arXiv subjects
Publications and source records attributed to Jie Han.
Let $k\ge 2$ and $n_1\ge n_2\ge n_3\ge n_4$ be integers such that $n_4$ is sufficiently larger than $k$. We determine the maximum number of edges of a 4-partite graph with parts of sizes $n_1,\dots, n_4$ that does not contain $k$ vertex-disjoint triangles. For any $r> t\ge 3$, we give a conjecture on the maximum number of edges of an $r$-partite graph that does not contain $k$ vertex-disjoint cliques $K_t$.
The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: Given integers $\ell,r$ and $n$ with $n\in r\mathbb{N}$, is it true that every $n$-vertex graph $G$ with $δ(G) \ge \max \{ \frac{1}{2},\frac{r - \ell}{r} \}n + o(n)$ and $α_{\ell}(G) = o(n) $ contains a $K_{r}$-factor? We give a negative answer for the case when $\ell\ge \frac{3r}{4}$ by giving a family of constructions using the so-called cover thresholds and show that the minimum degree condition given by our construction is asymptotically best possible. That is, for all integers $r,\ell$ with $r > \ell \ge \frac{3}{4}r$ and $μ>0$, there exist $α> 0$ and $N$ such that for every $n\in r\mathbb{N}$ with $n>N$, every $n$-vertex graph $G$ with $δ(G) \ge \left( \frac{1}{2-\varrho_{\ell}(r-1)} + μ\right)n $ and $α_{\ell}(G) \le αn$ contains a $K_{r}$-factor. Here $\varrho_{\ell}(r-1)$ is the Ramsey--Turán density for $K_{r-1}$ under the $\ell$-independence number condition.
Let $Y_{3,2}$ be the $3$-uniform hypergraph with two edges intersecting in two vertices. Our main result is that any $n$-vertex 3-uniform hypergraph with at least $\binom{n}{3} - \binom{n-m+1}{3} + o(n^3)$ edges contains a collection of $m$ vertex-disjoint copies of $Y_{3,2}$, for $m\le n/7$. The bound on the number of edges is asymptotically best possible. This problem generalizes the Matching Conjecture of Erdős. We then use this result combined with the absorbing method to determine the asymptotically best possible minimum $(k-3)$-degree threshold for $\ell$-Hamiltonicity in $k$-graphs, where $k\ge 7$ is odd and $\ell=(k-1)/2$. Moreover, we give related results on $ Y_{k,b} $-tilings and Hamilton $ \ell $-cycles with $ d $-degree for some other values of $ k,\ell,d $.
We investigate the emergence of spanning structures in sparse pseudo-random $k$-uniform hypergraphs, using the following comparatively weak notion of pseudo-randomness. A $k$-uniform hypergraph $H$ on $n$ vertices is called $(p,α,ε)$-pseudo-random if for all (not necessarily disjoint) vertex subsets $A_1,\dots, A_k{\subseteq} V(H)$ with $|A_1|\cdots |A_k|{\geq}αn^{k}$ we have $$e(A_1,\dots, A_k)=(1\pmε)p |A_1|\cdots |A_k|.$$ For any linear $k$-uniform $F$ we provide a bound on $α=α(n)$ in terms of $p=p(n)$ and $F$, such that (under natural divisibility assumptions on $n$) any $k$-uniform $\big(p,α, o(1)\big)$-pseudo-random $n$-vertex hypergraph $H$ with a mild minimum vertex degree condition contains an $F$-factor. The approach also enables us to establish the existence of loose Hamilton cycles in sufficiently pseudo-random hypergraphs and all results imply corresponding bounds for stronger notions of hypergraph pseudo-randomness such as jumbledness or large spectral gap. As a consequence, $\big(p,α, o(1)\big)$-pseudo-random $k$-graphs as above contain: $(i)$ a perfect matching if $α=o(p^{k})$ and $(ii)$ a loose Hamilton cycle if $α=o(p^{k-1})$. This extends the works of Lenz--Mubayi, and Lenz--Mubayi--Mycroft who studied the analogous problems in the dense setting.
Given a hypergraph $H$, the size-Ramsey number $\hat{r}_2(H)$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that in any colouring of the edges of $G$ with two colours there is a monochromatic copy of $H$. We prove that the size-Ramsey number of the $3$-uniform tight path on $n$ vertices $P^{(3)}_n$ is linear in $n$, i.e., $\hat{r}_2(P^{(3)}_n) = O(n)$. This answers a question by Dudek, Fleur, Mubayi, and Rödl for $3$-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417-434], who proved $\hat{r}_2(P^{(3)}_n) = O(n^{3/2} \log^{3/2} n)$.
