arXiv subjects

Zihan Tan

Publications and source records attributed to Zihan Tan.

At least 19 recordsLinked to original sources

arXiv2026

MetaRSI / RSI2: A Meta-Recursive Self-Improving System for Recursive Self-Improving Systems Themselves

Recursive self-improvement (RSI) lets a system improve the model-building machinery from its own failures, so every later model inherits the gain. Yet RSI has been validated almost exclusively on coding and formal benchmarks such as science QA and mathematics. This format bound limits RSI to improvement within a machine-checkable slice, not general capability where questions are open and correctness is settled by argument, replication, or measurement. We argue RSI must next operate across real, diverse scientific, engineering, and meta-scientific domains, not where formal evaluation is merely tractable. To that end we present MetaRSI-v1, where improvement is the scheduled composition of three typed operators over one unified paradigm. Data-RSI amplifies existing competence and marks its boundary; Harness-RSI edits a five-slot scaffold without touching weights; Model-RSI internalizes capability into parameters through bounded training. Sharing one loop kernel and artifact vocabulary, they make data, scaffold, and model changes composable rather than exclusive. A two-axis optimizer jointly decides operator order and each operator's proposal policy, while a meta-level policy revises the schedule across terms. We validate MetaRSI-v1 under the field's standard evaluations, on code and closed-form science, with no external teacher: the target model plays every role in its own loop. MetaRSI-v1 reframes self-improvement from a single-surface edit to a composition across the full model-production pipeline, opening two paths: a model route internalizing capability through training, and a harness route leaving weights untouched and thus extending self-improvement to any model reachable through an interface, with Data-RSI redefined as the shared substrate feeding both. The framework further yields refutable laws on where loops exist, how operators compose, and what supervision buys.

cs.LG↗

arXiv2026

Paths and Intersections: Repelling Pairs

We study two inverse problems for shortest-path metrics of Okamura-Seymour instances: recognizing metrics realizable by outerplanar graphs, and computing minimum-edge Okamura-Seymour realizations. We introduce the notion of \emph{repelling pairs}, a metric certificate that the shortest paths corresponding to two terminal pairs must be vertex-disjoint in every realization. Our central structural result is that, for an Okamura-Seymour metric with a prescribed cyclic order, a terminal path structure in an Okamura-Seymour instance can be realized by nonnegative edge lengths if and only if the paths assigned to every repelling pair are vertex-disjoint. Building on the notion of repelling pairs, we give algorithmic answers to the inverse problems. First, we design an algorithm that, given a metric, decides in polynomial time whether or it admits an outerplanar realization and constructs one when one exists. Second, given an Okamura-Seymour metric, we efficiently compute a canonical medial template whose crossing number equals the minimum number of edges in any Okamura-Seymour realization. The minimum-edge graph structures are exactly the primal graphs of arrangements of this template, and each can be assigned realizing edge lengths in polynomial time.

cs.DS↗

arXiv2026

Paths and Intersections: Recognizing Outerplanar Metrics

We study the following distance realization problem: given a metric $D$ on a set $T$ of terminals, does there exist an (edge-weighted) outerplanar graph $G$, such that $T\subseteq V(G)$, and for every pair $t,t'\in T$, $\textsf{dist}_G(t,t')=D(t,t')$? We first prove that there is no ``local characterization'', forming a contrast with trees and Okamura-Seymour instances. Our main result is an efficient algorithm for this problem whose running time is polynomial in $|T|$. Both our proof and our algorithm utilize a recent new approach of analyzing graph structures, by viewing graphs as paths and their intersections, which we believe is of independent interest.

cs.DS↗

arXiv2026

Quasi-two-dimensional dispersions of Brownian particles with competitive interactions: Dynamical clustering, non-Gaussianity and hydrodynamic correlations

We conduct a comprehensive dynamical analysis of quasi-two-dimensional (Q2D) dispersions of Brownian particles with competing short-range attractive (SA) and long-range repulsive (LR) interactions using Langevin dynamics (LD) and multiparticle collision dynamics (MPC). As the attractive interaction is strengthened, self-diffusion is significantly suppressed, and clustering gives rise to pronounced subdiffusive behavior. We find that cluster lifetimes are influenced more strongly by attraction strength than by particle concentration. Two dynamical criteria for the transition from non-clustered to clustered phases are identified in terms of the mean cluster lifetime and the relaxation time of local hexagonal order, respectively. Moreover, clustered Q2D-SALR systems exhibit pronounced non-Gaussian dynamics. In particular, the self-van Hove function in the equilibrium-cluster phase displays an approximately exponential form, consistent with an underlying diffusing-diffusivity mechanism. Importantly, MPC simulations reveal the critical role of hydrodynamic interactions (HIs) in collective dynamics. We observe that the anomalously enhanced large-scale collective diffusion characteristic of hydrodynamically interacting Q2D systems is qualitatively preserved in Q2D-SALR dispersions. However, this enhancement suppresses the intermediate-range-order peak in the hydrodynamic function compared to its three-dimensional counterpart. Furthermore, by analyzing the time-dependent evolution of hydrodynamic function and the sound mode in hydrodynamic correlations, we find that clustering in Q2D-SALR systems leads to an earlier onset of HIs than in Q2D hard-sphere reference systems, implying HIs become relevant already on inertial timescales.

