arXiv subjects

Hung Le

Publications and source records attributed to Hung Le.

169 records · Page 10Linked to original sources

arXiv2018

Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind

In this paper, we present results about the existence and uniqueness of solutions of elliptic equations with transmission and Wentzell boundary conditions. We provide Schauder estimates and existence results in Hölder spaces. As an application, we develop an existence theory for small-amplitude two-dimensional traveling waves in an air-water system with surface tension. The water region is assumed to be irrotational and of finite depth, and we permit a general distribution of vorticity in the atmosphere.

math.AP↗

arXiv2017

Greedy spanners are optimal in doubling metrics

We show that the greedy spanner algorithm constructs a $(1+ε)$-spanner of weight $ε^{-O(d)}w(\mathrm{MST})$ for a point set in metrics of doubling dimension $d$, resolving an open problem posed by Gottlieb. Our result generalizes the result by Narasimhan and Smid who showed that a point set in $d$-dimension Euclidean space has a $(1+ε)$-spanner of weight at most $ε^{-O(d)}w(\mathrm{MST})$. Our proof only uses the packing property of doubling metrics and thus implies a much simpler proof for the same result in Euclidean space.

cs.CG↗

arXiv2017

Minor-free graphs have light spanners

We show that every $H$-minor-free graph has a light $(1+ε)$-spanner, resolving an open problem of Grigni and Sissokho and proving a conjecture of Grigni and Hung. Our lightness bound is \[O\left(\frac{σ_H}{ε^3}\log \frac{1}ε\right)\] where $σ_H = |V(H)|\sqrt{\log |V(H)|}$ is the sparsity coefficient of $H$-minor-free graphs. That is, it has a practical dependency on the size of the minor $H$. Our result also implies that the polynomial time approximation scheme (PTAS) for the Travelling Salesperson Problem (TSP) in $H$-minor-free graphs by Demaine, Hajiaghayi and Kawarabayashi is an efficient PTAS whose running time is $2^{O_H\left(\frac{1}{ε^4}\log \frac{1}ε\right)}n^{O(1)}$ where $O_H$ ignores dependencies on the size of $H$. Our techniques significantly deviate from existing lines of research on spanners for $H$-minor-free graphs, but build upon the work of Chechik and Wulff-Nilsen for spanners of general graphs.

cs.DS↗

arXiv2017

Light spanners for bounded treewidth graphs imply light spanners for $H$-minor-free graphs

Grigni and Hung~\cite{GH12} conjectured that H-minor-free graphs have $(1+ε)$-spanners that are light, that is, of weight $g(|H|,ε)$ times the weight of the minimum spanning tree for some function $g$. This conjecture implies the {\em efficient} polynomial-time approximation scheme (PTAS) of the traveling salesperson problem in $H$-minor free graphs; that is, a PTAS whose running time is of the form $2^{f(ε)}n^{O(1)}$ for some function $f$. The state of the art PTAS for TSP in H-minor-free-graphs has running time $n^{1/ε^c}$. We take a further step toward proving this conjecture by showing that if the bounded treewidth graphs have light greedy spanners, then the conjecture is true. We also prove that the greedy spanner of a bounded pathwidth graph is light and discuss the possibility of extending our proof to bounded treewidth graphs.

cs.DS↗

arXiv2017

Embedded-width: A variant of treewidth for plane graphs

We define a special case of tree decompositions for planar graphs that respect a given embedding of the graph. We study the analogous width of the resulting decomposition we call the embedded-width of a plane graph. We show both upper bounds and lower bounds for the embedded-width of a graph in terms of its treewidth and describe a fixed parameter tractable algorithm to calculate the embedded-width of a plane graph. To do so, we give novel bounds on the size of matchings in planar graphs.

cs.DM↗

arXiv2017

A better bound on the largest induced forests in triangle-free planar graphs

It is well-known that there exists a triangle-free planar graph of $n$ verticess such that the largest induced forest has size at most $\frac{5n}{8}$. Salavatipour proved that there is a forest of size at least $\frac{5n}{9.41}$ in any triangle-free planar graph of $n$ vertices. Dross, Montassier and Pinlou improved Salavatipour's bound to $\frac{5n}{9.17}$. In this work, we further improve the bound to $\frac{5n}{9}$. Our technique is inspired by the recent ideas from Lukot'ka, Maz{á}k and Zhu.

math.CO↗

arXiv2015

Optimal dynamic program for r-domination problems over tree decompositions

There has been recent progress in showing that the exponential dependence on treewidth in dynamic programming algorithms for solving NP-hard problems are optimal under the Strong Exponential Time Hypothesis (SETH). We extend this work to $r$-domination problems. In $r$-dominating set, one wished to find a minimum subset $S$ of vertices such that every vertex of $G$ is within $r$ hops of some vertex in $S$. In connected $r$-dominating set, one additionally requires that the set induces a connected subgraph of $G$. We give a $O((2r+1)^{\mathrm{tw}} n)$ time algorithm for $r$-dominating set and a $O((2r+2)^{\mathrm{tw}} n^{O(1)})$ time algorithm for connected $r$-dominating set in $n$-vertex graphs of treewidth $\mathrm{tw}$. We show that the running time dependence on $r$ and $\mathrm{tw}$ is the best possible under SETH. This adds to earlier observations that a "+1" in the denominator is required for connectivity constraints.

cs.DS↗