arXiv subjects

Pan Peng

Publications and source records attributed to Pan Peng.

At least 19 recordsLinked to original sources

arXiv2026

Testing for a Large Induced Expander: Sublinear Algorithms and Query Lower Bounds

Testing expansion asks to distinguish graphs of large conductance from graphs that are far from having large conductance. We study a relaxation in which only part of the graph is required to expand: does an $n$-vertex graph of bounded degree contain an induced expander on a constant fraction of its vertices, when the rest of the graph, including the edges leaving the expander, is arbitrary? Let $P_{α,ϕ}$ denote the property of containing an induced subgraph on at least $αn$ vertices with inner conductance at least $ϕ$. For every constant $κ\in(0,1)$, we give a tester that accepts graphs in $P_{α,ϕ}$ and rejects graphs outside $P_{κα,μϕ^2}$, where $μ=Θ_{α,κ}(1/\log n)$, using $O_{α,κ}(\sqrt n\,ϕ^{-2}\log n)$ queries. For fixed $α\in(0,1]$, $κ\in(0,1)$, and $0<ϕ\le9/20$, we further show that distinguishing these two classes requires $Ω_{α,κ,ϕ}(n^{1/2+c_{κ,ϕ}μ})$ queries for every sufficiently large $n$ and every $0<μ\le1$, where $c_{κ,ϕ}>0$. The bound holds for adaptive algorithms, uniformly in $μ$. Hence, for testers with $\sqrt n\,\mathrm{polylog}(n)$ queries, the logarithmic loss in conductance is necessary up to a $\log\log n$ factor. The same tester also permits distance-based rejection at a weaker conductance while retaining the original core size. Our tester compares the endpoint distributions of random walks from different vertices with each other, rather than with the uniform distribution. Its analysis uses the trace chain, which observes the walk only at its visits to the unknown expanding core. Our lower bound builds on the noisy-parity construction of [CKK+18], using vector-valued parity labels in $\mathbb F_2^r$ to create many hidden classes, each too small to contain a qualifying induced expander.

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arXiv2026

Near-Linear Lower Bounds for Testing Directed Acyclicity

We study testing acyclicity of directed graphs in the bounded-degree incidence-lists model, where a tester can query the outneighbors of a vertex but not its inneighbors. Despite significant effort, no tester with $o(n)$ queries is known, while the best established lower bounds are $\widetildeΩ(n^{5/9})$ for one-sided testers (Chen, Randolph, Servedio, and Sun, 2022) and $Ω(n^{1/3})$ for two-sided testers (Bender and Ron, 2002). We show that a query exponent arbitrarily close to one is necessary: for every fixed $η\in(0,1)$, there exists a constant $\varepsilon_η>0$ such that, for every constant $d\ge2$, every randomized adaptive two-sided $\varepsilon_η$-tester for acyclicity of simple $n$-vertex directed graphs whose indegrees and outdegrees are at most $d$ makes $Ω(n^{1-η})$ queries. Our proof overcomes two barriers. First, in previous constructions, the acyclic part of the graph is easy to recognize, which helps an algorithm stay inside the cyclic part. We hide a high-girth cyclic core inside a hierarchy of acyclic masking regions, in which each region imitates the levels below it and can be told apart from them only through rare events, and an iterated information-theoretic argument shows that every exploration observes only small, low-diameter forest components of the core. Second, a two-sided tester may exploit the statistics of collisions without ever seeing a cycle, and at a near-linear budget collisions are unavoidable. We compare complete transcripts with those of an acyclic input built on a fixed core by counting injective embeddings of the observed forest: in high-girth cores, all connected collision patterns of bounded order contribute identically, which makes the two transcript distributions statistically indistinguishable.

