Karthik C. S.

David P. Woodruff, Vincent Cohen-Addad, Lalit Jain et al.

Recent advances in large language models (LLMs) have opened new avenues for accelerating scientific research. While models are increasingly capable of assisting with routine tasks, their ability to contribute to novel, expert-level mathematical discovery is less understood. We present a collection of case studies demonstrating how researchers have successfully collaborated with advanced AI models, specifically Google's Gemini-based models (in particular Gemini Deep Think and its advanced variants), to solve open problems, refute conjectures, and generate new proofs across diverse areas in theoretical computer science, as well as other areas such as economics, optimization, and physics. Based on these experiences, we extract common techniques for effective human-AI collaboration in theoretical research, such as iterative refinement, problem decomposition, and cross-disciplinary knowledge transfer. While the majority of our results stem from this interactive, conversational methodology, we also highlight specific instances that push beyond standard chat interfaces. These include deploying the model as a rigorous adversarial reviewer to detect subtle flaws in existing proofs, and embedding it within a "neuro-symbolic" loop that autonomously writes and executes code to verify complex derivations. Together, these examples highlight the potential of AI not just as a tool for automation, but as a versatile, genuine partner in the creative process of scientific discovery.

Lotte Blank, Karl Bringmann, Parinya Chalermsook et al.

In the (continuous) Euclidean $k$-center problem, given $n$ points in $\mathbb{R}^d$ and an integer $k$, the goal is to find $k$ center points in $\mathbb{R}^d$ that minimize the maximum Euclidean distance from any input point to its closest center. In this paper, we establish conditional lower bounds for this problem in constant dimensions in two settings. $\bullet$ Parameterized by $k$: Assuming the Exponential Time Hypothesis (ETH), we show that there is no $f(k)n^{o(k^{1-1/d})}$-time algorithm for the Euclidean $k$-center problem. This result shows that the algorithm of Agarwal and Procopiuc [SODA 1998; Algorithmica 2002] is essentially optimal. Furthermore, our lower bound rules out any $(1+\varepsilon)$-approximation algorithm running in time $(k/\varepsilon)^{o(k^{1-1/d})}n^{O(1)}$, thereby establishing near-optimality of the corresponding approximation scheme by the same authors. $\bullet$ Small $k$: Assuming the 3-SUM hypothesis, we prove that for any $\varepsilon>0$ there is no $O(n^{2-\varepsilon})$-time algorithm for the Euclidean $2$-center problem in $\mathbb{R}^3$. This settles an open question posed by Agarwal, Ben Avraham, and Sharir [SoCG 2010; Computational Geometry 2013]. In addition, under the same hypothesis, we prove that for any $\varepsilon > 0$, the Euclidean $6$-center problem in $\mathbb{R}^2$ also admits no $O(n^{2-\varepsilon})$-time algorithm. The technical core of all our proofs is a novel geometric embedding of a system of linear equations. We construct a point set where each variable corresponds to a specific collection of points, and the geometric structure ensures that a small-radius clustering is possible if and only if the system has a valid solution.

Sudatta Bhattacharya, Sanjana Dey, Elazar Goldenberg et al.

A function $φ:\{0,1\}^n \to \{0,1\}^N$ is called an isometric embedding of the $n$-dimensional Hamming metric space to the $N$-dimensional edit metric space if, for all $x,y\in\{0,1\}^n$, the Hamming distance between $x$ and $y$ is equal to the edit distance between $φ(x)$ and $φ(y)$. The rate of such an embedding is defined as the ratio $n/N$. It is well known in the literature how to construct isometric embeddings with rate $Ω(1/\log n)$. However, achieving even near-isometric embeddings with positive constant rate has remained elusive until now. In this paper, we present an isometric embedding with rate $1/8$ by discovering connections to synchronization strings, which were studied in the context of insertion-deletion codes (Haeupler-Shahrasbi [JACM'21]). At a technical level, we introduce a framework for obtaining high-rate isometric embeddings using a novel object called misaligners. As an immediate consequence of our constant-rate isometric embedding, we improve known conditional lower bounds for various optimization problems in the edit metric, now with optimal dependence on the dimension. We complement our results by showing that no isometric embedding $φ:\{0,1\}^n \to \{0,1\}^N$ can have rate greater than $15/32$ for all positive integers $n$. En route to proving this upper bound, we uncover fundamental structural properties necessary for every Hamming-to-edit isometric embedding. We also prove similar upper and lower bounds for embeddings over larger alphabets. Finally, we consider embeddings $φ:Σ_{\mathrm{in}}^n \to Σ_{\mathrm{out}}^N$ between different input and output alphabets, where the rate is given by $\frac{n\log|Σ_{\mathrm{in}}|}{N\log|Σ_{\mathrm{out}}|}$. In this setting, we show that the rate can be made arbitrarily close to $1$.

