Abhishek Dhawan

Abhishek Dhawan, Cheng Mao, Ashwin Pananjady

We consider a symmetric mixture of linear regressions with random samples from the pairwise comparison design, which can be seen as a noisy version of a type of Euclidean distance geometry problem. We analyze the expectation-maximization (EM) algorithm locally around the ground truth and establish that the sequence converges linearly, providing an $\ell_\infty$-norm guarantee on the estimation error of the iterates. Furthermore, we show that the limit of the EM sequence achieves the sharp rate of estimation in the $\ell_2$-norm, matching the information-theoretically optimal constant. We also argue through simulation that convergence from a random initialization is much more delicate in this setting, and does not appear to occur in general. Our results show that the EM algorithm can exhibit several unique behaviors when the covariate distribution is suitably structured.

Abhishek Dhawan, Nhi U. Dinh, Eren C. Kızıldağ et al.

We study the algorithmic tractability of finding large independent sets in dense random hypergraphs. In the sparse regime, much of the natural algorithms can be formulated within either the local or the low-degree polynomial (LDP) framework, and a rich literature has subsequently identified nearly sharp algorithmic thresholds within these classes by exploiting their stability. In the dense setting, however, the algorithmic paradigms are fundamentally different: they are online and thus need not be stable. Perhaps more crucially, even for the classical Erdős-Rényi random graph $G(n,p)$, LDPs are conjectured to fail in the 'easy' regime accessible to online algorithms, thereby challenging their viability for dense models. Our focus is on two models: (i) finding large independent sets in dense $r$-uniform Erdős-Rényi hypergraphs, and (ii) the more challenging problem of finding large $γ$-balanced independent sets in dense $r$-uniform $r$-partite hypergraphs, where the $i$-th coordinate of $γ\in\mathbb{Q}^r$ specifies the proportion of vertices from $V_i$ in the independent set. For both models, we pinpoint the size of the largest independent set and design online algorithms that achieve a multiplicative approximation factor of $r^{1/(r-1)}$ in the uniform and $(\max_i γ_i)^{-1/(r-1)}$ in the $r$-partite model. Furthermore, we establish matching algorithmic lower bounds, showing that these computational gaps are sharp: no online algorithms can breach these gaps.

Abhishek Dhawan

In recent work, Martinsson and Steiner showed that every $K_3$-free $d$-degenerate graph $G$ has fractional chromatic number $χ_f(G) = O\left(\frac{d}{\log d}\right)$. In this paper, we extend the result in two ways, employing an approach rooted in the analysis of the entropy of certain probability distributions. Our argument provides a template to tackle other problems, so it is of independent interest. First, we consider locally $r$-colorable graphs $G$, i.e., where $χ(G[N(v)]) \leq r$ for each vertex $v$. We show that $d$-degenerate locally $r$-colorable graphs $G$ satisfy $χ_f(G) = O\left(\frac{d\log (2r)}{\log d}\right)$, strengthening a result of Alon (1996) on the independence number of such graphs. Second, we extend Martinsson and Steiner's result to $r$-uniform $d$-degenerate hypergraphs $H$ of girth at least $4$. We show that such hypergraphs satisfy $χ_f(H) \leq c_r\left(\frac{d}{\log d}\right)^{\frac{1}{r-1}}$, implying a strict generalization of a seminal result of Ajtai, Komlós, Pintz, Spencer, and Szemerédi (1982) on the independence number of uncrowded hypergraphs. As a corollary, we obtain the same growth rate for the fractional chromatic number of $d$-degenerate linear hypergraphs. Our approach is constructive, yielding efficient algorithms to sample independent sets in each of the settings we consider.

Richard Bi, Peter Bradshaw, Abhishek Dhawan et al.

A strong edge coloring of a graph $G$ is an edge coloring $ϕ\,:\,E(G) \rightarrow \mathbb N$ such that each color class forms an induced matching in $G$. The strong chromatic index of $G$, written $χ'_s(G)$, is the minimum number of colors needed for a strong edge coloring of $G$. Erdős and Nešetřil conjectured in 1985 that if $G$ has maximum degree $d$, then $χ'_s(G) \leq \frac 54 d^2$. Mahdian showed in 2000 that if $G$ is $C_4$-free, then $χ'_s(G) \leq (2+o(1)) \frac{d^2}{\log d}$, and he conjectured that the same upper bound holds for $K_{t,t}$-free graphs. In this paper, we prove this conjecture and improve upon it to show the following: every $K_{t,t}$-free graph $G$ of maximum degree $d$ satisfies $χ'_s(G) \leq (1+o(1)) \frac{d^2}{\log d}$. We employ a variant of the Rödl nibble method to prove this result. The key new ingredient in our adaptation of the method is an application of the Kővári-Sós-Turán theorem to show that $H := L(G)^2$ satisfies certain structural properties. These properties, in conjunction with a variant of Talagrand's inequality to handle exceptional outcomes, allow us to concentrate the sizes of certain vertex sets through the nibble, even when these vertex sets have order smaller than the maximum codegree of $H$. We encapsulate these structural properties into a more general statement on list coloring that we believe to be of independent interest. In light of the conjectured computational threshold for coloring random graphs arising in average-case complexity theory, we suspect that our result is best possible using this approach.