In this paper, we focus on the multi-object tracking (MOT) problem of automatic driving and robot navigation. Most existing MOT methods track multiple objects using a singular RGB camera, which are prone to camera field-of-view and suffer tracking failures in complex scenarios due to background clutters and poor light conditions. To meet these challenges, we propose a MultiModality PAnoramic multi-object Tracking framework (MMPAT), which takes both 2D panorama images and 3D point clouds as input and then infers target trajectories using the multimodality data. The proposed method contains four major modules, a panorama image detection module, a multimodality data fusion module, a data association module and a trajectory inference model. We evaluate the proposed method on the JRDB dataset, where the MMPAT achieves the top performance in both the detection and tracking tasks and significantly outperforms state-of-the-art methods by a large margin (15.7 and 8.5 improvement in terms of AP and MOTA, respectively).
We determine, up to a multiplicative constant, the optimal number of random edges that need to be added to a $k$-graph $H$ with minimum vertex degree $Ω(n^{k-1})$ to ensure an $F$-factor with high probability, for any $F$ that belongs to a certain class $\mathcal{F}$ of $k$-graphs, which includes, e.g., all $k$-partite $k$-graphs, $K_4^{(3)-}$ and the Fano plane. In particular, taking $F$ to be a single edge, this settles a problem of Krivelevich, Kwan and Sudakov [Combin. Probab. Comput. 25 (2016), 909--927]. We also address the case in which the host graph $H$ is not dense, indicating that starting from certain such $H$ is essentially the same as starting from an empty graph (namely, the purely random model).
For any even integer $k\ge 6$, integer $d$ such that $k/2\le d\le k-1$, and sufficiently large $n\in (k/2)\mathbb N$, we find a tight minimum $d$-degree condition that guarantees the existence of a Hamilton $(k/2)$-cycle in every $k$-uniform hypergraph on $n$ vertices. When $n\in k\mathbb N$, the degree condition coincides with the one for the existence of perfect matchings provided by Rödl, Ruciński and Szemerédi (for $d=k-1$) and Treglown and Zhao (for $d\ge k/2$), and thus our result strengthens theirs in this case.
Boosting Trees are one of the most successful statistical learning approaches that involve sequentially growing an ensemble of simple regression trees (i.e., "weak learners"). However, gradient boosted trees are not yet available for spatially correlated data. This paper proposes a new gradient Boosted Trees algorithm for Spatial Data (Boost-S) with covariate information. Boost-S integrates the spatial correlation structure into the classical framework of gradient boosted trees. Each tree is grown by solving a regularized optimization problem, where the objective function involves two penalty terms on tree complexity and takes into account the underlying spatial correlation. A computationally-efficient algorithm is proposed to obtain the ensemble trees. The proposed Boost-S is applied to the spatially-correlated FDG-PET (fluorodeoxyglucose-positron emission tomography) imaging data collected during cancer chemoradiotherapy. Our numerical investigations successfully demonstrate the advantages of the proposed Boost-S over existing approaches for this particular application.
We determine Turán numbers for the family of 3-uniform minimal paths of length four \emph{for all $n$}. We also establish the second and third order Turán numbers and use them to compute the corresponding Ramsey numbers for up to four colors.
A perfect $K_r$-tiling in a graph $G$ is a collection of vertex-disjoint copies of $K_r$ that together cover all the vertices in $G$. In this paper we consider perfect $K_r$-tilings in the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze and Martin where one starts with a dense graph and then adds $m$ random edges to it. Specifically, given any fixed $0< α<1-1/r$ we determine how many random edges one must add to an $n$-vertex graph $G$ of minimum degree $δ(G) \geq αn$ to ensure that, asymptotically almost surely, the resulting graph contains a perfect $K_r$-tiling. As one increases $α$ we demonstrate that the number of random edges required `jumps' at regular intervals, and within these intervals our result is best-possible. This work therefore closes the gap between the seminal work of Johansson, Kahn and Vu (which resolves the purely random case, i.e., $α=0$) and that of Hajnal and Szemerédi (which demonstrates that for $α\geq 1-1/r$ the initial graph already houses the desired perfect $K_r$-tiling).
We show that for any fixed $α>0$, cherry-quasirandom 3-graphs of positive density and sufficiently large order $n$ with minimum vertex degree $α\binom n2$ have a tight Hamilton cycle. This solves a conjecture of Aigner-Horev and Levy.
Keevash and Mycroft [\emph{Mem.~Amer.~Math.~Soc., 2015}] developed a geometric theory for hypergraph matchings and characterized the dense simplicial complexes that contain a perfect matching. Their proof uses the hypergraph regularity method and the hypergraph blow-up lemma recently developed by Keevash. In this note we give a new proof of their results, which avoids these complex tools. In particular, our proof uses the lattice-based absorbing method developed by the author and a recent probabilistic argument of Kohayakawa, Person and the author.