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arXiv2026

The Power of Interaction in Locally Edge-Private Subgraph Counting

We study (non-induced) subgraph counting under interactive local edge differential privacy (LEDP). Given a fixed pattern graph $H$ with $k$ vertices, $\ell\ge1$ edges, and $q$ connected components, our goal is to approximately count its copies in an arbitrary $n$-vertex undirected graph while preserving edge privacy. We give two complementary accuracy guarantees. First, for every $H$, we give Linear Noise Cancellation (LNC), a two-round $\varepsilon$-LEDP algorithm with additive error and root mean square error both $O(n^{k-3/2}/\varepsilon^\ell)$ on every $n$-vertex graph. For fixed sufficiently small $\varepsilon$, this improves the optimal noninteractive additive error for triangles by a factor of $\sqrt n$ and gives the same improvement in worst-case root mean square error for all cliques on at least three vertices. To our knowledge, this is the first separation between interactive and noninteractive LEDP subgraph counting, showing that a single additional round can improve worst-case accuracy polynomially. Second, we determine how accuracy improves when the graph has maximum degree at most a public bound $d$. Our Predict and Correct (PC) framework gives an $(\ell+2)$-round $\varepsilon$-LEDP algorithm with additive error $O(n^{q-1/2}d^{k-q-1}\log^{3\ell-2}n/\varepsilon^\ell)$. For connected patterns, its polynomial factor is $\sqrt n\,d^{k-2}$, giving sublinear error on constant-degree graphs at fixed privacy. We prove matching lower bounds in $n$ and $d$, up to logarithmic factors at fixed privacy, even under arbitrary interaction and approximate privacy with sufficiently small $δ$. At $d=Θ(n)$, the lower bound also establishes the optimality of LNC's dependence on $n$.

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arXiv2026

Quantum Lower Bounds for Collision Problems

We establish tight lower bounds on the adaptive and non-adaptive quantum query complexities of Promised $k$-to-one Finding on an exactly $k$-to-one input string $x\in\mathbb Z_q^n$, for every fixed $k\ge2$ and every feasible alphabet size $q\ge n/k$. The lower bounds are $Ω_k(n^{1/2-1/(2(2^k-1))})$ for unrestricted adaptive algorithms and $Ω_k(n^{1-1/k})$ for one-round non-adaptive algorithms. The former matches the adaptive quantum algorithm of Liu and Zhandry [LZ19], whereas the latter matches classical birthday sampling even when the parallel queries are arbitrarily entangled. We also establish the non-adaptive lower bound $Ω_k(\sqrt{n/k})$ for $k$-to-one Testing when $q\ge n$, resolving an open question posed by Koiran, Landes, Portier, and Yao [KLPY10]. Our proofs generate the balanced input from a uniformly random permutation and use compressed permutation oracles to describe the information learned by the algorithm. The adaptive proof bounds the growth of partial collisions through successive queries. The one-round proof instead analyzes the state after the parallel queries directly, bounding the probability that the compressed database contains a full collision. In both proofs, a short verification step accounts for correct outputs that include unrecorded positions. For the decision problem, we also compare one-round states with independent and distinct output labels.

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arXiv2026

Sublinear Algorithms for Estimating Single-Linkage Clustering Costs

Single-linkage clustering (SLC) is a fundamental method for hierarchical data analysis. In the distance setting, a $k$-clustering produced by SLC can be obtained by computing a minimum spanning tree (MST) and deleting its $k-1$ heaviest edges. This naturally induces a cost profile for the SLC hierarchy: for each $k\in[n]$, we define $\mathrm{cost}_k$ to be the weight of the resulting $k$-component spanning forest, equivalently, the minimum total weight of any spanning forest with exactly $k$ connected components. The corresponding \emph{SLC cost profile} is $(\mathrm{cost}_1,\ldots,\mathrm{cost}_n)$, and the scalar quantity $\mathrm{cost}(G)=\sum_{k=1}^{n}\mathrm{cost}_k$ is the area under this profile. We study the problem of approximating these quantities in sublinear time. We assume that the input is a weighted graph $G$ of average degree $d$ with edge weights in $\{1,\dots,W\}$, accessed through adjacency-list queries; missing edges are treated as having infinite distance. Our main result is a sampling-based algorithm that outputs a succinct sketch of the entire SLC cost profile in the distance setting. The algorithm runs in $\widetilde{O}(d\sqrt{W}/\varepsilon^3)$ time and returns a sketch from which one can derive estimates $(\widehat{\mathrm{cost}}_1,\ldots,\widehat{\mathrm{cost}}_n)$ satisfying $\sum_{k=1}^{n}\bigl|\widehat{\mathrm{cost}}_k-\mathrm{cost}_k\bigr| \le \varepsilon\,\mathrm{cost}(G)$. Thus, we obtain an $\ell_1$ approximation to the full profile whose error is at most an $\varepsilon$-fraction of the area under the true profile. In particular, this yields a $(1\pm\varepsilon)$-approximation to $\mathrm{cost}(G)$ within the same running time. We also prove a nearly matching lower bound of $Ω(d\sqrt{W}/\varepsilon^2)$ queries for estimating $\mathrm{cost}(G)$.

