Konstantin Makarychev

Dionysis Arvanitakis, Vaggos Chatziafratis, Yiyuan Luo et al.

Quartet Reconstruction, the task of recovering a phylogenetic tree from smaller trees on four species called \textit{quartets}, is a well-studied problem in theoretical computer science with far-reaching connections to statistics, graph theory and biology. Given a random sample containing $m$ noisy quartets, labeled by an unknown ground-truth tree $T$ on $n$ taxa, we want to output a tree $\widehat T$ that is \textit{close} to $T$ in terms of quartet distance and can predict unseen quartets. Unfortunately, the empirical risk minimizer corresponds to the $\mathsf{NP}$-hard problem of finding a tree that maximizes agreements with the sampled quartets, and earlier works in approximation algorithms gave $(1-\eps)$-approximation schemes (PTAS) for dense instances with $m=Θ(n^4)$ quartets, or for $m=Θ(n^2\log n)$ quartets \textit{randomly} sampled from $T$. Prior to our work, it was unknown how many samples are information-theoretically required to learn the tree, and whether there is an efficient reconstruction algorithm. We present optimal results for reconstructing an unknown phylogenetic tree $T$ from a random sample of $m=Θ(n)$ quartets, corrupted under the Random Classification Noise (RCN) model. This matches the $Ω(n)$ lower bound required for any meaningful tree reconstruction. Our contribution is twofold: first, we give a tree reconstruction algorithm that, not only achieves a $(1-\eps)$-approximation, but most importantly \textit{recovers} a tree close to $T$ in quartet distance; second, we show a new $Θ(n)$ bound on the Natarajan dimension of phylogenies (an analog of VC dimension in multiclass classification). Our analysis relies on a new \textit{Quartet-based Embedding and Detection} procedure that identifies and removes well-clustered subtrees from the (unknown) ground-truth $T$ via semidefinite programming.

Suprovat Ghoshal, Neng Huang, Euiwoong Lee et al.

Max-Cut is a classical graph-partitioning problem where given a graph $G = (V,E)$, the objective is to find a cut $(S,S^c)$ which maximizes the number of edges crossing the cut. In a seminal work, Goemans and Williamson gave an $α_{GW} \approx 0.87856$-factor approximation algorithm for the problem, which was later shown to be tight by the work of Khot, Kindler, Mossel, and O'Donnell. Since then, there has been a steady progress in understanding the approximability at even finer levels, and a fundamental goal in this context is to understand how the structure of the underlying graph affects the approximability of the Max-Cut problem. In this work, we investigate this question by exploring how the chromatic structure of a graph affects the Max-Cut problem. While it is well-known that Max-Cut can be solved perfectly and near-perfectly in $2$-colorable and almost $2$-colorable graphs in polynomial time, here we explore its approximability under much weaker structural conditions such as when the graph is $3$-colorable or contains a large independent set. Our main contributions in this context are as follows: 1. We show Max-Cut is $α_{GW}$-hard to approximate for $3$-colorable graphs. 2. We identify a natural threshold $α^*$ such that the following holds. Firstly, for graphs which contain an independent set of size up to $α^*$, Max-Cut continues to be $α_{GW}$-factor hard to approximate. Furthermore, for any graph that contains an independent set of size $> α^*$, there exists an efficient $>α_{GW}$-approximation algorithm for Max-Cut. Our hardness results are derived using various analytical tools and novel variants of the Majority-Is-Stablest theorem, which might be of independent interest. Our algorithmic results are based on a novel SDP relaxation, which is then rounded and analyzed using interval arithmetic.

Konstantin Makarychev, Ilias Papanikolaou, Liren Shan

We study the problem of explainable k-medians clustering introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (2020). In this problem, the goal is to construct a threshold decision tree that partitions data into k clusters while minimizing the k-medians objective. These trees are interpretable because each internal node makes a simple decision by thresholding a single feature, allowing users to trace and understand how each point is assigned to a cluster. We present the first algorithm for explainable k-medians under lp norm for every finite p >= 1. Our algorithm achieves an O(p(log k)^{1 + 1/p - 1/p^2}) approximation to the optimal k-medians cost for any p >= 1. Previously, algorithms were known only for p = 1 and p = 2. For p = 2, our algorithm improves upon the existing bound of O(log^{3/2}k), and for p = 1, it matches the tight bound of log k + O(1) up to a multiplicative O(log log k) factor. We show how to implement our algorithm in a dynamic setting. The dynamic algorithm maintains an explainable clustering under a sequence of insertions and deletions, with amortized update time O(d log^3 k) and O(log k) recourse, making it suitable for large-scale and evolving datasets.

Mina Dalirrooyfard, Konstantin Makarychev, Slobodan Mitrović

We revisit the complexity analysis of the recursive version of the randomized greedy algorithm for computing a maximal independent set (MIS), originally analyzed by Yoshida, Yamamoto, and Ito (2009). They showed that, on average per vertex, the expected number of recursive calls made by this algorithm is upper bounded by the average degree of the input graph. While their analysis is clever and intricate, we provide a significantly simpler alternative that achieves the same guarantee. Our analysis is inspired by the recent work of Dalirrooyfard, Makarychev, and Mitrović (2024), who developed a potential-function-based argument to analyze a new algorithm for correlation clustering. We adapt this approach to the MIS setting, yielding a more direct and arguably more transparent analysis of the recursive randomized greedy MIS algorithm.

