Euiwoong Lee

Nairen Cao, Vincent Cohen-Addad, Euiwoong Lee et al.

Correlation clustering is a well-studied problem, first proposed by Bansal, Blum, and Chawla [Mach. Learn. '04]. The input is an unweighted, undirected graph. The problem is to cluster the vertices so as to minimize the number of edges between vertices in different clusters and missing edges between vertices inside the same cluster. This problem has a wide application in data mining and machine learning. We introduce a general framework that transforms existing static correlation clustering algorithms into fully-dynamic ones that work against an adaptive adversary. We show how to apply our framework to known efficient correlation clustering algorithms, starting from the classic 3-approximate Pivot algorithm from Ailon, Charikar and Newman [JACM'08]. Applied to the most recent sublinear $1.485$-approximation algorithm from Cao, Cohen-Addad, Lee, Li, Lolck, Newman, Thorup, Vogl, Yan and Zhang [STOC'25], we get a $1.485$-approximation fully-dynamic algorithm that works with worst-case constant update time. The original static algorithm gets its approximation factor with constant probability, and we get the same against an adaptive adversary in the sense that for any given update step, not known to our algorithm, our solution is a $1.485$-approximation with constant probability when we reach this update. Most of previous dynamic algorithms, including the celebrated result from Behnezhad, Charikar, Ma and Tan [FOCS'19], had approximation factors around $3$ in expectation, and they could only handle an oblivious adversary. A recent algorithm by Braverman, Dharangutte, Pai, Shah, and Wang [AISTATS'25] could handle an adaptive adversary, but it has a large unspecified constant approximation ratio. This contrasts with our general transformation, which works with all the best approximation factors known for the static case.

Hadley Black, Euiwoong Lee, Arya Mazumdar et al.

Recovering the underlying clustering of a set $U$ of $n$ points by asking pair-wise same-cluster queries has garnered significant interest in the last decade. Given a query $S \subset U$, $|S|=2$, the oracle returns yes if the points are in the same cluster and no otherwise. For adaptive algorithms with pair-wise queries, the number of required queries is known to be $Θ(nk)$, where $k$ is the number of clusters. However, non-adaptive schemes require $Ω(n^2)$ queries, which matches the trivial $O(n^2)$ upper bound attained by querying every pair of points. To break the quadratic barrier for non-adaptive queries, we study a generalization of this problem to subset queries for $|S|>2$, where the oracle returns the number of clusters intersecting $S$. Allowing for subset queries of unbounded size, $O(n)$ queries is possible with an adaptive scheme (Chakrabarty-Liao, 2024). However, the realm of non-adaptive algorithms is completely unknown. In this paper, we give the first non-adaptive algorithms for clustering with subset queries. Our main result is a non-adaptive algorithm making $O(n \log k \cdot (\log k + \log\log n)^2)$ queries, which improves to $O(n \log \log n)$ when $k$ is a constant. We also consider algorithms with a restricted query size of at most $s$. In this setting we prove that $Ω(\max(n^2/s^2,n))$ queries are necessary and obtain algorithms making $\tilde{O}(n^2k/s^2)$ queries for any $s \leq \sqrt{n}$ and $\tilde{O}(n^2/s)$ queries for any $s \leq n$. We also consider the natural special case when the clusters are balanced, obtaining non-adaptive algorithms which make $O(n \log k) + \tilde{O}(k)$ and $O(n\log^2 k)$ queries. Finally, allowing two rounds of adaptivity, we give an algorithm making $O(n \log k)$ queries in the general case and $O(n \log \log k)$ queries when the clusters are balanced.

Vincent Cohen-Addad, Fabrizio Grandoni, Euiwoong Lee et al.