For integers $k\ge 3$ and $1\le \ell\le k-1$, we prove that for any $α>0$, there exist $ε>0$ and $C>0$ such that for sufficiently large $n\in (k-\ell)\mathbb{N}$, the union of a $k$-uniform hypergraph with minimum vertex degree $αn^{k-1}$ and a binomial random $k$-uniform hypergraph $\mathbb{G}^{(k)}(n,p)$ with $p\ge n^{-(k-\ell)-ε}$ for $\ell\ge 2$ and $p\ge C n^{-(k-1)}$ for $\ell=1$ on the same vertex set contains a Hamiltonian $\ell$-cycle with high probability. Our result is best possible up to the values of $ε$ and $C$ and answers a question of Krivelevich, Kwan and Sudakov.
The Adiabatic Quantum-Flux-Parametron (AQFP) superconducting technology has been recently developed, which achieves the highest energy efficiency among superconducting logic families, potentially huge gain compared with state-of-the-art CMOS. In 2016, the successful fabrication and testing of AQFP-based circuits with the scale of 83,000 JJs have demonstrated the scalability and potential of implementing large-scale systems using AQFP. As a result, it will be promising for AQFP in high-performance computing and deep space applications, with Deep Neural Network (DNN) inference acceleration as an important example. Besides ultra-high energy efficiency, AQFP exhibits two unique characteristics: the deep pipelining nature since each AQFP logic gate is connected with an AC clock signal, which increases the difficulty to avoid RAW hazards; the second is the unique opportunity of true random number generation (RNG) using a single AQFP buffer, far more efficient than RNG in CMOS. We point out that these two characteristics make AQFP especially compatible with the \emph{stochastic computing} (SC) technique, which uses a time-independent bit sequence for value representation, and is compatible with the deep pipelining nature. Further, the application of SC has been investigated in DNNs in prior work, and the suitability has been illustrated as SC is more compatible with approximate computations. This work is the first to develop an SC-based DNN acceleration framework using AQFP technology.
Given $3 \leq k \leq s$, we say that a $k$-uniform hypergraph $C^k_s$ is a tight cycle on $s$ vertices if there is a cyclic ordering of the vertices of $C^k_s$ such that every $k$ consecutive vertices under this ordering form an edge. We prove that if $k \ge 3$ and $s \ge 2k^2$, then every $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2 + o(1))n$ has the property that every vertex is covered by a copy of $C^k_s$. Our result is asymptotically best possible for infinitely many pairs of $s$ and $k$, e.g. when $s$ and $k$ are coprime. A perfect $C^k_s$-tiling is a spanning collection of vertex-disjoint copies of $C^k_s$. When $s$ is divisible by $k$, the problem of determining the minimum codegree that guarantees a perfect $C^k_s$-tiling was solved by a result of Mycroft. We prove that if $k \ge 3$ and $s \ge 5k^2$ is not divisible by $k$ and $s$ divides $n$, then every $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2 + 1/(2s) + o(1))n$ has a perfect $C^k_s$-tiling. Again our result is asymptotically best possible for infinitely many pairs of $s$ and $k$, e.g. when $s$ and $k$ are coprime with $k$ even.
Given two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, a perfect $F$-tiling (or an $F$-factor) in $H$ is a set of vertex disjoint copies of $F$ that together cover the vertex set of $H$. For all complete $k$-partite $k$-graphs $K$, Mycroft proved a minimum codegree condition that guarantees a $K$-factor in an $n$-vertex $k$-graph, which is tight up to an error term $o(n)$. In this paper we improve the error term in Mycroft's result to a sub-linear term that relates to the Turán number of $K$ when the differences of the sizes of the vertex classes of $K$ are co-prime. Furthermore, we find a construction which shows that our improved codegree condition is asymptotically tight in infinitely many cases thus disproving a conjecture of Mycroft. At last, we determine exact minimum codegree conditions for tiling $K^{(k)}(1, \dots, 1, 2)$ and tiling loose cycles thus generalizing results of Czygrinow, DeBiasio, and Nagle, and of Czygrinow, respectively.
We solve a problem of Krivelevich, Kwan and Sudakov [SIAM Journal on Discrete Mathematics 31 (2017), 155-171] concerning the threshold for the containment of all bounded degree spanning trees in the model of randomly perturbed dense graphs. More precisely, we show that, if we start with a dense graph $G_α$ on $n$ vertices with $δ(G_α)\ge αn$ for $α>0$ and we add to it the binomial random graph $G(n,C/n)$, then with high probability the graph $G_α\cup G(n,C/n)$ contains copies of all spanning trees with maximum degree at most $Δ$ simultaneously, where $C$ depends only on $α$ and $Δ$.