arXiv subjects

Huong Luu

Publications and source records attributed to Huong Luu.

4 recordsLinked to original sources

arXiv2026

Two Complexity Results on Spanning-Tree Congestion Problems

In the spanning-tree congestion problem ($\mathsf{STC}$), we are given a graph $G$, and the objective is to compute a spanning tree of $G$ that minimizes the maximum edge congestion. While $\mathsf{STC}$ is known to be $\mathbb{NP}$-hard, even for some restricted graph classes, several key questions regarding its computational complexity remain open, and we address some of these in our paper. (i) For graphs of degree at most $\Delta$, it is known that $\mathsf{STC}$ is $\mathbb{NP}$-hard when $\Delta\ge 8$. We provide a complete resolution of this variant, by showing that $\mathsf{STC}$ remains $\mathbb{NP}$-hard for each degree bound $\Delta\ge 3$. (ii) In the decision version of $\mathsf{STC}$, given an integer $K$, the goal is to determine whether the congestion of $G$ is at most $K$. We prove that this variant is polynomial-time solvable for $K$-edge-connected graphs.

cs.DS↗

arXiv2023

A Note on Local Convergence of Iterative Processes for Pipe Network Analysis

Analysis of pipe networks involves computing flow rates and pressure differences on pipe segments in the network, given the external inflow/outflow values. This analysis can be conducted using iterative methods, among which the algorithms of Hardy Cross and Newton-Raphson have historically been applied in practice. In this note, we address the mathematical analysis of the local convergence of these algorithms. The loop-based Newton-Raphson algorithm converges quadratically fast, and we provide estimates for its convergence radius to correct some estimates in the previous literature. In contrast, we show that the convergence of the Hardy Cross algorithm is only linear. This provides theoretical confirmation of experimental observations reported earlier in the literature.

cs.CE↗

arXiv2023

Better Hardness Results for the Minimum Spanning Tree Congestion Problem

In the spanning tree congestion problem, given a connected graph $G$, the objective is to compute a spanning tree $T$ in $G$ that minimizes its maximum edge congestion, where the congestion of an edge $e$ of $T$ is the number of edges in $G$ for which the unique path in $T$ between their endpoints traverses $e$. The problem is known to be $\mathbb{NP}$-hard, but its approximability is still poorly understood. In the decision version of this problem, denoted $K-\textsf{STC}$, we need to determine if $G$ has a spanning tree with congestion at most $K$. It is known that $K-\textsf{STC}$ is $\mathbb{NP}$-complete for $K\ge 8$. On the other hand, $3-\textsf{STC}$ can be solved in polynomial time, with the complexity status of this problem for $K\in \{4,5,6,7\}$ remaining an open problem. We substantially improve the earlier hardness results by proving that $K-\textsf{STC}$ is $\mathbb{NP}$-complete for $K\ge 5$. This leaves only the case $K=4$ open, and improves the lower bound on the approximation ratio to $1.2$. Motivated by evidence that minimizing congestion is hard even for graphs of small constant radius, we consider $K-\textsf{STC}$ restricted to graphs of radius $2$, and we prove that this variant is $\mathbb{NP}$-complete for all $K\ge 6$. Exploring further in this direction, we also examine the variant, denoted $K-\textsf{STC}D$, where the objective is to determine if the graph has a depth-$D$ spanning three of congestion at most $K$. We prove that $6-\textsf{STC}2$ is $\mathbb{NP}$-complete even for bipartite graphs. For bipartite graphs we establish a tight bound, by also proving that $5-\textsf{STC}2$ is polynomial-time solvable. Additionally, we complement this result with polynomial-time algorithms for two special cases that involve bipartite graphs and restrictions on vertex degrees.

cs.CC↗

arXiv2019

Modeling Fluid Mixing in Microfluidic Grids

We describe an approach for modeling fluid concentration profiles in grid-based microfluidic chips for fluid mixing. This approach provides an algorithm that predicts fluid concentrations at the chip outlets. Our algorithm significantly outperforms COMSOL finite element simulations in term of runtime while still producing results that closely approximate those of COMSOL.

cs.CE↗