Sayak Chakrabarty, Konstantin Makarychev

We show that a simple single-pass semi-streaming variant of the Pivot algorithm for Correlation Clustering gives a (3 + ε)-approximation using O(n/ε) words of memory. This is a slight improvement over the recent results of Cambus, Kuhn, Lindy, Pai, and Uitto, who gave a (3 + ε)-approximation using O(n log n) words of memory, and Behnezhad, Charikar, Ma, and Tan, who gave a 5-approximation 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.

Konstantin Makarychev, Liren Shan

We provide a new bi-criteria $\tilde{O}(\log^2 k)$ competitive algorithm for explainable $k$-means clustering. Explainable $k$-means 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 $k$-means 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 bi-criteria algorithm for explainable clustering $\tilde{O}(k)$ competitive, and this bound is tight. Our randomized bi-criteria algorithm constructs a threshold decision tree that partitions the data set into $(1+δ)k$ clusters (where $δ\in (0,1)$ is a parameter of the algorithm). The cost of this clustering is at most $\tilde{O}(1/ δ\cdot \log^2 k)$ times the cost of the optimal unconstrained $k$-means clustering. We show that this bound is almost optimal.

Jafar Jafarov, Sanchit Kalhan, Konstantin Makarychev et al.

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 $\mathcal{C}$ of graph $G$, a similar edge is in disagreement with $\mathcal{C}$, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with $\mathcal{C}$ if its endpoints belong to the same cluster. The disagreements vector, $\text{dis}$, is a vector indexed by the vertices of $G$ such that the $v$-th coordinate $\text{dis}_v$ equals the weight of all disagreeing edges incident on $v$. The goal is to produce a clustering that minimizes the $\ell_p$ norm of the disagreements vector for $p\geq 1$. We study the $\ell_p$ objective in Correlation Clustering under the following assumption: Every similar edge has weight in the range of $[α\mathbf{w},\mathbf{w}]$ and every dissimilar edge has weight at least $α\mathbf{w}$ (where $α\leq 1$ and $\mathbf{w}>0$ is a scaling parameter). We give an $O\left((\frac{1}α)^{\frac{1}{2}-\frac{1}{2p}}\cdot \log\frac{1}α\right)$ approximation algorithm for this problem. Furthermore, we show an almost matching convex programming integrality gap.

Jafar Jafarov, Sanchit Kalhan, Konstantin Makarychev et al.

In the Correlation Clustering problem, we are given a weighted graph $G$ with its edges labeled as "similar" or "dissimilar" by a binary classifier. The goal is to produce a clustering that minimizes the weight of "disagreements": the sum of the weights of "similar" edges across clusters and "dissimilar" edges within clusters. We study the correlation clustering problem under the following assumption: Every "similar" edge $e$ has weight $\mathbf{w}_e\in[α\mathbf{w}, \mathbf{w}]$ and every "dissimilar" edge $e$ has weight $\mathbf{w}_e\geq α\mathbf{w}$ (where $α\leq 1$ and $\mathbf{w}>0$ is a scaling parameter). We give a $(3 + 2 \log_e (1/α))$ approximation algorithm for this problem. This assumption captures well the scenario when classification errors are asymmetric. Additionally, we show an asymptotically matching Linear Programming integrality gap of $Ω(\log 1/α)$.

Konstantin Makarychev, Liren Shan

We consider the problem of explainable $k$-medians and $k$-means 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 $k$-medians or $k$-means 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 $\tilde O(\log k)$ competitive with $k$-medians with $\ell_1$ norm and $\tilde O(k)$ competitive with $k$-means. 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 $k$-medians with $\ell_2$ norm. Our first algorithm is near-optimal: Dasgupta et al (2020) showed a lower bound of $Ω(\log k)$ for $k$-medians; in this work, we prove a lower bound of $\tildeΩ(k)$ for $k$-means. We also provide a lower bound of $Ω(\log k)$ for $k$-medians with $\ell_2$ norm.

Konstantin Makarychev, Aravind Reddy, Liren Shan

In this paper, we study k-means++ and k-means++ parallel, the two most popular algorithms for the classic k-means clustering problem. We provide novel analyses and show improved approximation and bi-criteria approximation guarantees for k-means++ and k-means++ parallel. Our results give a better theoretical justification for why these algorithms perform extremely well in practice. We also propose a new variant of k-means++ parallel algorithm (Exponential Race k-means++) that has the same approximation guarantees as k-means++.

Sepideh Mahabadi, Konstantin Makarychev, Yury Makarychev et al.

