapadhyay et al., ICDCN 2018) Mettu-Plaxton style algorithms are more easily amenable to being implemented efficiently in distributed and large-scale models of computation.

We study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. Our work is motivated by the following central question: can we break the long-standing Θ(log n) time-complexity barrier and the Θ(m) message-complexity barrier in the Congest model for MIS or closely-related symmetry breaking problems?

We study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. The time (round) complexity of MIS (and ruling sets) have attracted much attention in the Local model. Indeed, recent results (Barenboim et al., FOCS 2012, Ghaffari SODA 2016) for the MIS problem have tried to break the long-standing O(log n)-round "barrier" achieved by Luby's algorithm, but these yield o(log n)-round complexity only when the maximum degree Delta is somewhat small relative to n. More importantly, these results apply only in the Local model. In fact, the best known time bound in the Congest model is still O(log n) (via Luby's algorithm) even for moderately small Delta (i.e., for Delta = Omega(log n) and Delta = o(n)). Furthermore, message complexity has been largely ignored in the context of local symmetry breaking. Luby's algorithm takes O(m) messages on m-edge graphs and this is the best known bound with respect to messages. Our work is motivated by the following central question: can we break the Theta(log n) time complexity barrier and the Theta(m) message complexity barrier in the Congest model for MIS or closely-related symmetry breaking problems? This paper presents progress towards this question for the distributed ruling set problem in the Congest model. A beta-ruling set is an independent set such that every node in the graph is at most beta hops from a node in the independent set. We present the following results: - Time Complexity: We show that we can break the O(log n) "barrier" for 2- and 3-ruling sets. We compute 3-ruling sets in O(log n/log log n) rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in O(log Delta (log n)^(1/2 + epsilon) + log n/log log n) rounds for any epsilon > 0, which is o(log n) for a wide range of Delta values (e.g., Delta = 2^(log n)^(1/2-epsilon)). These are the first 2- and 3-ruling set algorithms to improve over the O(log n)-round complexity of Luby's algorithm in the Congest model. - Message Complexity: We show an Omega(n^2) lower bound on the message complexity of computing an MIS (i.e., 1-ruling set) which holds also for randomized algorithms and present a contrast to this by showing a randomized algorithm for 2-ruling sets that, whp, uses only O(n log^2 n) messages and runs in O(Delta log n) rounds. This is the first message-efficient algorithm known for ruling sets, which has message complexity nearly linear in n (which is optimal up to a polylogarithmic factor).