arXiv subjects

Jie Han

Publications and source records attributed to Jie Han.

At least 19 recordsLinked to original sources

arXiv2026

Clique-factors in randomly perturbed hypergraphs

In this paper we establish the following result for clique-factors in randomly perturbed hypergraphs when the cliques are large enough. For integers $k \geq 3$ and $t\geq t_0(k)$, we show that for every $α> 0$, there exists $C>0$ such that for sufficiently large $n\in(t+1)\mathbb{N}$, the union of a $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $αn$ and a binomial random $k$-uniform hypergraph $H^{(k)}(n,p)$ with $p\geq Cn^{-{t/\binom{t+1}{k}}}$ on the same vertex set contains a $K_{t+1}^{(k)}$-factor asymptotically almost surely. The bound on $p$ is asymptotically best possible and extends the graph case, resolved by Balogh, Treglown and Wagner. In particular, we have $t_0(3)=3$, which implies that our result applies to every $t\ge 3$ for $3$-uniform hypergraphs. Together with the previously established result for perfect matchings, this covers all possible clique sizes for $3$-uniform hypergraphs.

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arXiv2026

EDISCO: Equivariant DIScrete Diffusion for Euclidean Combinatorial Optimization

Euclidean combinatorial optimization problems (ECOPs), such as the Traveling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP), possess inherent symmetries under the two-dimensional Euclidean group E(2), including rotations, reflections, and translations. Existing learning-based methods, including recent diffusion-based methods, rely on data augmentation or regularization to approximate E(2)-equivariance. This paper presents EDISCO, the first discrete diffusion model for ECOPs with exact E(2)-invariant generative distributions over node-index solutions. EDISCO introduces an E(2)-equivariant edge-score network coupled with a categorical continuous-time Markov chain over discrete edge variables, and exact posterior sampling provides efficient multi-step inference. This design gives EDISCO a local geometric inductive bias: edge neighborhoods with the same relative geometry and combinatorial context are represented consistently regardless of absolute position or orientation, making learning more efficient and inference more robust than non-equivariant methods. EDISCO outperforms previous learning-based state-of-the-art solvers on synthetic TSP from 100 to 10000 nodes and CVRP from 50 to 2000 customers, while using only 33-50% of the training instances. Trained only on uniform synthetic data, EDISCO also outperforms competing learning-based baselines under spatial distribution shift and CVRP constraint-tightness shift. Code is available at https://github.com/ValleyC/EDISCO.

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arXiv2026

Alon's conjecture on intersecting families of graphs

Given a graph $F$, a family $\mathcal F$ of graphs on $[n]$ is \emph{$F$-intersecting} if $G\cap H$ contains a copy of $F$ for every $G,H\in\mathcal F$. We prove that there exists an absolute constant $\varepsilon>0$ such that every $P_4$-intersecting family $\mathcal F$ satisfies $|\mathcal F|\le\left(\frac12-\varepsilon\right)2^{\binom n2}$, which resolves a conjecture of Alon. Combined with Alon's reduction, this proves that a graph $F$ admits $F$-intersecting families of asymptotic density $1/2$ if and only if $F$ is a star forest. Multi-colored versions are also considered.

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arXiv2026

Perfect Matching in $k$-Partite $k$-Uniform Hypergraphs

A balanced $k$-partite $k$-graph is a $k$-uniform hypergraph whose vertex set is partitioned into $k$ classes of the same size and whose edges meet every class in exactly one vertex. Lo and Markström (2014) determined the minimum vertex-degree threshold for perfect matchings when $k=3$, and Lu, Wang and Yuan recently determined it when $k=4$. We prove the corresponding exact result for every fixed $k\ge5$ and all sufficiently large class sizes. The close case follows from the general theorem of Lu, Wang and Yuan. For the non-closed case, we extend their stability result from the 3-partite setting to arbitrary partite uniformity, using the probability-tail rigidity theorem of Cao, Liu and Zhang, thereby replacing the earlier weighted lemma.

