About

I am a Professor of Computer Science at Northwestern University. I am interested in designing efficient algorithms for computationally hard problems. The aim of my research is to introduce new core techniques and design general principles for developing and analyzing algorithms that work in theory and practice. My research interests include approximation algorithms, beyond worst-case analysis, theory of machine learning, and applications of high-dimension geometry in computer science.

Research topics

• Computer Science
• Algorithm
• Mathematics
• Discrete mathematics
• Artificial Intelligence
• Mathematical analysis
• Geometry
• Statistics
• Combinatorics
• Arithmetic
• Physics

Selected publications

• A Simple Average-case Analysis of Recursive Randomized Greedy MIS

  ArXiv.org · 2026-04-01

  articleOpen access

  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.

  PublisherOA PDF

• On the Approximability of Max-Cut on 3-Colorable Graphs and Graphs with Large Independent Sets

  arXiv (Cornell University) · 2026-04-11

  articleOpen access

  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.

  PublisherOA PDF

• A Simple Average-case Analysis of Recursive Randomized Greedy MIS

  Society for Industrial and Applied Mathematics eBooks · 2026-01-01

  book-chapter

  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 upperbounded 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.

  PublisherDOI

• On the Approximability of Max-Cut on 3-Colorable Graphs and Graphs with Large Independent Sets

  arXiv (Cornell University) · 2026-04-11

  preprintOpen access

  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.

  PublisherDOI

• A Simple Average-case Analysis of Recursive Randomized Greedy MIS

  arXiv (Cornell University) · 2026-04-01

  preprintOpen access

  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.

  PublisherDOI

• Optimal Phylogenetic Reconstruction from Sampled Quartets

  ArXiv.org · 2026-04-19

  articleOpen accessSenior author

  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.

  PublisherOA PDF

• Optimal Phylogenetic Reconstruction from Sampled Quartets

  arXiv (Cornell University) · 2026-04-19

  preprintOpen accessSenior author

  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.

  PublisherDOI

• SPARSE-PIVOT: Dynamic correlation clustering for node insertions

  ArXiv.org · 2025-07-02

  preprintOpen access

  We present a new Correlation Clustering algorithm for a dynamic setting where nodes are added one at a time. In this model, proposed by Cohen-Addad, Lattanzi, Maggiori, and Parotsidis (ICML 2024), the algorithm uses database queries to access the input graph and updates the clustering as each new node is added. Our algorithm has the amortized update time of $O_ε(\log^{O(1)}(n))$. Its approximation factor is $20+\varepsilon$, which is a substantial improvement over the approximation factor of the algorithm by Cohen-Addad et al. We complement our theoretical findings by empirically evaluating the approximation guarantee of our algorithm. The results show that it outperforms the algorithm by Cohen-Addad et al.~in practice.

  PublisherOA PDFDOI

• Dynamic Algorithm for Explainable k-medians Clustering under lp Norm

  ArXiv.org · 2025-12-01

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

• Constraint Satisfaction Problems with Advice

  arXiv (Cornell University) · 2024-03-04

  preprintOpen access

  We initiate the study of algorithms for constraint satisfaction problems with ML oracle advice. We introduce two models of advice and then design approximation algorithms for Max Cut, Max $2$-Lin, and Max $3$-Lin in these models. In particular, we show the following. 1. For Max-Cut and Max $2$-Lin, we design an algorithm that yields near-optimal solutions when the average degree is larger than a threshold degree, which only depends on the amount of advice and is independent of the instance size. We also give an algorithm for nearly satisfiable Max $3$-Lin instances with quantitatively similar guarantees. 2. Further, we provide impossibility results for algorithms in these models. In particular, under standard complexity assumptions, we show that Max $3$-Lin is still $1/2 + η$ hard to approximate given access to advice, when there are no assumptions on the instance degree distribution. Additionally, we also show that Max $4$-Lin is $1/2 + η$ hard to approximate even when the average degree of the instance is linear in the number of variables.

  PublisherOA PDFDOI

Recent grants

• Collaborative Research: AF: Medium: Design and Analysis of Models and Algorithms for Real-life Problems

  NSF · $724k · 2020–2026

Frequent coauthors

• David A. Bader

  162 shared

• Guojing Cong

  Oak Ridge National Laboratory

  159 shared

• Srinivas Aluru

  150 shared

• Yury Makarychev

  122 shared

• Felix Wolf

  Technical University of Darmstadt

  82 shared

• Robert Henschel

  81 shared

• Sandhya Dwarkadas

  81 shared

• Matthias Müller

  81 shared

Labs

• Konstantin Makarychev LabPI

Education

• Ph.D.

  Princeton University

  2007

• B.S.

  Department of Mathematics at Moscow State University

• Other

  Moscow Math High School #57