cond-mat.soft↗

arXiv2026

Lower Bounds on Flow Sparsifiers with Steiner Nodes

Given a large graph $G$ with a set of its $k$ vertices called terminals, a \emph{quality-$q$ flow sparsifier} is a small graph $G'$ that contains the terminals and preserves all multicommodity flows between them up to some multiplicative factor $q\ge 1$, called the \emph{quality}. Constructing flow sparsifiers with good quality and small size ($|V(G')|$) has been a central problem in graph compression. The most common approach of constructing flow sparsifiers is contraction: first compute a partition of the vertices in $V(G)$, and then contract each part into a supernode to obtain $G'$. When $G'$ is only allowed to contain all terminals, the best quality is shown to be $O(\log k/\log\log k)$ and $\Omega(\sqrt{\log k/\log\log k})$. In this paper, we show that allowing a few Steiner nodes does not help much in improving the quality. Specifically, there exist $k$-terminal graphs such that, even if we allow $k\cdot 2^{(\log k)^{\Omega(1)}}$ Steiner nodes in its contraction-based flow sparsifier, the quality is still $\Omega\big((\log k)^{0.3}\big)$.

cs.DS↗

arXiv2025

Cutting Planarians: Planar Emulators for String Graphs

In this paper we construct distance sketches for intersection graphs of arbitrary path-connected regions in the plane (known as the string graphs) in the constant and $1+\varepsilon$ distortion regimes. Furthermore, the distance sketches themselves are planar graphs. First, we show that every unweighted string graph $G$ has an $O(1)$-distortion planar emulator: that is, there exists an edge-weighted planar graph $H$ containing every vertex in $G$, such that every pair of vertices $(u,v)$ satisfies $\delta_G(u,v) \le \delta_H(u,v) \le O(1) \cdot \delta_G(u,v)$. Furthermore, we show that for any constant $\varepsilon > 0$, there is an edge-weighted planar graph $H'$ such that every pair of vertices $(u,v)$ satisfies $\delta_G(u,v) \le \delta_{H'}(u,v) \le (1+\varepsilon) \cdot \delta_G(u,v) + O(\varepsilon^{-4}\textrm{poly}\log n)$. No previous constructions of sparse distance sketches were known even for intersection graphs of simple shapes like axis-parallel rectangles or fat convex polygons. As applications, we construct the first $(1+\varepsilon, +O(1))$ mixed-distortion tree cover and distance oracle for arbitrary string graphs, as well as the first additive $+(\varepsilon\Delta+O(1))$-distortion embedding of string graphs $G$ with diameter $\Delta$ into graphs of constant treewidth $O(\varepsilon^{-4})$.

cs.DS↗

arXiv2025

Simple Length-Constrained Expander Decompositions

Length-constrained expander decompositions are a new graph decomposition that has led to several recent breakthroughs in fast graph algorithms. Roughly, an $(h, s)$-length $\phi$-expander decomposition is a small collection of length increases to a graph so that nodes within distance $h$ can route flow over paths of length $hs$ while using each edge to an extent at most $1/\phi$. Prior work showed that every $n$-node and $m$-edge graph admits an $(h, s)$-length $\phi$-expander decomposition of size $\log n \cdot s n^{O(1/s)} \cdot \phi m$. In this work, we give a simple proof of the existence of $(h, s)$-length $\phi$-expander decompositions with an improved size of $s n^{O(1/s)}\cdot \phi m$. Our proof is a straightforward application of the fact that the union of sparse length-constrained cuts is itself a sparse length-constrained cut. In deriving our result, we improve the loss in sparsity when taking the union of sparse length-constrained cuts from $\log ^3 n\cdot s^3 n^{O(1/s)}$ to $s\cdot n^{O(1/s)}$.