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arXiv2026

Optimally pseudorandom $K_4$-free graphs

We show that optimally pseudorandom $K_4$-free graphs of order $n$ and degree $d = Θ(n^{4/5})$ exist by constructing a graph in the split Cayley hexagon, matching the known upper bound. This resolves the first open case for $K_k$-free graphs after $k=3$ for which Alon gave a tight construction in 1994. This has a variety of implications for $K_4$-free pseudorandom graphs. Furthermore, it implies an explicit lower bound on the Ramsey number of $r(4, t) \geq t^{1.\overline{6} - o(1)}$, improving the previous record by Kostochka, Pudlák, and Rödl of $r(4, t) \geq t^{1.6-o(1)}$.

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arXiv2026

Exact Minimum $d$-Degree Thresholds for Hypergraph Perfect Matchings

For fixed integers $k\ge3$ and $1\le d\le k-1$ and sufficiently large $n\in k\mathbb N$, we establish the sharp minimum $d$-degree thresholds that forces perfect matching in every $n$-vertex $k$-uniform hypergraphs. This was conjectued by Treglown and Zhao, and the $d=1$ case was conjectued by Kühn, Osthus and Treglown.

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arXiv2026

Tight Hamiltonian Cycles in Uniformly Dense $3$-Graphs

We study minimum degree conditions for tight Hamiltonian cycles in uniformly dense $3$-uniform hypergraphs. We prove that for every $d,α>0$, every sufficiently large $(ρ,d)$-dense $3$-graph on $n$ vertices with minimum codegree at least $(1/3+α)n$ contains a tight Hamiltonian cycle. This resolves a problem of Aigner-Horev and Levy in a stronger form, and the constant $1/3$ is asymptotically best possible. We also show that uniform density does not lower the asymptotic vertex-degree threshold: there are $(ρ,d)$-dense $3$-graphs with minimum vertex degree $(5/9-o(1))\binom{n}{2}$ and no tight Hamiltonian cycle. Finally, we construct $(ρ,2-\sqrt{3})$-dense examples with minimum codegree $(2-\sqrt{3}-o(1))n$ and no tight Hamiltonian cycle, answering negatively a question of Ara{ú}jo, Piga and Schacht.

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arXiv2026

PPAPlace: Differentiable Cross-Stage Objectives for Chip Placement Optimization

Macro placement significantly affects a chip's post-route performance, power, and area (PPA). Most placement methods optimize half-perimeter wirelength (HPWL) as the primary objective. However, recent benchmarking shows a near-zero correlation between HPWL and post-route timing metrics such as the worst negative slack (WNS) and total negative slack (TNS). As a result, all six evaluated artificial intelligence (AI) placers degraded PPA relative to the hierarchical baseline. Recent efforts have tried to train cross-stage predictors to close this gap. However, existing methods focus on macro-only representations and use pre-route metrics as training labels. A label fidelity study of ten circuits at four design flow stages reveals that HPWL and pre-route timing poorly reflect final post-route timing rankings. In contrast, post-global-routing achieves the best balance between final timing fidelity and label generation cost-effectiveness. Based on this finding, PPAPlace is a timing-driven differentiable surrogate predicting post-route PPA from macro and standard-cell placements. The surrogate is a dual-stream predictor that combines graph attention over the chip netlist with spatial convolution over the placement grid. It is trained on post-global-routing labels. The predicted WNS and TNS gradients flow end-to-end back to cell coordinates. PPAPlace exploits these gradients in two ways: as a co-objective injected into an analytical placer's optimization loop (PPAPlace-CoOpt), and as a post-placement refinement step that adjusts macro positions via projected gradient descent (PPAPlace-Refine). On five ChiPBench test circuits excluded from training, PPAPlace improves average WNS and TNS by 22\% and 51\% over the hierarchical baseline while preserving power and routability, using the same predictor without test-circuit retraining. Code is available at https://github.com/ValleyC/PPAPlace.