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arXiv2026

Learning-Augmented Streaming Algorithms for Approximating Boolean Max-CSPs

We study learning-augmented streaming algorithms for estimating the value of Boolean Maximum Constraint Satisfaction Problems (Max-CSPs). Specifically, we consider streaming algorithms equipped with an $\varepsilon$-accurate oracle: for each variable, the oracle outputs a label in $\{-1,1\}$ that agrees with a fixed optimal assignment with probability $1/2+\varepsilon$, and disagrees otherwise. Prior work by Dong, Peng, and Vakilian [ITCS 2025] showed that, for the special case of Max-Cut, such an oracle enables a $(1/2+Ω(\varepsilon^2))$-approximation using $\operatorname{poly}(1/\varepsilon)$ words of space in insertion-only streams, and $\operatorname{poly}(1/\varepsilon,\log n)$ words of space in dynamic streams. We substantially generalize and strengthen their result, provided that a highly accurate estimate of $\varepsilon$ is available. For any Boolean Max-$2$CSP and any $η>0$ independent of $\varepsilon$, we obtain, with high constant probability, a single-pass $(1-η)$-approximation using $\operatorname{poly}(1/\varepsilon,1/η)$ words of space in insertion-only streams, and $\operatorname{poly}(1/\varepsilon,1/η,\log n)$ words of space in dynamic streams, where $n$ is the number of variables. We further extend our techniques to general Boolean Max-$k$CSPs for $k\ge 3$, with slightly worse space complexity.

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arXiv2026

A proof of the framed LMOV conjecture for links

We prove the Labastida--Mariño--Ooguri--Vafa conjecture for oriented links with arbitrary integer framings. Consequently, the framed reformulated invariants have finite integral LMOV expansions for every integer framing vector.

math.GT↗

arXiv2026

Quantum Property Testing for Bounded-Degree Directed Graphs

We study quantum property testing of directed graphs whose maximum in-degree and out-degree are bounded by a fixed constant $d$. For a proximity parameter $\varepsilon$, we prove that every property testable with $O_{\varepsilon,d}(1)$ quantum queries in the bidirectional model, where both incoming and outgoing neighbors are accessible, can also be tested in the quantum unidirectional model, where only outgoing neighbors are accessible, using $n^{1/2-Ω_{\varepsilon,d}(1)}$ queries. This gives an almost quadratic quantum speedup over the best known generic classical transformation. Our proof has two main ingredients. First, we show that, in the bidirectional model for bounded-degree digraphs, the class of properties testable with constantly many classical queries coincides with the class testable with constantly many quantum queries. Second, we give a transformation from classical bidirectional testers to quantum unidirectional testers by designing a quantum unidirectional algorithm for estimating the frequency vector of constant-radius rooted discs. The algorithm combines quantum counting and Grover search with a correction procedure that removes false local appearances of smaller disc types inside larger neighborhoods. We further show that this transformation is essentially tight. For every sufficiently small fixed $\varepsilon>0$, there is a degree bound $d=d(\varepsilon)$ and an explicit property $P_\varepsilon$ for $d$-bounded-degree digraphs that is $\varepsilon$-testable with $O_\varepsilon(1)$ classical, and hence quantum, bidirectional queries, but requires $ \widetildeΩ_\varepsilon \left(n^{1/2-f'(\varepsilon)}\right) $ quantum queries in the unidirectional model, where $f'(\varepsilon)\to 0$ as $\varepsilon\to 0$.

