arXiv subjects

Jie Xue

Publications and source records attributed to Jie Xue.

At least 55 records · Page 3Linked to original sources

arXiv2022

The maximum spectral radius of irregular bipartite graphs

A bipartite graph is subcubic if it is an irregular bipartite graph with maximum degree three. In this paper, we prove that the asymptotic value of maximum spectral radius over subcubic bipartite graphs of order $n$ is $3-\varTheta(\frac{π^{2}}{n^{2}})$. Our key approach is taking full advantage of the eigenvalues of certain tridiagonal matrices, due to Willms [SIAM J. Matrix Anal. Appl. 30 (2008) 639--656]. Moreover, for large maximum degree, i.e., the maximum degree is at least $\lfloor n/2 \rfloor$, we characterize irregular bipartite graphs with maximum spectral radius. For general maximum degree, we present an upper bound on the spectral radius of irregular bipartite graphs in terms of the order and maximum degree.

math.CO↗

arXiv2022

Subexponential Parameterized Algorithms for Cut and Cycle Hitting Problems on H-Minor-Free Graphs

We design the first subexponential-time (parameterized) algorithms for several cut and cycle-hitting problems on $H$-minor free graphs. In particular, we obtain the following results (where $k$ is the solution-size parameter). 1. $2^{O(\sqrt{k}\log k)} \cdot n^{O(1)}$ time algorithms for Edge Bipartization and Odd Cycle Transversal; 2. a $2^{O(\sqrt{k}\log^4 k)} \cdot n^{O(1)}$ time algorithm for Edge Multiway Cut and a $2^{O(r \sqrt{k} \log k)} \cdot n^{O(1)}$ time algorithm for Vertex Multiway Cut, where $r$ is the number of terminals to be separated; 3. a $2^{O((r+\sqrt{k})\log^4 (rk))} \cdot n^{O(1)}$ time algorithm for Edge Multicut and a $2^{O((\sqrt{rk}+r) \log (rk))} \cdot n^{O(1)}$ time algorithm for Vertex Multicut, where $r$ is the number of terminal pairs to be separated; 4. a $2^{O(\sqrt{k} \log g \log^4 k)} \cdot n^{O(1)}$ time algorithm for Group Feedback Edge Set and a $2^{O(g \sqrt{k}\log(gk))} \cdot n^{O(1)}$ time algorithm for Group Feedback Vertex Set, where $g$ is the size of the group. 5. In addition, our approach also gives $n^{O(\sqrt{k})}$ time algorithms for all above problems with the exception of $n^{O(r+\sqrt{k})}$ time for Edge/Vertex Multicut and $(ng)^{O(\sqrt{k})}$ time for Group Feedback Edge/Vertex Set. We obtain our results by giving a new decomposition theorem on graphs of bounded genus, or more generally, an $h$-almost-embeddable graph for any fixed constant $h$. In particular we show the following. Let $G$ be an $h$-almost-embeddable graph for a constant $h$. Then for every $p\in\mathbb{N}$, there exist disjoint sets $Z_1,\dots,Z_p \subseteq V(G)$ such that for every $i \in \{1,\dots,p\}$ and every $Z'\subseteq Z_i$, the treewidth of $G/(Z_i\backslash Z')$ is $O(p+|Z'|)$. Here $G/(Z_i\backslash Z')$ is the graph obtained from $G$ by contracting edges with both endpoints in $Z_i \backslash Z'$.

cs.DS↗

arXiv2022

Multiwinner Elections under Minimax Chamberlin-Courant Rule in Euclidean Space

We consider multiwinner elections in Euclidean space using the minimax Chamberlin-Courant rule. In this setting, voters and candidates are embedded in a $d$-dimensional Euclidean space, and the goal is to choose a committee of $k$ candidates so that the rank of any voter's most preferred candidate in the committee is minimized. (The problem is also equivalent to the ordinal version of the classical $k$-center problem.) We show that the problem is NP-hard in any dimension $d \geq 2$, and also provably hard to approximate. Our main results are three polynomial-time approximation schemes, each of which finds a committee with provably good minimax score. In all cases, we show that our approximation bounds are tight or close to tight. We mainly focus on the $1$-Borda rule but some of our results also hold for the more general $r$-Borda.

cs.GT↗

arXiv2022

A spatial-temporal short-term traffic flow prediction model based on dynamical-learning graph convolution mechanism