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arXiv2026

On the number of factorable induced subgraphs

Let $F$ be an $r$-vertex graph. In this paper, we study the $F$-factor problem in random induced subgraphs of dense graphs. We show that for any $r$-vertex graph $F$ and $γ>0$, if $H$ is an $n$-vertex graph with minimum degree at least $(1-1/χ_{cr}(F)+γ)n$, then for every fixed $p \in (0,1)$, the random induced subgraph $H[p]$ contains an $F$-factor with probability at least $1/(rq)-o_n(1)$, where $q\in \mathbb{N}$ is the order of certain coset group defined from $H$. The probability is asymptotically best possible for infinitely many $F$ and $H$ and yields that a $1/(rq)-o_n(1)$ proportion of the subsets of $H$ induce $F$-factors, interestingly, regardless of whether $H$ itself admits an $F$-factor. Similar results are obtained for perfect matchings in hypergraphs under minimum degree conditions. Our proof combines concentration inequalities, lattice point counting in $\mathbb{Z}^d$ and structural theorems for $F$-factors in dense (hyper)graphs.

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arXiv2026

Perfect Matchings in Random Sparsifications of Dense Hypergraphs

Given $1\le\ell 1-(1-1/k)^{k-\ell}$ and this was recently verified by the work of Gan--Han, together with a very recent work of Fu et al. In this paper we study the existence of perfect matchings in the random $p$-sparsification of such $k$-uniform hypergraphs, that is, for $p=p(n)\in [0,1]$, each edge is selected independently with probability $p$. Building on the structural theory of Gan and Han, we show that the corresponding dense perfect matching results are robust under random sparsification. As consequences, we obtain deterministic polynomial-time algorithms that asymptotically almost surely solve the associated decision problems, as well as lower bounds on the number of perfect matchings in such hypergraphs -- interestingly, such hypergraphs either have no perfect matching, or have $(Ω(n))^{(1-1/k)n}$ perfect matchings. Moreover, we also establish analogous results for the $F$-factor problem in graphs. Our proofs combine a partial exposure algorithm, the lattice-based absorption method, and a random redistribution method of Kelly, Müyesser and Pokrovskiy, via the framework of spread distributions. A key new ingredient is a lattice-preparation step that separates the contributions of the two classes of robust index vectors arising in the Gan--Han structural theory. Together with the random redistribution method, this allows us to establish the desired spread property in the family of perfect matchings.

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arXiv2026

CODA: Cascaded Online Discontinuity-Aware Alignment for Real-Time Image-Based Score Following

Real-time score following from sheet images remains chal- lenging because the model must process streaming au- dio while resolving highly repetitive visual patterns un- der strict latency constraints. Recent image-based meth- ods have attempted to use multi-resolution prediction by simultaneously predicting the positions of the active sys- tem, bar, and note. However, their predictions across these different levels of notation are independent, which makes the predictions unstable and introduces unnecessary ex- tra search space for bar- and note-level predictions. Most existing methods also lack mechanisms to recover from score discontinuities, such as repeats, da capo (D.C.), or coda jumps. This paper proposes CODA, to the best of our knowledge, the first real-time score following system that addresses both gaps. CODA explicitly exploits the cascaded structure of music scores: it first selects the ac- tive system, then the active bar within it, and finally the active note within the selected bar. This enforces pre- diction consistency across resolutions. A silence-driven break mode enables recovery from arbitrary score discon- tinuities without requiring knowledge of the repeat struc- ture. Evaluated on the Multimodal Sheet Music Dataset (MSMD) piano benchmarks, CODA achieves state-of-the- art tracking accuracy and discontinuity-recovery perfor- mance under real-time throughput. Code is available at https://github.com/ValleyC/CODA.