quant-ph↗

arXiv2026

Testing Induced-Subgraph Freeness in Outerplanar Graphs under the Random-Neighbor Oracle

We prove that, for every fixed nonempty graph $H$, induced-$H$-freeness is testable with $\varepsilon^{-O_H(1)}$ queries on outerplanar graphs with no maximum-degree bound in the $\textit{random-neighbor model}$, where each query at a vertex returns a uniformly random neighbor. Thus, the query complexity is polynomial in $1/\varepsilon$ and independent of the number $n$ of vertices. Previously, the best bound known for this problem was the $\operatorname{poly}(\log n)$-query guarantee that follows from the general outerplanar-graph tester of Babu, Khoury, and Newman (2016) in the stronger $\textit{adjacency-list model}$, which provides exact degree queries and indexed access to neighbors. Our tester has $\textit{two-sided error}$, which is necessary in general: induced-$P_3$-freeness has no one-sided constant-query tester in the random-neighbor model, even on outerplanar graphs of maximum degree two.

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arXiv2026

Streaming Hypergraph Coloring via Palette Sparsification

For every fixed $k\ge2$, we give a randomized one-pass insertion-only algorithm that colors an $n$-vertex $k$-uniform hypergraph of maximum degree $Δ$ with $O(Δ^{1/(k-1)})$ colors using $\widetilde O_k(n)$ bits of working memory. As a graph-theoretic result of independent interest, we also prove a tight palette-sparsification theorem for general uniform hypergraphs. Independently sampled lists of $Θ(\sqrt{\log n})$ colors from a palette of size $O(Δ^{1/(k-1)})$ preserve colorability with high probability; the list-size dependence is asymptotically optimal. These results extend to bounded-rank hypergraphs. We complement the algorithm with a deterministic lower bound: for every fixed polylogarithmic semi-streaming space bound, there are polylogarithmic values of $Δ$ for which any deterministic one-pass algorithm requires $\exp(Δ^{Ω(1)})$ colors.

math.CO↗

arXiv2026

Private Approximation of Graph Spectra and Cuts via Spectral Amplifiers

We study the problem of releasing a synthetic graph that approximates the sizes of all cuts of an input graph under edge-level differential privacy. If one insists on purely additive error, the optimal worst-case error is $\widetildeΘ(n^{3/2})$. If one allows a small multiplicative slack, an information-theoretic exponential-time mechanism achieves nearly linear additive error, but the best known polynomial-time algorithms have substantially larger error. We give a polynomial-time $(\varepsilon,δ)$-differentially private algorithm which, for every $n$-vertex unweighted graph $G$, outputs a non-negative weighted synthetic graph $\widetilde G$ such that, with high probability, every cut $S\subseteq V(G)$ satisfies \[ |w_G(S)-w_{\widetilde G}(S)| \le γw_G(S)+\widetilde O_{\varepsilon,δ,γ}(n^{13/12+o(1)}). \] This improves the previous polynomial-time worst-case bound $\widetilde O(n^{5/4+o(1)})$ of Aamand et al. (ICML 2025) for mixed multiplicative/additive private cut approximation. The main technical ingredient is a new set of private spectral primitives for bounded-degree graphs, one of them gives spectral error $\widetilde O_δ((nd)^{1/4}/\sqrt\varepsilon)$ in estimating the graph Laplacian for graphs of maximum degree $d$, being the first to beat the standard $\min\{2d,\widetilde O_δ(\sqrt{n}/\varepsilon)\}$ baseline in the high-degree regime. We further develop a primitive with a sharper error dependence on $n$ and $d$ for the downstream cut approximation. Combined with a new edge-sensitive terminal cut oracle with additive error $\widetilde O(n+(n^2M)^{1/3})$ on graphs with $M$ edges, this yields the final worst-case $\widetilde O(n^{13/12+o(1)})$ private cut-release error.