Short-term traffic flow prediction is a vital branch of the Intelligent Traffic System (ITS) and plays an important role in traffic management. Graph convolution network (GCN) is widely used in traffic prediction models to better deal with the graphical structure data of road networks. However, the influence weights among different road sections are usually distinct in real life, and hard to be manually analyzed. Traditional GCN mechanism, relying on manually-set adjacency matrix, is unable to dynamically learn such spatial pattern during the training. To deal with this drawback, this paper proposes a novel location graph convolutional network (Location-GCN). Location-GCN solves this problem by adding a new learnable matrix into the GCN mechanism, using the absolute value of this matrix to represent the distinct influence levels among different nodes. Then, long short-term memory (LSTM) is employed in the proposed traffic prediction model. Moreover, Trigonometric function encoding is used in this study to enable the short-term input sequence to convey the long-term periodical information. Ultimately, the proposed model is compared with the baseline models and evaluated on two real word traffic flow datasets. The results show our model is more accurate and robust on both datasets than other representative traffic prediction models.

cs.LG↗

arXiv2022

Point Separation and Obstacle Removal by Finding and Hitting Odd Cycles

Suppose we are given a pair of points $s, t$ and a set $S$ of $n$ geometric objects in the plane, called obstacles. We show that in polynomial time one can construct an auxiliary (multi-)graph $G$ with vertex set $S$ and every edge labeled from $\{0, 1\}$, such that a set $S_d \subseteq S$ of obstacles separates $s$ from $t$ if and only if $G[S_d]$ contains a cycle whose sum of labels is odd. Using this structural characterization of separating sets of obstacles we obtain the following algorithmic results. In the Obstacle-Removal problem the task is to find a curve in the plane connecting s to t intersecting at most q obstacles. We give a $2.3146^qn^{O(1)}$ algorithm for Obstacle-Removal, significantly improving upon the previously best known $q^{O(q^3)} n^{O(1)}$ algorithm of Eiben and Lokshtanov (SoCG'20). We also obtain an alternative proof of a constant factor approximation algorithm for Obstacle-Removal, substantially simplifying the arguments of Kumar et al. (SODA'21). In the Generalized Points-Separation problem, the input consists of the set S of obstacles, a point set A of k points and p pairs $(s_1, t_1),... (s_p, t_p)$ of points from A. The task is to find a minimum subset $S_r \subseteq S$ such that for every $i$, every curve from $s_i$ to $t_i$ intersects at least one obstacle in $S_r$. We obtain $2^{O(p)} n^{O(k)}$-time algorithm for Generalized Points-Separation problem. This resolves an open problem of Cabello and Giannopoulos (SoCG'13), who asked about the existence of such an algorithm for the special case where $(s_1, t_1), ... (s_p, t_p)$ contains all the pairs of points in A. Finally, we improve the running time of our algorithm to $f(p,k) n^{O(\sqrt{k})}$ when the obstacles are unit disks, where $f(p,k) = 2^O(p) k^{O(k)}$, and show that, assuming the Exponential Time Hypothesis (ETH), the running time dependence on $k$ of our algorithms is essentially optimal.

cs.CG↗

arXiv2021

Enumerations of Universal Cycles for $k$-Permutations

Universal cycle for $k$-permutations is a cyclic arrangement in which each $k$-permutation appears exactly once as $k$ consecutive elements. Enumeration problem of universal cycles for $k$-permutations is discussed and one new enumerating method is proposed in this paper. Accurate enumerating formulae are provided when $k=2,3$.

math.CO↗

arXiv2021

Dynamic Geometric Set Cover, Revisited

Geometric set cover is a classical problem in computational geometry, which has been extensively studied in the past. In the dynamic version of the problem, points and ranges may be inserted and deleted, and our goal is to efficiently maintain a set cover solution (satisfying certain quality requirement). In this paper, we give a plethora of new dynamic geometric set cover data structures in 1D and 2D, which significantly improve and extend the previous results: 1. The first data structure for $(1+\varepsilon)$-approximate dynamic interval set cover with polylogarithmic amortized update time. Specifically, we achieve an update time of $O(\log^3 n/\varepsilon)$, improving the $O(n^δ/\varepsilon)$ bound of Agarwal et al. [SoCG'20], where $δ>0$ denotes an arbitrarily small constant. 2. A data structure for $O(1)$-approximate dynamic unit-square set cover with $2^{O(\sqrt{\log n})}$ amortized update time, substantially improving the $O(n^{1/2+δ})$ update time of Agarwal et al. [SoCG'20]. 3. A data structure for $O(1)$-approximate dynamic square set cover with $O(n^{1/2+δ})$ randomized amortized update time, improving the $O(n^{2/3+δ})$ update time of Chan and He [SoCG'21]. 4. A data structure for $O(1)$-approximate dynamic 2D halfplane set cover with $O(n^{17/23+δ})$ randomized amortized update time. The previous solution for halfplane set cover by Chan and He [SoCG'21] is slower and can only report the size of the approximate solution. 5. The first sublinear results for the \textit{weighted} version of dynamic geometric set cover. Specifically, we give a data structure for $(3+o(1))$-approximate dynamic weighted interval set cover with $2^{O(\sqrt{\log n \log\log n})}$ amortized update time and a data structure for $O(1)$-approximate dynamic weighted unit-square set cover with $O(n^δ)$ amortized update time.

