arXiv subjects

Abhishek Dhawan

Publications and source records attributed to Abhishek Dhawan.

29 records · Page 2Linked to original sources

arXiv2025

Fast and Simple $(1+ε)Δ$-Edge-Coloring of Dense Graphs

Let $ε\in (0, 1)$ and $n, Δ\in \mathbb N$ be such that $Δ= Ω\left(\max\left\{\frac{\log n}ε,\, \left(\frac{1}ε\log \frac{1}ε\right)^2\right\}\right)$. Given an $n$-vertex $m$-edge simple graph $G$ of maximum degree $Δ$, we present a randomized $O\left(m\,\log^3 Δ\,/\,ε^2\right)$-time algorithm that computes a proper $(1+ε)Δ$-edge-coloring of $G$ with high probability. This improves upon the best known results for a wide range of the parameters $ε$, $n$, and $Δ$. Our approach combines a flagging strategy from earlier work of the author with a shifting procedure employed by Duan, He, and Zhang for dynamic edge-coloring. The resulting algorithm is simple to implement and may be of practical interest.

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arXiv2025

A linear-time algorithm for $(1+ε)Δ$-edge-coloring

We present a randomized algorithm that, given a constant $ε> 0$, outputs a proper $(1+ε)Δ$-edge-coloring of an $m$-edge simple graph $G$ of maximum degree $Δ\geq 1/ε$ in $O(m)$ time with high probability. This is the first linear-time algorithm for this problem covering the full range of possible values of $Δ$. Indeed, even for edge-coloring with $2Δ- 1$ colors (i.e., meeting the "greedy" bound), no such linear-time algorithm has been previously known.

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arXiv2025

Multigraph edge-coloring with local list sizes

Let $G$ be a multigraph and $L\,:\,E(G) \to 2^\mathbb{N}$ be a list assignment on the edges of $G$. Suppose additionally, for every vertex $x$, the edges incident to $x$ have at least $f(x)$ colors in common. We consider a variant of local edge-colorings wherein the color received by an edge $e$ must be contained in $L(e)$. The locality appears in the function $f$, i.e., $f(x)$ is some function of the local structure of $x$ in $G$. Such a notion is a natural generalization of traditional local edge-coloring. Our main results include sufficient conditions on the function $f$ to construct such colorings. As corollaries, we obtain local analogs of Vizing and Shannon's theorems, recovering a recent result of Conley, Grebík and Pikhurko.

math.CO↗

arXiv2024

Palette Sparsification for Graphs with Sparse Neighborhoods

A seminal palette sparsification result of Assadi, Chen, and Khanna states that in every $n$-vertex graph of maximum degree $Δ$, sampling $Θ(\log n)$ colors per vertex from $\{1, \ldots, Δ+1\}$ almost certainly allows for a proper coloring from the sampled colors. Alon and Assadi extended this work proving a similar result for $O\left(Δ/\log Δ\right)$-coloring triangle-free graphs. Apart from being interesting results from a combinatorial standpoint, their results have various applications to the design of graph coloring algorithms in different models of computation. In this work, we focus on locally sparse graphs, i.e., graphs with sparse neighborhoods. We say a graph $G = (V, E)$ is $k$-locally-sparse if for each vertex $v \in V$, the subgraph $G[N(v)]$ contains at most $k$ edges. A celebrated result of Alon, Krivelevich, and Sudakov shows that such graphs are $O(Δ/\log (Δ/\sqrt{k}))$-colorable. For any $α\in (0, 1)$ and $k \ll Δ^{2α}$, let $G$ be a $k$-locally-sparse graph. For $q = Θ\left(Δ/\log \left(Δ^α/\sqrt{k}\right)\right)$, we show that sampling $O\left(Δ^α+ \sqrt{\log n}\right)$ colors per vertex is sufficient to obtain a proper $q$-coloring of $G$ from the sampled colors. Setting $k = 1$ recovers the aforementioned result of Alon and Assadi for triangle-free graphs. A key element in our proof is a proposition regarding correspondence coloring in the so-called color-degree setting, which improves upon recent work of Anderson, Kuchukova, and the author and is of independent interest.