Vincent Cohen-Addad, Karthik C. S., David Saulpic et al.

The $k$-means problem is a classic objective for modeling clustering in a metric space. Given a set of points in a metric space, the goal is to find $k$ representative points so as to minimize the sum of the squared distances from each point to its closest representative. In this work, we study the approximability of $k$-means in Euclidean spaces parameterized by the number of clusters, $k$. In seminal works, de la Vega, Karpinski, Kenyon, and Rabani [STOC'03] and Kumar, Sabharwal, and Sen [JACM'10] showed how to obtain a $(1+\varepsilon)$-approximation for high-dimensional Euclidean $k$-means in time $2^{(k/\varepsilon)^{O(1)}} \cdot dn^{O(1)}$. In this work, we introduce a new fine-grained hypothesis called Exponential Time for Expanders Hypothesis (XXH) which roughly asserts that there are no non-trivial exponential time approximation algorithms for the vertex cover problem on near perfect vertex expanders. Assuming XXH, we close the above long line of work on approximating Euclidean $k$-means by showing that there is no $2^{(k/\varepsilon)^{1-o(1)}} \cdot n^{O(1)}$ time algorithm achieving a $(1+\varepsilon)$-approximation for $k$-means in Euclidean space. This lower bound is tight as it matches the algorithm given by Feldman, Monemizadeh, and Sohler [SoCG'07] whose runtime is $2^{\tilde{O}(k/\varepsilon)} + O(ndk)$. Furthermore, assuming XXH, we show that the seminal $O(n^{kd+1})$ runtime exact algorithm of Inaba, Katoh, and Imai [SoCG'94] for $k$-means is optimal for small values of $k$.

Vincent Cohen-Addad, Karthik C. S., David Saulpic et al.

The $k$-median and $k$-means clustering objectives are classic objectives for modeling clustering in a metric space. Given a set of points in a metric space, the goal of the $k$-median (resp. $k$-means) problem is to find $k$ representative points so as to minimize the sum of the distances (resp. sum of squared distances) from each point to its closest representative. Cohen-Addad, Feldmann, and Saulpic [JACM'21] showed how to obtain a $(1+\varepsilon)$-factor approximation in low-dimensional Euclidean metric for both the $k$-median and $k$-means problems in near-linear time $2^{(1/\varepsilon)^{O(d^2)}} n \cdot \text{polylog}(n)$ (where $d$ is the dimension and $n$ is the number of input points). We improve this running time to $2^{\tilde{O}(1/\varepsilon)^{d-1}} \cdot n \cdot \text{polylog}(n)$, and show an almost matching lower bound: under the Gap Exponential Time Hypothesis for 3-SAT, there is no $2^{{o}(1/\varepsilon^{d-1})} n^{O(1)}$ algorithm achieving a $(1+\varepsilon)$-approximation for $k$-means.

Surya Teja Gavva, Karthik C. S., Sharath Punna

Over the last three decades, researchers have intensively explored various clustering tools for categorical data analysis. Despite the proposal of various clustering algorithms, the classical k-modes algorithm remains a popular choice for unsupervised learning of categorical data. Surprisingly, our first insight is that in a natural generative block model, the k-modes algorithm performs poorly for a large range of parameters. We remedy this issue by proposing a soft rounding variant of the k-modes algorithm (SoftModes) and theoretically prove that our variant addresses the drawbacks of the k-modes algorithm in the generative model. Finally, we empirically verify that SoftModes performs well on both synthetic and real-world datasets.

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 $γ>0$.

Amir Abboud, Nick Fischer, Elazar Goldenberg et al.

We study the fundamental problem of finding the best string to represent a given set, in the form of the Closest String problem: Given a set $X \subseteq Σ^d$ of $n$ strings, find the string $x^*$ minimizing the radius of the smallest Hamming ball around $x^*$ that encloses all the strings in $X$. In this paper, we investigate whether the Closest String problem admits algorithms that are faster than the trivial exhaustive search algorithm. We obtain the following results for the two natural versions of the problem: $\bullet$ In the continuous Closest String problem, the goal is to find the solution string $x^*$ anywhere in $Σ^d$. For binary strings, the exhaustive search algorithm runs in time $O(2^d poly(nd))$ and we prove that it cannot be improved to time $O(2^{(1-ε) d} poly(nd))$, for any $ε> 0$, unless the Strong Exponential Time Hypothesis fails. $\bullet$ In the discrete Closest String problem, $x^*$ is required to be in the input set $X$. While this problem is clearly in polynomial time, its fine-grained complexity has been pinpointed to be quadratic time $n^{2 \pm o(1)}$ whenever the dimension is $ω(\log n) < d < n^{o(1)}$. We complement this known hardness result with new algorithms, proving essentially that whenever $d$ falls out of this hard range, the discrete Closest String problem can be solved faster than exhaustive search. In the small-$d$ regime, our algorithm is based on a novel application of the inclusion-exclusion principle. Interestingly, all of our results apply (and some are even stronger) to the natural dual of the Closest String problem, called the Remotest String problem, where the task is to find a string maximizing the Hamming distance to all the strings in $X$.