cs.CG↗

arXiv2020

On Dominance-Free Samples of a (Colored) Stochastic Dataset

A point $p \in \mathbb{R}^d$ is said to dominate another point $q \in \mathbb{R}^d$ if the coordinate of $p$ is greater than or equal to the coordinate of $q$ in every dimension. A set of points in $\mathbb{R}^d$ is dominance-free if any two points do not dominate each other. We consider the problem of counting the dominance-free subsets of a given dataset in $\mathbb{R}^d$, or more generally, computing the probability that a random sample of a stochastic dataset in $\mathbb{R}^d$ where each point is sampled independently with its existence probability is dominance-free. In fact, we investigate a colored generalization of the problem, in which the points in the given stochastic dataset are colored and we are interested in the random samples that are inter-color dominance-free (i.e., any two points with different colors do not dominate each other). We propose the first algorithm that solves the problem for $d=2$ in near-quadratic time. On the other hand, we show that the problem is #P-hard for any $d \geq 3$, even if the points have a restricted color pattern; this implies the #P-hardness of the uncolored version (i.e., computing the dominance-free probability of a uncolored stochastic dataset) for $d \geq 3$. In addition, we show that even when the existence probabilities of the points are all equal to 0.5, the problem remains #P-hard for any $d \geq 7$; this implies the #P-hardness of counting dominance-free subsets for $d \geq 7$. In order to prove our hardness results, we establish some results about embedding the vertices a graph into low-dimensional Euclidean space such that two vertices are connected by an edge in the graph iff they form a dominance pair in the embedding. These results may be of independent interest and can possibly be applied to other problems.

cs.CG↗

arXiv2020

A spectral extremal problem on graphs with given size and matching number

Brualdi and Hoffman (1985) proposed the problem of determining the maximal spectral radius of graphs with given size. In this paper, we consider the Brualdi-Hoffman type problem of graphs with given matching number. The maximal $Q$-spectral radius of graphs with given size and matching number is obtained, and the corresponding extremal graphs are also determined.

math.CO↗

arXiv2020

Dynamic geometric set cover and hitting set

We investigate dynamic versions of geometric set cover and hitting set where points and ranges may be inserted or deleted, and we want to efficiently maintain an (approximately) optimal solution for the current problem instance. While their static versions have been extensively studied in the past, surprisingly little is known about dynamic geometric set cover and hitting set. For instance, even for the most basic case of one-dimensional interval set cover and hitting set, no nontrivial results were known. The main contribution of our paper are two frameworks that lead to efficient data structures for dynamically maintaining set covers and hitting sets in $\mathbb{R}^1$ and $\mathbb{R}^2$. The first framework uses bootstrapping and gives a $(1+\varepsilon)$-approximate data structure for dynamic interval set cover in $\mathbb{R}^1$ with $O(n^α/\varepsilon)$ amortized update time for any constant $α> 0$; in $\mathbb{R}^2$, this method gives $O(1)$-approximate data structures for unit-square (and quadrant) set cover and hitting set with $O(n^{1/2+α})$ amortized update time. The second framework uses local modification, and leads to a $(1+\varepsilon)$-approximate data structure for dynamic interval hitting set in $\mathbb{R}^1$ with $\widetilde{O}(1/\varepsilon)$ amortized update time; in $\mathbb{R}^2$, it gives $O(1)$-approximate data structures for unit-square (and quadrant) set cover and hitting set in the \textit{partially} dynamic settings with $\widetilde{O}(1)$ amortized update time.

cs.CG↗

arXiv2020

On the $A_α$-spectra of graphs

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $α\in [0,1]$, Nikiforov \cite{VN1} defined the matrix $A_α(G)$ as $$A_α(G)=αD(G)+(1-α)A(G).$$ In this paper, we give some results on the eigenvalues of $A_α(G)$ with $α>1/2$. In particular, we show that for each $e\notin E(G)$, $λ_i(A_α(G+e))\geqλ_i(A_α(G))$. By utilizing the result, we prove have $λ_k(A_α(G))\leqαn-1$ for $2\leq k\leq n$. Moreover, we characterize the extremal graphs with equality holding. Finally, we show that $λ_n(A_α({G}))\geq 2α-1$ if $G$ contains no isolated vertices.