cs.DS↗

arXiv2025

Lower Bounds on Tree Covers

Given an $n$-point metric space $(X,d_X)$, a tree cover $\mathcal{T}$ is a set of $|\mathcal{T}|=k$ trees on $X$ such that every pair of vertices in $X$ has a low-distortion path in one of the trees in $\mathcal{T}$. Tree covers have been playing a crucial role in graph algorithms for decades, and the research focus is the construction of tree covers with small size $k$ and distortion. When $k=1$, the best distortion is known to be $\Theta(n)$. For a constant $k\ge 2$, the best distortion upper bound is $\tilde O(n^{\frac 1 k})$ and the strongest lower bound is $\Omega(\log_k n)$, leaving a gap to be closed. In this paper, we improve the lower bound to $\Omega(n^{\frac{1}{2^{k-1}}})$. Our proof is a novel analysis on a structurally simple grid-like graph, which utilizes some combinatorial fixed-point theorems. We believe that they will prove useful for analyzing other tree-like data structures as well.

cs.DS↗

arXiv2025

Paths and Intersections: Exact Emulators for Planar Graphs

We study vertex sparsification for preserving distances in planar graphs. Given an edge-weighted planar graph with $k$ terminals, the goal is to construct an emulator, which is a smaller edge-weighted planar graph that contains the terminals and exactly preserves the pairwise distances between them. We construct exact planar emulators of size $O(f^2k^2)$ in the setting where terminals lie on $f$ faces in the planar embedding of the input graph. Our result generalizes and interpolates between the previous results of Chang and Ophelders and Goranci, Henzinger, and Peng which is an $O(k^2)$ bound in the setting where all terminals lie on a single face (i.e., $f=1$), and the result of Krauthgamer, Nguyen, and Zondiner, which is an $O(k^4)$ bound for the general case (i.e., $f=k$). Our construction follows a recent new way of analyzing graph structures, by viewing graphs as paths and their intersections, which we believe is of independent interest.

cs.DS↗

arXiv2025

S2FGL: Spatial Spectral Federated Graph Learning

Federated Graph Learning (FGL) combines the privacy-preserving capabilities of federated learning (FL) with the strong graph modeling capability of Graph Neural Networks (GNNs). Current research addresses subgraph-FL from the structural perspective, neglecting the propagation of graph signals on spatial and spectral domains of the structure. From a spatial perspective, subgraph-FL introduces edge disconnections between clients, leading to disruptions in label signals and a degradation in the semantic knowledge of the global GNN. From a spectral perspective, spectral heterogeneity causes inconsistencies in signal frequencies across subgraphs, which makes local GNNs overfit the local signal propagation schemes. As a result, spectral client drift occurs, undermining global generalizability. To tackle the challenges, we propose a global knowledge repository to mitigate the challenge of poor semantic knowledge caused by label signal disruption. Furthermore, we design a frequency alignment to address spectral client drift. The combination of Spatial and Spectral strategies forms our framework S2FGL. Extensive experiments on multiple datasets demonstrate the superiority of S2FGL. The code is available at https://github.com/Wonder7racer/S2FGL.git.

cs.LG↗

arXiv2025

Metric Distortion for Tournament Voting and Beyond

In the well-studied metric distortion problem in social choice, we have voters and candidates located in a shared metric space, and the objective is to design a voting rule that selects a candidate with minimal total distance to the voters. However, the voting rule has limited information about the distances in the metric, such as each voter's ordinal rankings of the candidates in order of distances. The central question is whether we can design rules that, for any election and underlying metric space, select a candidate whose total cost deviates from the optimal by only a small factor, referred to as the distortion. A long line of work resolved the optimal distortion of deterministic rules, and recent work resolved the optimal distortion of randomized (weighted) tournament rules, which only use the aggregate preferences between pairs of candidates. In both cases, simple rules achieve the optimal distortion of $3$. Can we achieve the best of both worlds: a deterministic tournament rule matching the lower bound of $3$? Prior to our work, the best rules have distortion $2 + \sqrt{5} \approx 4.2361$. In this work, we establish a lower bound of $3.1128$ on the distortion of any deterministic tournament rule, even when there are only 5 candidates, and improve the upper bound with a novel rule guaranteeing distortion $3.9312$. We then generalize tournament rules to the class of $k$-tournament rules which obtain the aggregate preferences between $k$-tuples of candidates. We show that there is a family of deterministic $k$-tournament rules that achieves distortion approaching $3$ as $k$ grows. Finally, we show that even with $k = 3$, a randomized $k$-tournament rule can achieve distortion less than $3$, which had been a longstanding barrier even for the larger class of ranked voting rules.

