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Given a graph with positive and negative edge labels, the correlation clustering problem aims to cluster the nodes so to minimize the total number of betweencluster positive and withincluster negative edges. This problem has many applications in data mining, particularly in unsupervised learning. Inspired by the prevalence of large graphs and constantly changing data in modern applications, we study correlation clustering in dynamic, parallel (MPC), and local computation (LCA) settings. We design an approach that improves stateoftheart runtime complexities in all these settings. In particular, we provide the first fully dynamic algorithm that runs in an expected amortized constant time, without any dependence on the graph size. Moreover, our algorithm essentially matches the approximation guarantee of the celebrated PIVOT algorithm.more » « lessFree, publiclyaccessible full text available June 30, 2025

Motivated by applications to classification problems on metric data, we study Weighted Metric Clustering problem: given a metric d over n points and a k x k symmetric matrix A with nonnegative entries, the goal is to find a kpartition of these points into clusters C1,...,Ck, while minimizing the sum of A[i,j] * d(u,v) over all pairs of clusters Ci and Cj and all pairs of points u from Ci and v from Cj. Specific choices of A lead to Weighted Metric Clustering capturing wellstudied graph partitioning problems in metric spaces, such as MinUncut, MinkSum, MinkCut, and more.Our main result is that Weighted Metric Clustering admits a polynomialtime approximation scheme (PTAS). Our algorithm handles all the above problems using the SheraliAdams linear programming relaxation. This subsumes several prior works, unifies many of the techniques for various metric clustering objectives, and yields a PTAS for several new problems, including metric clustering on manifolds and a new family of hierarchical clustering objectives. Our experiments on the hierarchical clustering objective show that it better captures the groundtruth structural information compared to the popular Dasgupta's objective.
Free, publiclyaccessible full text available March 25, 2025 
We prove a new generalization of the higherorder Cheeger inequality for partitioning with buffers. Consider a graph G=(V,E). The buffered expansion of a set S⊆V with a buffer B⊆V∖S is the edge expansion of S after removing all the edges from set S to its buffer B. An εbuffered kpartitioning is a partitioning of a graph into disjoint components Pi and buffers Bi, in which the size of buffer Bi for Pi is small relative to the size of Pi: Bi≤εPi. The buffered expansion of a buffered partition is the maximum of buffered expansions of the k sets Pi with buffers Bi. Let hk,εG be the buffered expansion of the optimal εbuffered kpartitioning, then for every δ>0, hk,εG≤Oδ(1)⋅(logkε)⋅λ⌊(1+δ)k⌋, where λ⌊(1+δ)k⌋ is the ⌊(1+δ)k⌋th smallest eigenvalue of the normalized Laplacian of G. Our inequality is constructive and avoids the ``squareroot loss'' that is present in the standard Cheeger inequalities (even for k=2). We also provide a complementary lower bound, and a novel generalization to the setting with arbitrary vertex weights and edge costs. Moreover our result implies and generalizes the standard higherorder Cheeger inequalities and another recent Cheegertype inequality by Kwok, Lau, and Lee (2017) involving robust vertex expansion.more » « lessFree, publiclyaccessible full text available January 1, 2025

We show that a simple singlepass semistreaming variant of the Pivot algorithm for Correlation Clustering gives a (3 + epsilon)approximation using O(n/epsilon) words of memory. This is a slight improvement over the recent results of Cambus, Kuhn, Lindy, Pai, and Uitto, who gave a (3 + epsilon)approximation using O(n log n) words of memory, and Behnezhad, Charikar, Ma, and Tan, who gave a 5approximation using O(n) words of memory. One of the main contributions of this paper is that both the algorithm and its analysis are very simple, and also the algorithm is easy to implement.more » « lessFree, publiclyaccessible full text available December 10, 2024

We show that the RandomCoordinateCut algorithm gives the optimal competitive ratio for explainable kmedians in l1. The problem of explainable kmedians was introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian in 2020. Several groups of authors independently proposed a simple polynomialtime randomized algorithm for the problem and showed that this algorithm is O(log k loglog k) competitive. We provide a tight analysis of the algorithm and prove that its competitive ratio is upper bounded by 2ln k +2. This bound matches the Omega(log k) lower bound by Dasgupta et al (2020).more » « lessFree, publiclyaccessible full text available December 10, 2024