math.CO↗

arXiv2020

Fractional matching number and spectral radius of nonnegative matrix of graphs

A fractional matching of a graph $G$ is a function $f:E(G) \to [0,1]$ such that for any $v\in V(G)$, $\sum_{e\in E_G(v)}f(e)\leq 1$ where $E_G(v) = \{e \in E(G): e$ is incident with $v$ in $G\}$. The fractional matching number of $G$ is $μ_{f}(G) = \max\{\sum_{e\in E(G)} f(e): f$ is fractional matching of $G\}$. For any real numbers $a \ge 0$ and $k \in (0, n)$, it is observed that if $n = |V(G)|$ and $δ(G) > \frac{n-k}{2}$, then $μ_{f}(G)>\frac{n-k}{2}$. We determine a function $φ(a, n,δ, k)$ and show that for a connected graph $G$ with $n = |V(G)|$, $δ(G) \leq\frac{n-k}{2}$, spectral radius $λ_1(G)$ and complement $\overline{G}$, each of the following holds. (i) If $λ_{1}(aD(G)+A(G))<φ(a, n, δ, k),$ then $μ_{f}(G)>\frac{n-k}{2}.$ (ii) If $λ_{1}(aD(\overline{G})+A(\overline{G}))<(a+1)(δ+k-1),$ then $μ_{f}(G)>\frac{n-k}{2}.$ As corollaries, sufficient spectral condition for fractional perfect matchings and analogous results involving $Q$-index and $A_α$-spectral radius are obtained, and former spectral results in [European J. Combin. 55 (2016) 144-148] are extended.

math.CO↗

arXiv2019

Range closest-pair search in higher dimensions

Range closest-pair (RCP) search is a range-search variant of the classical closest-pair problem, which aims to store a given set $S$ of points into some space-efficient data structure such that when a query range $Q$ is specified, the closest pair in $S \cap Q$ can be reported quickly. RCP search has received attention over years, but the primary focus was only on $\mathbb{R}^2$. In this paper, we study RCP search in higher dimensions. We give the first nontrivial RCP data structures for orthogonal, simplex, halfspace, and ball queries in $\mathbb{R}^d$ for any constant $d$. Furthermore, we prove a conditional lower bound for orthogonal RCP search for $d \geq 3$.

cs.CG↗

arXiv2019

Improved Algorithms for the Bichromatic Two-Center Problem for Pairs of Points

We consider a bichromatic two-center problem for pairs of points. Given a set $S$ of $n$ pairs of points in the plane, for every pair, we want to assign a red color to one point and a blue color to the other, in such a way that the value $\max\{r_1,r_2\}$ is minimized, where $r_1$ (resp., $r_2$) is the radius of the smallest enclosing disk of all red (resp., blue) points. Previously, an exact algorithm of $O(n^3\log^2 n)$ time and a $(1+\varepsilon)$-approximate algorithm of $O(n + (1/\varepsilon)^6 \log^2 (1/\varepsilon))$ time were known. In this paper, we propose a new exact algorithm of $O(n^2\log^2 n)$ time and a new $(1+\varepsilon)$-approximate algorithm of $O(n + (1/\varepsilon)^3 \log^2 (1/\varepsilon))$ time.

cs.CG↗

arXiv2019

Searching for the closest-pair in a query translate

We consider a range-search variant of the closest-pair problem. Let $\varGamma$ be a fixed shape in the plane. We are interested in storing a given set of $n$ points in the plane in some data structure such that for any specified translate of $\varGamma$, the closest pair of points contained in the translate can be reported efficiently. We present results on this problem for two important settings: when $\varGamma$ is a polygon (possibly with holes) and when $\varGamma$ is a general convex body whose boundary is smooth. When $\varGamma$ is a polygon, we present a data structure using $O(n)$ space and $O(\log n)$ query time, which is asymptotically optimal. When $\varGamma$ is a general convex body with a smooth boundary, we give a near-optimal data structure using $O(n \log n)$ space and $O(\log^2 n)$ query time. Our results settle some open questions posed by Xue et al. [SoCG 2018].