Chengyuan Deng, Surya Teja Gavva, Karthik C. S. et al.

Over the last few years Explainable Clustering has gathered a lot of attention. Dasgupta et al. [ICML'20] initiated the study of explainable $k$-means and $k$-median clustering problems where the explanation is captured by a threshold decision tree which partitions the space at each node using axis parallel hyperplanes. Recently, Laber et al. [Pattern Recognition'23] made a case to consider the depth of the decision tree as an additional complexity measure of interest. In this work, we prove that even when the input points are in the Euclidean plane, then any depth reduction in the explanation incurs unbounded loss in the $k$-means and $k$-median cost. Formally, we show that there exists a data set $X\subseteq \mathbb{R}^2$, for which there is a decision tree of depth $k-1$ whose $k$-means/$k$-median cost matches the optimal clustering cost of $X$, but every decision tree of depth less than $k-1$ has unbounded cost w.r.t. the optimal cost of clustering. We extend our results to the $k$-center objective as well, albeit with weaker guarantees.

Amir Abboud, Mohammad Hossein Bateni, Vincent Cohen-Addad et al.

We consider the classic 1-center problem: Given a set $P$ of $n$ points in a metric space find the point in $P$ that minimizes the maximum distance to the other points of $P$. We study the complexity of this problem in $d$-dimensional $\ell_p$-metrics and in edit and Ulam metrics over strings of length $d$. Our results for the 1-center problem may be classified based on $d$ as follows. $\bullet$ Small $d$: Assuming the hitting set conjecture (HSC), we show that when $d=ω(\log n)$, no subquadratic algorithm can solve 1-center problem in any of the $\ell_p$-metrics, or in edit or Ulam metrics. $\bullet$ Large $d$: When $d=Ω(n)$, we extend our conditional lower bound to rule out subquartic algorithms for 1-center problem in edit metric (assuming Quantified SETH). On the other hand, we give a $(1+ε)$-approximation for 1-center in Ulam metric with running time $\tilde{O_{\varepsilon}}(nd+n^2\sqrt{d})$. We also strengthen some of the above lower bounds by allowing approximations or by reducing the dimension $d$, but only against a weaker class of algorithms which list all requisite solutions. Moreover, we extend one of our hardness results to rule out subquartic algorithms for the well-studied 1-median problem in the edit metric, where given a set of $n$ strings each of length $n$, the goal is to find a string in the set that minimizes the sum of the edit distances to the rest of the strings in the set.

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).

Vincent Cohen-Addad, Karthik C. S., Guillaume Lagarde

A classic problem in unsupervised learning and data analysis is to find simpler and easy-to-visualize representations of the data that preserve its essential properties. A widely-used method to preserve the underlying hierarchical structure of the data while reducing its complexity is to find an embedding of the data into a tree or an ultrametric. The most popular algorithms for this task are the classic linkage algorithms (single, average, or complete). However, these methods on a data set of $n$ points in $Ω(\log n)$ dimensions exhibit a quite prohibitive running time of $Θ(n^2)$. In this paper, we provide a new algorithm which takes as input a set of points $P$ in $\mathbb{R}^d$, and for every $c\ge 1$, runs in time $n^{1+\fracρ{c^2}}$ (for some universal constant $ρ>1$) to output an ultrametric $Δ$ such that for any two points $u,v$ in $P$, we have $Δ(u,v)$ is within a multiplicative factor of $5c$ to the distance between $u$ and $v$ in the "best" ultrametric representation of $P$. Here, the best ultrametric is the ultrametric $\tildeΔ$ that minimizes the maximum distance distortion with respect to the $\ell_2$ distance, namely that minimizes $\underset{u,v \in P}{\max}\ \frac{\tildeΔ(u,v)}{\|u-v\|_2}$. We complement the above result by showing that under popular complexity theoretic assumptions, for every constant $\varepsilon>0$, no algorithm with running time $n^{2-\varepsilon}$ can distinguish between inputs in $\ell_\infty$-metric that admit isometric embedding and those that incur a distortion of $\frac{3}{2}$. Finally, we present empirical evaluation on classic machine learning datasets and show that the output of our algorithm is comparable to the output of the linkage algorithms while achieving a much faster running time.