We study the natural problem of Triplet Reconstruction (also known as Rooted Triplets Consistency or Triplet Clustering), originally motivated by applications in computational biology and relational databases (Aho, Sagiv, Szymanski, and Ullman, 1981) [2]: given n datapoints, we want to embed them onto the n leaves of a rooted binary tree (also known as a hierarchical clustering, or ultrametric embedding) such that a given set of m triplet constraints is satisfied. A triplet constraint i j · k for points i, j, k indicates that 'i, j are more closely related to each other than to k,' (in terms of distances d(i, j) ≤ d(i, k) and d(i, j) ≤ d(j, k)) and we say that a tree satisfies the triplet i j · k if the distance in the tree between i, j is smaller than the distance between i, k (or j, k). Among all possible trees with n leaves, can we efficiently find one that satisfies a large fraction of the m given triplets? Aho et al. (1981) [2] studied the decision version and gave an elegant polynomialtime algorithm that determines whether or not there exists a tree that satisfies all of the m constraints. Moreover, it is straightforward to see that a random binary tree achieves a constant 13approximation, since there are only 3 distinct triplets i jk, i k j, j k · i (each will be satisfied w.p. 13). Unfortunately, despite more than four decades of research by various communities, there is no better approximation algorithm for this basic Triplet Reconstruction problem.Our main theoremwhich captures Triplet Reconstruction as a special caseis a general hardness of approximation result about Constraint Satisfaction Problems (CSPs) over infinite domains (CSPs where instead of boolean values {0,1} or a fixedsize domain, the variables can be mapped to any of the n leaves of a tree). Specifically, we prove that assuming the Unique Games Conjecture [57], Triplet Reconstruction and more generally, every Constraint Satisfaction Problem (CSP) over hierarchies is approximation resistant, i.e., there is no polynomialtime algorithm that does asymptotically better than a biased random assignment.Our result settles the approximability not only for Triplet Reconstruction, but for many interesting problems that have been studied by various scientific communities such as the popular Quartet Reconstruction and Subtree/Supertree Aggregation Problems. More broadly, our result significantly extends the list of approximation resistant predicates by pointing to a large new family of hard problems over hierarchies. Our main theorem is a generalization of Guruswami, Håstad, Manokaran, Raghavendra, and Charikar (2011) [36], who showed that ordering CSPs (CSPs over permutations of n elements, e.g., Max Acyclic Subgraph, Betweenness, NonBetweenness) are approximation resistant. The main challenge in our analyses stems from the fact that trees have topology (in contrast to permutations and ordering CSPs) and it is the tree topology that determines whether a given constraint on the variables is satisfied or not. As a byproduct, we also present some of the first CSPs where their approximation resistance is proved against biased random assignments, instead of uniformly random assignments.more » « lessFree, publiclyaccessible full text available November 6, 2024

Gørtz, Inge Li ; FarachColton, Martin ; Puglisi, Simon J ; Herman, Grzegorz (Ed.)We consider variants of the classic Multiway Cut problem. Multiway Cut asks to partition a graph G into k parts so as to separate k given terminals. Recently, Chandrasekaran and Wang (ESA 2021) introduced l_pnorm Multiway Cut, a generalization of the problem, in which the goal is to minimize the l_p norm of the edge boundaries of k parts. We provide an O(log^{1/2} n log^{1/2 + 1/p} k) approximation algorithm for this problem, improving upon the approximation guarantee of O(log^{3/2} n log^{1/2} k) due to Chandrasekaran and Wang. We also introduce and study Norm Multiway Cut, a further generalization of Multiway Cut. We assume that we are given access to an oracle, which answers certain queries about the norm. We present an O(log^{1/2} n log^{7/2} k) approximation algorithm with a weaker oracle and an O(log^{1/2} n log^{5/2} k) approximation algorithm with a stronger oracle. Additionally, we show that without any oracle access, there is no n^{1/4ε} approximation algorithm for every ε > 0 assuming the Hypergraph DensevsRandom Conjecture.more » « less

We provide a new bicriteria O(log2k) competitive algorithm for explainable kmeans clustering. Explainable kmeans was recently introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). It is described by an easy to interpret and understand (threshold) decision tree or diagram. The cost of the explainable kmeans clustering equals to the sum of costs of its clusters; and the cost of each cluster equals the sum of squared distances from the points in the cluster to the center of that cluster. The best non bicriteria algorithm for explainable clustering O(k) competitive, and this bound is tight. Our randomized bicriteria algorithm constructs a threshold decision tree that partitions the data set into (1+δ)k clusters (where δ∈(0,1) is a parameter of the algorithm). The cost of this clustering is at most O(1/δ⋅log2k) times the cost of the optimal unconstrained kmeans clustering. We show that this bound is almost optimal.more » « less

null (Ed.)In the Correlation Clustering problem, we are given a complete weighted graph G with its edges labeled as “similar" and “dissimilar" by a noisy binary classifier. For a clustering C of graph G, a similar edge is in disagreement with C, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with C if its endpoints belong to the same cluster. The disagreements vector, Agree, is a vector indexed by the vertices of G such that the vth coordinate Disagre equals the weight of all disagreeing edges incident on v. The goal is to produce a clustering that minimizes the ℓp norm of the disagreements vector for p≥1. We study the ℓ_p objective in Correlation Clustering under the following assumption: Every similar edge has weight in [αw,w] and every dissimilar edge has weight at least αw (where α≤1 and w>0 is a scaling parameter). We give an O((1/α)^{1/2−1/2p}⋅log(1/α)) approximation algorithm for this problem. Furthermore, we show an almost matching convex programming integrality gap.more » « less

null (Ed.)We consider the problem of explainable kmedians and kmeans introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). In this problem, our goal is to find a threshold decision tree that partitions data into k clusters and minimizes the kmedians or kmeans objective. The obtained clustering is easy to interpret because every decision node of a threshold tree splits data based on a single feature into two groups. We propose a new algorithm for this problem which is O(log k) competitive with kmedians with ℓ1 norm and O(k) competitive with kmeans. This is an improvement over the previous guarantees of O(k) and O(k^2) by Dasgupta et al (2020). We also provide a new algorithm which is O(log^{3}{2}k) competitive for kmedians with ℓ2 norm. Our first algorithm is nearoptimal: Dasgupta et al (2020) showed a lower bound of Ω(log k) for kmedians; in this work, we prove a lower bound of Ω(k) for kmeans. We also provide a lower bound of Ω(log k) for kmedians with ℓ2 norm.more » « less