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arXiv2026

An Exponential Lower Bound for Spectral Density Estimation on Unweighted Graphs

We study lower bounds for estimating the spectral density of the normalized adjacency matrix of a graph. Previously, Cohen-Steiner et al. [KDD 2018] proposed an algorithm for $\varepsilon$-approximate spectral density estimation in the Wasserstein-1 distance, using $2^{O(1/\varepsilon)}$ random walks initiated from uniformly random nodes in the graph. Later, Jin et al. [COLT 2023] established a nearly matching exponential lower bound for \emph{weighted} graphs, assuming the algorithm has access to samples from random walks started at random nodes. It was left open whether this lower bound could be extended to \emph{unweighted} graphs. In this paper, we answer this question in the affirmative by proving an exponential lower bound for unweighted graphs. Specifically, we show that no algorithm can compute an $\varepsilon$-approximation to the spectrum of a normalized graph adjacency matrix with constant success probability, even when given the full transcripts of $2^{Ω(1/\varepsilon^{1/6})}$ random walks, each of length $2^{Ω(1/\varepsilon^{1/6})}$, started from uniformly random nodes.

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arXiv2026

Four-Cycle Counting in Low-Degeneracy Graph Streams

We study the problem of $(1+\varepsilon)$-approximating the number of four-cycles in graphs given as arbitrary order edge streams. We propose two new algorithms based on sampling induced subgraphs. Our first contribution is a two-pass algorithm that uses $\widetilde{O}(κm / \sqrt{T})$ space, where $m$ is the number of edges, $T$ is the number of four-cycles, and $κ$ is the graph's degeneracy. This algorithm improves upon existing theoretical bounds and is provably optimal for constant-degeneracy graphs, matching the known $Ω(m/\sqrt{T})$ lower bound up to lower-order factors. Our second contribution is a one-pass algorithm that remains accurate when four-cycles are not highly concentrated around individual nodes, edges, or wedges; this structural property is common in sparse social and collaboration networks. We evaluate both algorithms on a variety of real-world graph streams. The two-pass algorithm consistently outperforms state-of-the-art methods, using substantially less space to achieve a desired accuracy. The one-pass algorithm is competitive when four-cycles are evenly distributed, matching our theoretical analysis. Unlike several recent works, our algorithms perform well even on non-bipartite graphs such as social networks.

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arXiv2026

Differentially Private Range Subgraph Counting

Subgraph counting is a fundamental problem in graph analysis. Motivated by practical scenarios where graph analytics are performed on subgraphs induced by selected vertices -- rather than on the entire graph -- and by growing privacy concerns, we initiate the study of differentially private range subgraph counting (DPRSC). The goal is to privately count occurrences of a fixed pattern graph within induced subgraphs defined by multi-dimensional attribute ranges. Unlike classical point counting, subgraph counting is inherently nonlinear and exhibits high sensitivity: a single edge modification can affect many subgraph occurrences. We present the first efficient algorithms for DPRSC with small additive error. Our approach introduces a subgraph projection that reduces DPRSC to weighted orthogonal range counting, enabling the use of range trees and local sensitivity estimation to achieve accurate private query answering. We complement our algorithms with matching lower bounds, obtained by reducing reconstruction attacks to DPRSC and leveraging discrepancy theory. In particular, we show that any differentially private algorithm for DPRSC must incur additive error exponential in the dimension. Empirical evaluations demonstrate that our algorithms significantly outperform baseline methods in accuracy and runtime while maintaining strong privacy guarantees.