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arXiv2024

Borel Vizing's Theorem for Graphs of Subexponential Growth

We show that every Borel graph $G$ of subexponential growth has a Borel proper edge-coloring with $Δ(G) + 1$ colors. We deduce this from a stronger result, namely that an $n$-vertex (finite) graph $G$ of subexponential growth can be properly edge-colored using $Δ(G) + 1$ colors by an $O(\log^\ast n)$-round deterministic distributed algorithm in the $\mathsf{LOCAL}$ model, where the implied constants in the $O(\cdot)$ notation are determined by a bound on the growth rate of $G$.

math.CO↗

arXiv2024

Balanced independent sets and colorings of hypergraphs

A $k$-uniform hypergraph $H = (V, E)$ is $k$-partite if $V$ can be partitioned into $k$ sets $V_1, \ldots, V_k$ such that every edge in $E$ contains precisely one vertex from each $V_i$. We call such a graph $n$-balanced if $|V_i| = n$ for each $i$. An independent set $I$ in $H$ is balanced if $|I\cap V_i| = |I\cap V_j|$ for each $1 \leq i, j \leq k$, and a coloring is balanced if each color class induces a balanced independent set in $H$. In this paper, we provide a lower bound on the balanced independence number $α_b(H)$ in terms of the average degree $D = |E|/n$, and an upper bound on the balanced chromatic number $χ_b(H)$ in terms of the maximum degree $Δ$. Our results recover those of recent work of Chakraborti for $k = 2$.

math.CO↗

arXiv2024

A Simple Algorithm for Near-Vizing Edge-Coloring in Near-Linear Time

We present a simple $(1+\varepsilon)Δ$-edge-coloring algorithm for graphs of maximum degree $Δ= Ω(\log n / \varepsilon)$ with running time $O\left(m\,\log^3 n/\varepsilon^3\right)$. Our algorithm improves upon that of [Duan, He, and Zhang; SODA19], which was the first near-linear time algorithm for this problem. While our results are weaker than the current state-of-the-art, our approach is significantly simpler, both in terms of analysis as well as implementation, and may be of practical interest.

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arXiv2023

Edge-Coloring Algorithms for Bounded Degree Multigraphs

In this paper, we consider algorithms for edge-coloring multigraphs $G$ of bounded maximum degree, i.e., $Δ(G) = O(1)$. Shannon's theorem states that any multigraph of maximum degree $Δ$ can be properly edge-colored with $\lfloor3Δ/2\rfloor$ colors. Our main results include algorithms for computing such colorings. We design deterministic and randomized sequential algorithms with running time $O(n\log n)$ and $O(n)$, respectively. This is the first improvement since the $O(n^2)$ algorithm in Shannon's original paper, and our randomized algorithm is optimal up to constant factors. We also develop distributed algorithms in the $\mathsf{LOCAL}$ model of computation. Namely, we design deterministic and randomized $\mathsf{LOCAL}$ algorithms with running time $\tilde O(\log^5 n)$ and $O(\log^2n)$, respectively. The deterministic sequential algorithm is a simplified extension of earlier work of Gabow et al. in edge-coloring simple graphs. The other algorithms apply the entropy compression method in a similar way to recent work by the author and Bernshteyn, where the authors design algorithms for Vizing's theorem for simple graphs. We also extend those results to Vizing's theorem for multigraphs.

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arXiv2023

Sharp analysis of EM for learning mixtures of pairwise differences

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.

math.ST↗

arXiv2023

Detection of Dense Subhypergraphs by Low-Degree Polynomials

Detection of a planted dense subgraph in a random graph is a fundamental statistical and computational problem that has been extensively studied in recent years. We study a hypergraph version of the problem. Let $G^r(n,p)$ denote the $r$-uniform Erdős-Rényi hypergraph model with $n$ vertices and edge density $p$. We consider detecting the presence of a planted $G^r(n^γ, n^{-α})$ subhypergraph in a $G^r(n, n^{-β})$ hypergraph, where $0< α< β< r-1$ and $0 < γ< 1$. Focusing on tests that are degree-$n^{o(1)}$ polynomials of the entries of the adjacency tensor, we determine the threshold between the easy and hard regimes for the detection problem. More precisely, for $0 < γ< 1/2$, the threshold is given by $α= βγ$, and for $1/2 \le γ< 1$, the threshold is given by $α= β/2 + r(γ- 1/2)$. Our results are already new in the graph case $r=2$, as we consider the subtle log-density regime where hardness based on average-case reductions is not known. Our proof of low-degree hardness is based on a conditional variant of the standard low-degree likelihood calculation.

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arXiv2022

Coloring graphs with forbidden bipartite subgraphs

A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph $F$, there is a constant $c_F > 0$ such that if $G$ is an $F$-free graph of maximum degree $Δ$, then $χ(G) \leq c_F Δ/ \logΔ$. Alon, Krivelevich, and Sudakov verified this conjecture for a class of graphs $F$ that includes all bipartite graphs. Moreover, it follows from recent work by Davies, Kang, Pirot, and Sereni that if $G$ is $K_{t,t}$-free, then $χ(G) \leq (t + o(1)) Δ/ \logΔ$ as $Δ\to \infty$. We improve this bound to $(1+o(1)) Δ/\log Δ$, making the constant factor independent of $t$. We further extend our result to the DP-coloring setting (also known as correspondence coloring), introduced by Dvořák and Postle.

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