In the classical NP-hard metric $k$-median problem, we are given a set of $n$ clients and centers with metric distances between them, along with an integer parameter $k\geq 1$. The objective is to select a subset of $k$ open centers that minimizes the total distance from each client to its closest open center. In their seminal work, Jain, Mahdian, Markakis, Saberi, and Vazirani presented the Greedy algorithm for facility location, which implies a $2$-approximation algorithm for $k$-median that opens $k$ centers in expectation. Since then, substantial research has aimed at narrowing the gap between their algorithm and the best achievable approximation by an algorithm guaranteed to open exactly $k$ centers. During the last decade, all improvements have been achieved by leveraging their algorithm or a small improvement thereof, followed by a second step called bi-point rounding, which inherently increases the approximation guarantee. Our main result closes this gap: for any $ε>0$, we present a $(2+ε)$-approximation algorithm for $k$-median, improving the previous best-known approximation factor of $2.613$. Our approach builds on a combination of two algorithms. First, we present a non-trivial modification of the Greedy algorithm that operates with $O(\log n/ε^2)$ adaptive phases. Through a novel walk-between-solutions approach, this enables us to construct a $(2+ε)$-approximation algorithm for $k$-median that consistently opens at most $k + O(\log n{/ε^2})$ centers. Second, we develop a novel $(2+ε)$-approximation algorithm tailored for stable instances, where removing any center from an optimal solution increases the cost by at least an $Ω(ε^3/\log n)$ fraction. Achieving this involves a sampling approach inspired by the $k$-means++ algorithm and a reduction to submodular optimization subject to a partition matroid.

Karthik C. S., Euiwoong Lee, Yuval Rabani et al.

The $\ell_2^2$ min-sum $k$-clustering problem is to partition an input set into clusters $C_1,\ldots,C_k$ to minimize $\sum_{i=1}^k\sum_{p,q\in C_i}\|p-q\|_2^2$. Although $\ell_2^2$ min-sum $k$-clustering is NP-hard, it is not known whether it is NP-hard to approximate $\ell_2^2$ min-sum $k$-clustering beyond a certain factor. In this paper, we give the first hardness-of-approximation result for the $\ell_2^2$ min-sum $k$-clustering problem. We show that it is NP-hard to approximate the objective to a factor better than $1.056$ and moreover, assuming a balanced variant of the Johnson Coverage Hypothesis, it is NP-hard to approximate the objective to a factor better than 1.327. We then complement our hardness result by giving a nearly linear time parameterized PTAS for $\ell_2^2$ min-sum $k$-clustering running in time $O\left(n^{1+o(1)}d\cdot \exp((k\cdot\varepsilon^{-1})^{O(1)})\right)$, where $d$ is the underlying dimension of the input dataset. Finally, we consider a learning-augmented setting, where the algorithm has access to an oracle that outputs a label $i\in[k]$ for input point, thereby implicitly partitioning the input dataset into $k$ clusters that induce an approximately optimal solution, up to some amount of adversarial error $α\in\left[0,\frac{1}{2}\right)$. We give a polynomial-time algorithm that outputs a $\frac{1+γα}{(1-α)^2}$-approximation to $\ell_2^2$ min-sum $k$-clustering, for a fixed constant $γ>0$.

Vincent Cohen-Addad, Karthik C. S., Euiwoong Lee

K-median and k-means are the two most popular objectives for clustering algorithms. Despite intensive effort, a good understanding of the approximability of these objectives, particularly in $\ell_p$-metrics, remains a major open problem. In this paper, we significantly improve upon the hardness of approximation factors known in literature for these objectives in $\ell_p$-metrics. We introduce a new hypothesis called the Johnson Coverage Hypothesis (JCH), which roughly asserts that the well-studied max k-coverage problem on set systems is hard to approximate to a factor greater than 1-1/e, even when the membership graph of the set system is a subgraph of the Johnson graph. We then show that together with generalizations of the embedding techniques introduced by Cohen-Addad and Karthik (FOCS '19), JCH implies hardness of approximation results for k-median and k-means in $\ell_p$-metrics for factors which are close to the ones obtained for general metrics. In particular, assuming JCH we show that it is hard to approximate the k-means objective: $\bullet$ Discrete case: To a factor of 3.94 in the $\ell_1$-metric and to a factor of 1.73 in the $\ell_2$-metric; this improves upon the previous factor of 1.56 and 1.17 respectively, obtained under UGC. $\bullet$ Continuous case: To a factor of 2.10 in the $\ell_1$-metric and to a factor of 1.36 in the $\ell_2$-metric; this improves upon the previous factor of 1.07 in the $\ell_2$-metric obtained under UGC. We also obtain similar improvements under JCH for the k-median objective. Additionally, we prove a weak version of JCH using the work of Dinur et al. (SICOMP '05) on Hypergraph Vertex Cover, and recover all the results stated above of Cohen-Addad and Karthik (FOCS '19) to (nearly) the same inapproximability factors but now under the standard NP$\neq$P assumption (instead of UGC).