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arXiv2026

CoEvoP&R: Co-Evolving Placement Objectives with Routing Feedback via Large Language Models

Analytical placers rely on differentiable objective functions to guide placement, typically combining intermediate surrogate metrics such as half-perimeter wirelength (HPWL) and cell-density penalties. However, these placement-stage surrogates remain misaligned with downstream routed and timing quality. Prior work reduces this gap with human-designed terms or learned black-box surrogates, but the former requires expert retuning and the latter is difficult to explain, debug, or deploy in analytical placement flows. CoEvoP&R addresses these limitations with a large language model (LLM)-based framework that automatically evolves analytical placement objectives. At each generation, the prompt combines the restricted objective interface, baseline context, and archived prior candidates with routing-related feedback from placement, timing proxy, and routing tools. The LLM proposes readable differentiable objectives, which are embedded and validated in DREAMPlace, evaluated through a timing proxy and an actual router, and stored with their feedback to guide later generations. Across eight ChiP-Bench Nangate45 designs and three seeds, CoEvoP&R reduces post-route routed wirelength and congestion by 16.9% and 36.7%, with gains of 0.70 ns in worst negative slack and a 912 ns reduction in total negative slack magnitude over native DREAMPlace. Across eight ICCAD 2015 Superblue designs, it reduces post-route routed wirelength and congestion by 5.4% and 23.2%. Code is available at https://github.com/FCHXWH823/CoEvoP-R.git.

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arXiv2026

Decision problem for Hamilton $2$-cycles in $4$-graphs

A $4$-uniform $2$-cycle in a $4$-uniform hypergraph of length $t$ is a cyclic ordering of $2t$ vertices $v_1v_2\cdots v_{2t}v_1$ such that $v_{2i+1}v_{2i+2}v_{2i+3}v_{2i+4}$ are edges for $0\le i\le t-1$ while the addition is modulo $2t$. For every $γ>0$ and large $n$, we characterize the $n$-vertex $4$-uniform hypergraphs such that every triple of vertices is contained in at least $(1/3+γ)n$ edges and admits a Hamilton $2$-cycle. Up to the error term $γn$, the assumption on the minimum codegree is best possible and verifies a conjecture of Garbe and Mycroft. As a consequence, this gives a polynomial-time algorithm that decides whether an $n$-vertex $4$-uniform hypergraph with minimum codegree $(1/3+γ)n$ contains a Hamilton $2$-cycle. This stands as a steep contrast to the graph case where such a hardness gap has size $o(n)$.

math.CO↗

arXiv2026

Finding matchings in dense hypergraphs

We consider the algorithmic decision problem that takes as input an $n$-vertex $k$-uniform hypergraph $H$ with minimum codegree at least $m-c$ and decides whether it has a matching of size $m$. We show that this decision problem is fixed parameter tractable with respect to $c$. Furthermore, our algorithm not only decides the problem, but actually either finds a matching of size $m$ or a certificate that no such matching exists. In particular, when $m=n/k$ and $c=O(\log n)$, this gives a polynomial-time algorithm, that given any $n$-vertex $k$-uniform hypergraph $H$ with minimum codegree at least $n/k-c$, finds either a perfect matching in $H$ or a certificate that no perfect matching exists.

math.CO↗

arXiv2026

Clique-factors in sparse pseudorandom graphs

We prove that for any $t\ge 3$ there exist constants $c>0$ and $n_0$ such that any $d$-regular $n$-vertex graph $G$ with $t\mid n\geq n_0$ and second largest eigenvalue in absolute value $λ$ satisfying $λ\le c d^{t}/n^{t-1}$ contains a $K_t$-factor, that is, vertex-disjoint copies of $K_t$ covering every vertex of $G$. The result generalizes to broader setting of jumbled graphs, which were introduced by Thomason in the eighties.

