arXiv subjects

Hung Thuan Nguyen

Publications and source records attributed to Hung Thuan Nguyen.

4 recordsLinked to original sources

arXiv2026

The Complexity of Distributed Minimum Weight Cycle Approximation

We study the Minimum Weight Cycle (MWC) problem in the $\mathsf{CONGEST}$ model of distributed computing. For undirected weighted graphs, we give a randomized $(k+1)$-approximation algorithm for every \underline{real number} $k \geq (1+\sqrt{5})/2 \approx 1.618$. The algorithm runs in \[ \tilde{O}\left(n^{\frac{k+1}{2k+1}} + D\right) \] rounds, where $n$ is the number of nodes and $D$ is the unweighted diameter of the graph. Varying $k$ therefore yields a smooth trade-off between approximation ratio and round complexity. On the lower-bound side, assuming the Erdős girth conjecture, we prove that for every \underline{integer} $k \geq 1$ and every $ε> 0$, any randomized $(k+1-ε)$-approximation algorithm for MWC requires \[ \tildeΩ\left(n^{\frac{k+1}{2k+1}}+D\right) \] rounds. The lower bound holds for both directed unweighted graphs and undirected weighted graphs, even on graphs of diameter $Θ(\log n)$. Consequently, for every integer $k \geq 2$, our upper and lower bounds for undirected weighted graphs match up to polylogarithmic factors. This gives a nearly tight characterization of the round complexity of approximate MWC across an infinite family of approximation ratios. These results improve the previous state of the art of Manoharan and Ramachandran (PODC 2024), who gave a $(2+ε)$-approximation algorithm for undirected weighted graphs in $\tilde{O}(n^{2/3}+D)$ rounds, and proved an $\tildeΩ(\sqrt{n})$ lower bound for arbitrary approximation ratios in directed unweighted and undirected weighted graphs.

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arXiv2025

Optimal Distributed Replacement Paths

We study the replacement paths problem in the $\mathsf{CONGEST}$ model of distributed computing. Given an $s$-$t$ shortest path $P$, the goal is to compute, for every edge $e$ in $P$, the shortest-path distance from $s$ to $t$ avoiding $e$. For unweighted directed graphs, we establish the tight randomized round complexity bound for this problem as $\widetildeΘ(n^{2/3} + D)$ by showing matching upper and lower bounds. Our upper bound extends to $(1+ε)$-approximation for weighted directed graphs. Our lower bound applies even to the second simple shortest path problem, which asks only for the smallest replacement path length. These results improve upon the very recent work of Manoharan and Ramachandran (SIROCCO 2024), who showed a lower bound of $\widetildeΩ(n^{1/2} + D)$ and an upper bound of $\widetilde{O}(n^{2/3} + \sqrt{n h_{st}} + D)$, where $h_{st}$ is the number of hops in the given $s$-$t$ shortest path $P$.

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arXiv2025

Round and Communication Efficient Graph Coloring

In the context of communication complexity, we explore protocols for graph coloring, focusing on the vertex and edge coloring problems in $n$-vertex graphs $G$ with a maximum degree $Δ$. We consider a scenario where the edges of $G$ are partitioned between two players. Our first contribution is a randomized protocol that efficiently finds a $(Δ+ 1)$-vertex coloring of $G$, utilizing $O(n)$ bits of communication in expectation and completing in $O(\log \log n \cdot \log Δ)$ rounds in the worst case. This advancement represents a significant improvement over the work of Flin and Mittal [Distributed Computing 2025], who achieved the same communication cost but required $O(n)$ rounds in expectation, thereby making a significant reduction in the round complexity. Our second contribution is a deterministic protocol to compute a $(2Δ- 1)$-edge coloring of $G$, which maintains the same $O(n)$ bits of communication and uses only $O(1)$ rounds. We complement the result with a tight $Ω(n)$-bit lower bound on the communication complexity of the $(2Δ-1)$-edge coloring problem, while a similar $Ω(n)$ lower bound for the $(Δ+1)$-vertex coloring problem has been established by Flin and Mittal [Distributed Computing 2025]. Our result implies a space lower bound of $Ω(n)$ bits for $(2Δ- 1)$-edge coloring in the $W$-streaming model, which is the first non-trivial space lower bound for edge coloring in the $W$-streaming model.

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arXiv2024

A Tight Lower Bound for 3-Coloring Grids in the Online-LOCAL Model

Recently, \citeauthor*{akbari2021locality}~(ICALP 2023) studied the locality of graph problems in distributed, sequential, dynamic, and online settings from a {unified} point of view. They designed a novel $O(\log n)$-locality deterministic algorithm for proper 3-coloring bipartite graphs in the $\mathsf{Online}$-$\mathsf{LOCAL}$ model. In this work, we establish the optimality of the algorithm by showing a \textit{tight} deterministic $Ω(\log n)$ locality lower bound, which holds even on grids. To complement this result, we have the following additional results: \begin{enumerate} \item We show a higher and {tight} $Ω(\sqrt{n})$ lower bound for 3-coloring toroidal and cylindrical grids. \item Considering the generalization of $3$-coloring bipartite graphs to $(k+1)$-coloring $k$-partite graphs, %where $k \geq 2$ is a constant, we show that the problem also has $O(\log n)$ locality when the input is a $k$-partite graph that admits a \emph{locally inferable unique coloring}. This special class of $k$-partite graphs covers several fundamental graph classes such as $k$-trees and triangular grids. Moreover, for this special class of graphs, we show a {tight} $Ω(\log n)$ locality lower bound. \item For general $k$-partite graphs with $k \geq 3$, we prove that the problem of $(2k-2)$-coloring $k$-partite graphs exhibits a locality of $Ω(n)$ in the $\onlineLOCAL$ model, matching the round complexity of the same problem in the $\LOCAL$ model recently shown by \citeauthor*{coiteux2023no}~(STOC 2024). Consequently, the problem of $(k+1)$-coloring $k$-partite graphs admits a locality lower bound of $Ω(n)$ when $k\geq 3$, contrasting sharply with the $Θ(\log n)$ locality for the case of $k=2$. \end{enumerate}

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