Vincent Cohen-Addad, Karthik C. S., Euiwoong Lee

The k-means objective is arguably the most widely-used cost function for modeling clustering tasks in a metric space. In practice and historically, k-means is thought of in a continuous setting, namely where the centers can be located anywhere in the metric space. For example, the popular Lloyd's heuristic locates a center at the mean of each cluster. Despite persistent efforts on understanding the approximability of k-means, and other classic clustering problems such as k-median and k-minsum, our knowledge of the hardness of approximation factors of these problems remains quite poor. In this paper, we significantly improve upon the hardness of approximation factors known in the literature for these objectives. We show that if the input lies in a general metric space, it is NP-hard to approximate: $\bullet$ Continuous k-median to a factor of $2-o(1)$; this improves upon the previous inapproximability factor of 1.36 shown by Guha and Khuller (J. Algorithms '99). $\bullet$ Continuous k-means to a factor of $4- o(1)$; this improves upon the previous inapproximability factor of 2.10 shown by Guha and Khuller (J. Algorithms '99). $\bullet$ k-minsum to a factor of $1.415$; this improves upon the APX-hardness shown by Guruswami and Indyk (SODA '03). Our results shed new and perhaps counter-intuitive light on the differences between clustering problems in the continuous setting versus the discrete setting (where the candidate centers are given as part of the input).

Frank Ban, Vijay Bhattiprolu, Karl Bringmann et al.

A number of recent works have studied algorithms for entrywise $\ell_p$-low rank approximation, namely, algorithms which given an $n \times d$ matrix $A$ (with $n \geq d$), output a rank-$k$ matrix $B$ minimizing $\|A-B\|_p^p=\sum_{i,j}|A_{i,j}-B_{i,j}|^p$ when $p > 0$; and $\|A-B\|_0=\sum_{i,j}[A_{i,j}\neq B_{i,j}]$ for $p=0$. On the algorithmic side, for $p \in (0,2)$, we give the first $(1+ε)$-approximation algorithm running in time $n^{\text{poly}(k/ε)}$. Further, for $p = 0$, we give the first almost-linear time approximation scheme for what we call the Generalized Binary $\ell_0$-Rank-$k$ problem. Our algorithm computes $(1+ε)$-approximation in time $(1/ε)^{2^{O(k)}/ε^{2}} \cdot nd^{1+o(1)}$. On the hardness of approximation side, for $p \in (1,2)$, assuming the Small Set Expansion Hypothesis and the Exponential Time Hypothesis (ETH), we show that there exists $δ:= δ(α) > 0$ such that the entrywise $\ell_p$-Rank-$k$ problem has no $α$-approximation algorithm running in time $2^{k^δ}$.

Kijung Shin, Euiwoong Lee, Dhivya Eswaran et al.

We consider goods that can be shared with k-hop neighbors (i.e., the set of nodes within k hops from an owner) on a social network. We examine incentives to buy such a good by devising game-theoretic models where each node decides whether to buy the good or free ride. First, we find that social inefficiency, specifically excessive purchase of the good, occurs in Nash equilibria. Second, the social inefficiency decreases as k increases and thus a good can be shared with more nodes. Third, and most importantly, the social inefficiency can also be significantly reduced by charging free riders an access cost and paying it to owners, leading to the conclusion that organizations and system designers should impose such a cost. These findings are supported by our theoretical analysis in terms of the price of anarchy and the price of stability; and by simulations based on synthetic and real social networks.