cs.CG↗

arXiv2019

Near-Optimal Algorithms for Shortest Paths in Weighted Unit-Disk Graphs

We revisit a classical graph-theoretic problem, the \textit{single-source shortest-path} (SSSP) problem, in weighted unit-disk graphs. We first propose an exact (and deterministic) algorithm which solves the problem in $O(n \log^2 n)$ time using linear space, where $n$ is the number of the vertices of the graph. This significantly improves the previous deterministic algorithm by Cabello and Jejčič [CGTA'15] which uses $O(n^{1+δ})$ time and $O(n^{1+δ})$ space (for any small constant $δ>0$) and the previous randomized algorithm by Kaplan et al. [SODA'17] which uses $O(n \log^{12+o(1)} n)$ expected time and $O(n \log^3 n)$ space. More specifically, we show that if the 2D offline insertion-only (additively-)weighted nearest-neighbor problem with $k$ operations (i.e., insertions and queries) can be solved in $f(k)$ time, then the SSSP problem in weighted unit-disk graphs can be solved in $O(n \log n+f(n))$ time. Using the same framework with some new ideas, we also obtain a $(1+\varepsilon)$-approximate algorithm for the problem, using $O(n \log n + n \log^2(1/\varepsilon))$ time and linear space. This improves the previous $(1+\varepsilon)$-approximate algorithm by Chan and Skrepetos [SoCG'18] which uses $O((1/\varepsilon)^2 n \log n)$ time and $O((1/\varepsilon)^2 n)$ space. More specifically, we show that if the 2D offline insertion-only weighted nearest-neighbor problem with $k_1$ operations in which at most $k_2$ operations are insertions can be solved in $f(k_1,k_2)$ time, then the $(1+\varepsilon)$-approximate SSSP problem in weighted unit-disk graphs can be solved in $O(n \log n+f(n,O(\varepsilon^{-2})))$ time. Because of the $Ω(n \log n)$-time lower bound of the problem (even when approximation is allowed), both of our algorithms are almost optimal.

cs.CG↗

arXiv2018

Colored range closest-pair problem under general distance functions

The range closest-pair (RCP) problem is the range-search version of the classical closest-pair problem, which aims to store a given dataset of points in some data structure such that when a query range $X$ is specified, the closest pair of points contained in $X$ can be reported efficiently. A natural generalization of the RCP problem is the {colored range closest-pair} (CRCP) problem in which the given data points are colored and the goal is to find the closest {bichromatic} pair contained in the query range. All the previous work on the RCP problem was restricted to the uncolored version and the Euclidean distance function. In this paper, we make the first progress on the CRCP problem. We investigate the problem under a general distance function induced by a monotone norm; in particular, this covers all the $L_p$-metrics for $p > 0$ and the $L_\infty$-metric. We design efficient $(1+\varepsilon)$-approximate CRCP data structures for orthogonal queries in $\mathbb{R}^2$, where $\varepsilon>0$ is a pre-specified parameter. The highlights are two data structures for answering rectangle queries, one of which uses $O(\varepsilon^{-1} n \log^4 n)$ space and $O(\log^4 n + \varepsilon^{-1} \log^3 n + \varepsilon^{-2} \log n)$ query time while the other uses $O(\varepsilon^{-1} n \log^3 n)$ space and $O(\log^5 n + \varepsilon^{-1} \log^4 n + \varepsilon^{-2} \log^2 n)$ query time. In addition, we also apply our techniques to the CRCP problem in higher dimensions, obtaining efficient data structures for slab, 2-box, and 3D dominance queries. Before this paper, almost all the existing results for the RCP problem were achieved in $\mathbb{R}^2$.

cs.CG↗

arXiv2018

A note on the $A_α$-spectral radius of graphs

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $α\in [0,1]$, Nikiforov [Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math. 11 (2017) 81--107] defined the matrix $A_α(G)$ as $A_α(G)=αD(G)+(1-α)A(G).$ Let $u$ and $v$ be two vertices of a connected graph $G$. Suppose that $u$ and $v$ are connected by a path $w_0(=v)w_1\cdots w_{s-1}w_s(=u)$ where $d(w_i)=2$ for $1\leq i\leq s-1$. Let $G_{p,s,q}(u,v)$ be the graph obtained by attaching the paths $P_p$ to $u$ and $P_q$ to $v$. Let $s=0,1$. Nikiforov and Rojo [On the $α$-index of graphs with pendent paths, Linear Algebra Appl. 550 (2018) 87--104] conjectured that $ρ_α(G_{p,s,q}(u,v))<ρ_α(G_{p-1,s,q+1}(u,v))$ if $p\geq q+2.$ In this paper, we confirm the conjecture. As applications, firstly, the extremal graph with maximal $A_α$-spectral radius with fixed order and cut vertices is characterized. Secondly, we characterize the extremal tree which attains the maximal $A_α$-spectral radius with fixed order and matching number. These results generalize some known results.

math.CO↗