We introduce and study the notion of an outer bi-Lipschitz extension of a map between Euclidean spaces. The notion is a natural analogue of the notion of a Lipschitz extension of a Lipschitz map. We show that for every map $f$ there exists an outer bi-Lipschitz extension $f'$ whose distortion is greater than that of $f$ by at most a constant factor. This result can be seen as a counterpart of the classic Kirszbraun theorem for outer bi-Lipschitz extensions. We also study outer bi-Lipschitz extensions of near-isometric maps and show upper and lower bounds for them. Then, we present applications of our results to prioritized and terminal dimension reduction problems. * We prove a prioritized variant of the Johnson-Lindenstrauss lemma: given a set of points $X\subset \mathbb{R}^d$ of size $N$ and a permutation ("priority ranking") of $X$, there exists an embedding $f$ of $X$ into $\mathbb{R}^{O(\log N)}$ with distortion $O(\log \log N)$ such that the point of rank $j$ has only $O(\log^{3 + \varepsilon} j)$ non-zero coordinates - more specifically, all but the first $O(\log^{3+\varepsilon} j)$ coordinates are equal to $0$; the distortion of $f$ restricted to the first $j$ points (according to the ranking) is at most $O(\log\log j)$. The result makes a progress towards answering an open question by Elkin, Filtser, and Neiman about prioritized dimension reductions. * We prove that given a set $X$ of $N$ points in $\mathbb{R}^d$, there exists a terminal dimension reduction embedding of $\mathbb{R}^d$ into $\mathbb{R}^{d'}$, where $d' = O\left(\frac{\log N}{\varepsilon^4}\right)$, which preserves distances $\|x-y\|$ between points $x\in X$ and $y \in \mathbb{R}^{d}$, up to a multiplicative factor of $1 \pm \varepsilon$. This improves a recent result by Elkin, Filtser, and Neiman. The dimension reductions that we obtain are nonlinear, and this nonlinearity is necessary.

Konstantin Makarychev, Yury Makarychev, Ilya Razenshteyn

Consider an instance of Euclidean $k$-means or $k$-medians clustering. We show that the cost of the optimal solution is preserved up to a factor of $(1+\varepsilon)$ under a projection onto a random $O(\log(k / \varepsilon) / \varepsilon^2)$-dimensional subspace. Further, the cost of every clustering is preserved within $(1+\varepsilon)$. More generally, our result applies to any dimension reduction map satisfying a mild sub-Gaussian-tail condition. Our bound on the dimension is nearly optimal. Additionally, our result applies to Euclidean $k$-clustering with the distances raised to the $p$-th power for any constant $p$. For $k$-means, our result resolves an open problem posed by Cohen, Elder, Musco, Musco, and Persu (STOC 2015); for $k$-medians, it answers a question raised by Kannan.

Konstantin Makarychev, Yury Makarychev, Aravindan Vijayaraghavan

We study the problem of learning communities in the presence of modeling errors and give robust recovery algorithms for the Stochastic Block Model (SBM). This model, which is also known as the Planted Partition Model, is widely used for community detection and graph partitioning in various fields, including machine learning, statistics, and social sciences. Many algorithms exist for learning communities in the Stochastic Block Model, but they do not work well in the presence of errors. In this paper, we initiate the study of robust algorithms for partial recovery in SBM with modeling errors or noise. We consider graphs generated according to the Stochastic Block Model and then modified by an adversary. We allow two types of adversarial errors, Feige---Kilian or monotone errors, and edge outlier errors. Mossel, Neeman and Sly (STOC 2015) posed an open question about whether an almost exact recovery is possible when the adversary is allowed to add $o(n)$ edges. Our work answers this question affirmatively even in the case of $k>2$ communities. We then show that our algorithms work not only when the instances come from SBM, but also work when the instances come from any distribution of graphs that is $εm$ close to SBM in the Kullback---Leibler divergence. This result also works in the presence of adversarial errors. Finally, we present almost tight lower bounds for two communities.

Konstantin Makarychev, Yury Makarychev, Aravindan Vijayaraghavan

In this paper, we propose and study a semi-random model for the Correlation Clustering problem on arbitrary graphs G. We give two approximation algorithms for Correlation Clustering instances from this model. The first algorithm finds a solution of value $(1+ δ) optcost + O_δ(n\log^3 n)$ with high probability, where $optcost$ is the value of the optimal solution (for every $δ> 0$). The second algorithm finds the ground truth clustering with an arbitrarily small classification error $η$ (under some additional assumptions on the instance).

Konstantin Makarychev, Yury Makarychev, Aravindan Vijayaraghavan

We propose and study a new semi-random semi-adversarial model for Balanced Cut, a planted model with permutation-invariant random edges (PIE). Our model is much more general than planted models considered previously. Consider a set of vertices V partitioned into two clusters $L$ and $R$ of equal size. Let $G$ be an arbitrary graph on $V$ with no edges between $L$ and $R$. Let $E_{random}$ be a set of edges sampled from an arbitrary permutation-invariant distribution (a distribution that is invariant under permutation of vertices in $L$ and in $R$). Then we say that $G + E_{random}$ is a graph with permutation-invariant random edges. We present an approximation algorithm for the Balanced Cut problem that finds a balanced cut of cost $O(|E_{random}|) + n \text{polylog}(n)$ in this model. In the regime when $|E_{random}| = Ω(n \text{polylog}(n))$, this is a constant factor approximation with respect to the cost of the planted cut.