math.CO↗

arXiv2026

Finding any given 2-factor in sparse pseudorandom graphs efficiently

Given an $n$-vertex pseudorandom graph $G$ and an $n$-vertex graph $H$ with maximum degree at most two, we wish to find a copy of $H$ in $G$, i.e.\ an embedding $φ\colon V(H)\to V(G)$ so that $φ(u)φ(v)\in E(G)$ for all $uv\in E(H)$. Particular instances of this problem include finding a triangle-factor and finding a Hamilton cycle in $G$. Here, we provide a deterministic polynomial time algorithm that finds a given $H$ in any suitably pseudorandom graph $G$. The pseudorandom graphs we consider are $(p,λ)$-bijumbled graphs of minimum degree which is a constant proportion of the average degree, i.e.\ $Ω(pn)$. A $(p,λ)$-bijumbled graph is characterised through the discrepancy property: $\left|e(A,B)-p|A||B|\right |<λ\sqrt{|A||B|}$ for any two sets of vertices $A$ and $B$. Our condition $λ=O(p^2n/\log n)$ on bijumbledness is within a log factor from being tight and provides a positive answer to a recent question of Nenadov. We combine novel variants of the absorption-reservoir method, a powerful tool from extremal graph theory and random graphs. Our approach builds on our previous work (\emph{European Journal of Combinatorics} \textbf{82} (2019), 102999), incorporating the work of Nenadov (\emph{Bulletin of the London Mathematical Society} \textbf{51} (3) (2019), pp.~421--430), together with additional ideas and simplifications.

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arXiv2026

A Ramsey-Turán theory for tilings in graphs

For a $k$-vertex graph $F$ and an $n$-vertex graph $G$, an $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$. For $r\in \mathbb{N}$, the $r$-independence number of $G$, denoted $α_r(G)$, is the largest size of a $K_r$-free set of vertices in $G$. In this paper, we discuss Ramsey--Turán-type theorems for tilings where one is interested in minimum degree and independence number conditions (and the interaction between the two) that guarantee the existence of optimal $F$-tilings. For cliques, we show that for any $k\geq 3$ and $η>0$, any graph $G$ on $n$ vertices with $δ(G)\ge ηn$ and $α_k(G)=o(n)$ has a $K_k$-tiling covering all but $\lfloor 1/η\rfloor(k-1)$ vertices. All conditions in this result are tight; the number of vertices left uncovered can not be improved and, for $η<\tfrac{1}{k}$, a condition of $α_{k-1}(G)=o(n)$ would not suffice. When $η>\tfrac{1}{k}$, we then show that $α_{k-1}(G)=o(n)$ does suffice, but not $α_{k-2}(G)=o(n)$. These results unify and generalise previous results of Balogh--Molla--Sharifzadeh, Nenadov--Pehova and Balogh--McDowell--Molla--Mycroft on the subject. We further explore the picture when $F$ is a tree or a cycle and discuss the effect of replacing the independence number condition with $α^*(G)=o(n)$ (meaning that any pair of disjoint linear sized sets induce an edge between them) where one can force perfect $F$-tilings covering all the vertices. Finally we discuss the consequences of these results in the randomly perturbed graph setting.

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arXiv2026

Note on the codegree version of the Erdős--Ko--Rado theorem

Kupavskii proved a codegree version of the Erdős--Ko--Rado theorem by showing that for an intersecting family $\mathcal{F} \subseteq \binom{[n]}{k}$ with $n \geq 2k +3d/(1-d/k)$, the minimum $d$-degree of $\mathcal{F}$ is at most $\binom{n-d-1}{k-d-1}$. Huang and Zhang improved the bound on $n$ to $n \geq 2k+2d-3$. In this short note, we prove that if $d = k-1$, then the bound on $n$ can be improved to $2k + \sqrt{2k} + O(1)$. In addition, we extend our method to show that the bound on $n$ can be improved to $2k + 7k^{2/3}+O(k^{1/3})$ when $d=k-2$.

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