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arXiv2026

Quantum Algorithms for Triangle Cut Sparsification

Triangles capture higher-order structures in graphs and are fundamental to applications such as clustering and network analysis. To enable efficient use of such structures at scale, we study the problem of triangle cut sparsification, which aims to reduce the graph size while approximately preserving triangle counts across every cut. We investigate quantum algorithms for this problem, using triangle listing as our main technical ingredient. In particular, we present a quantum algorithm for triangle listing that, for a graph with $n$ vertices, $m$ edges, and $t$ triangles, runs in time $T_{\mathrm{q\text{-}list}} =$ $\widetilde{O}\bigl(\min(n^{5/4}t^{7/12} + n^{7/6}t^{7/9}, m + m^{3/4}t^{1/2},$ $n^{3/2}t^{1/2})\bigr)$, improving upon the best known classical bounds over a broad range of parameters. Our algorithm is based on a heavy-light vertex partition and an extension of triangle detection via quantum walks and Grover search. Leveraging this result, we design a quantum algorithm for constructing $\varepsilon$-triangle cut sparsifiers of size $\widetilde{O}(n/\varepsilon^2)$ in time $\widetilde{O}(T_{\mathrm{q\text{-}list}} + \sqrt{mn}/\varepsilon)$. Finally, we demonstrate applications to clustering algorithms based on triangle-related measures and prove a lower bound of $Ω(n/\varepsilon^2)$ on the size of any $\varepsilon$-triangle cut sparsifiers.

quant-ph↗

arXiv2026

Streaming Max-Cut in General Metrics

Max-Cut is a fundamental combinatorial optimization problem that has been studied in various computational settings. We initiate the study of its streaming complexity in \emph{general metric spaces} with access to distance oracles. We give a $(1 + ε)$-approximate algorithm for estimating the Max-Cut value in \emph{sliding-window} streams using only poly-logarithmic space. This is the first sliding-window algorithm for Max-Cut even in Euclidean spaces, and it matches a known insertion-only space bound in the special case of Euclidean spaces [Chen, Jiang, Krauthgamer, STOC'23]. In sharp contrast, we give a $\poly(n)$-space lower bound in the \emph{dynamic} streaming setting. This yields a separation from the Euclidean case, where the polylogarithmic-space $(1+ε)$-approximation extends to dynamic streams. On the technical side, our sliding-window algorithm builds on the smooth histogram framework of [Braverman and Ostrovsky, SICOMP'10]. To make this framework applicable, we establish the first smoothness bound for metric Max-Cut. Moreover, we develop a streaming algorithm for metric Max-Cut in insertion-only streams, whose key ingredient is a new metric reservoir sampling technique.

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arXiv2026

Sublinear Spectral Clustering Oracle with Little Memory

We study the problem of designing \emph{sublinear spectral clustering oracles} for well-clusterable graphs. Such an oracle is an algorithm that, given query access to the adjacency list of a graph $G$, first constructs a compact data structure $\mathcal{D}$ that captures the clustering structure of $G$. Once built, $\mathcal{D}$ enables sublinear time responses to \textsc{WhichCluster}$(G,x)$ queries for any vertex $x$. A major limitation of existing oracles is that constructing $\mathcal{D}$ requires $Ω(\sqrt{n})$ memory, which becomes a bottleneck for massive graphs and memory-limited settings. In this paper, we break this barrier and establish a memory-time trade-off for sublinear spectral clustering oracles. Specifically, for well-clusterable graphs, we present oracles that construct $\mathcal{D}$ using much smaller than $O(\sqrt{n})$ memory (e.g., $O(n^{0.01})$) while still answering membership queries in sublinear time. We also characterize the trade-off frontier between memory usage $S$ and query time $T$, showing, for example, that $S\cdot T=\widetilde{O}(n)$ for clusterable graphs with a logarithmic conductance gap, and we show that this trade-off is nearly optimal (up to logarithmic factors) for a natural class of approaches. Finally, to complement our theory, we validate the performance of our oracles through experiments on synthetic networks.

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arXiv2026

Near-Optimal Four-Cycle Counting in Graph Streams

We study four-cycle counting in arbitrary order graph streams. We present a 3-pass algorithm for $(1+\varepsilon)$-approximating the number of four-cycles using $\widetilde{O}(m/\sqrt{T})$ space, where $m$ is the number of edges and $T$ the number of four-cycles in the graph. This improves upon a 3-pass algorithm by Vorotnikova using space $\widetilde{O}(m/T^{1/3})$ and matches a multi-pass lower bound of $Ω(m/\sqrt{T})$ by McGregor and Vorotnikova.

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