cs.GT↗

arXiv2025

Multiplicative Spanners in Minor-Free Graphs

In FOCS 2017, Borradaille, Le, and Wulff-Nilsen addressed a long-standing open problem by proving that minor-free graphs have light spanners. Specifically, they proved that every $K_h$-minor-free graph has a $(1+\epsilon)$-spanner of lightness $O_{\epsilon}(h \sqrt{\log h})$, hence constant when $h$ and $\epsilon$ are regarded as constants. We extend this result by showing that a more expressive size/stretch tradeoff is available. Specifically: for any positive integer $k$, every $n$-node, $K_h$-minor-free graph has a $(2k-1)$-spanner with sparsity \[O\left(h^{\frac{2}{k+1}} \cdot \text{polylog } h\right),\] and a $(1+\epsilon)(2k-1)$-spanner with lightness \[O_{\epsilon}\left(h^{\frac{2}{k+1}} \cdot \text{polylog } h \right).\] We further prove that this exponent $\frac{2}{k+1}$ is best possible, assuming the girth conjecture. At a technical level, our proofs leverage the recent improvements by Postle (2020) to the remarkable density increment theorem for minor-free graphs.

cs.DS↗

arXiv2024

Trypanosoma brucei moving in microchannels and through constrictions

Trypanosoma brucei (T. brucei), a single-celled parasite and natural microswimmer, is responsible for fatal sleeping sickness in infected mammals, including humans. Understanding how T. brucei interacts with fluid environments and navigates through confining spaces is crucial not only for medical and clinical applications but also for a fundamental understanding of how life organizes in a confined microscopic world. Using a hybrid multi-particle collision dynamics (MPCD)--molecular dynamics (MD) approach, we present our investigations on the locomotion of an in silico T. brucei in three types of fluid environments: bulk fluid, straight cylindrical microchannels, and microchannels with constrictions. We observe that the helical swimming trajectory of the in silico T. brucei becomes rectified in straight cylindrical channels compared to bulk fluid. The swimming speed for different channel widths is governed by the diameter of the helical trajectory. The speed first slightly increases as the channel narrows and then decreases when the helix diameter is compressed. An optimal swimming speed is achieved, when the channel width is approximately twice the bulk helix diameter. It results from an interplay of the trypanosome's hydrodynamic interactions with the cylindrical channel walls and the high deformability of the parasite. In microchannels with constrictions, the motions of the anterior and posterior ends, the end-to-end distance, and the log-rolling motion of the cell body are characterized and show salient differences compared to the straight-channel case. Depending on the constriction length and width, we observe characteristic slip, stuck, and stuck-slip motions of the model T. brucei within the constriction. Our findings may provide some mechanical insights into how T. brucei moves through blood vessels and tissues, and across the blood-brain barrier.

cond-mat.soft↗

arXiv2024

FedSSP: Federated Graph Learning with Spectral Knowledge and Personalized Preference

Personalized Federated Graph Learning (pFGL) facilitates the decentralized training of Graph Neural Networks (GNNs) without compromising privacy while accommodating personalized requirements for non-IID participants. In cross-domain scenarios, structural heterogeneity poses significant challenges for pFGL. Nevertheless, previous pFGL methods incorrectly share non-generic knowledge globally and fail to tailor personalized solutions locally under domain structural shift. We innovatively reveal that the spectral nature of graphs can well reflect inherent domain structural shifts. Correspondingly, our method overcomes it by sharing generic spectral knowledge. Moreover, we indicate the biased message-passing schemes for graph structures and propose the personalized preference module. Combining both strategies, we propose our pFGL framework FedSSP which Shares generic Spectral knowledge while satisfying graph Preferences. Furthermore, We perform extensive experiments on cross-dataset and cross-domain settings to demonstrate the superiority of our framework. The code is available at https://github.com/OakleyTan/FedSSP.

cs.LG↗

arXiv2024

Paths and Intersections: Characterization of Quasi-metrics in Directed Okamura-Seymour Instances

We study the following distance realization problem. Given a quasi-metric $D$ on a set $T$ of terminals, does there exist a directed Okamura-Seymour graph that realizes $D$ as the (directed) shortest-path distance metric on $T$? We show that, if we are further given the circular ordering of terminals lying on the boundary, then Monge property is a sufficient and necessary condition. This generalizes previous results for undirected Okamura-Seymour instances. With the circular ordering, we give a greedy algorithm for constructing a directed Okamura-Seymour instance that realizes the input quasi-metric. The algorithm takes the dual perspective concerning flows and routings, and is based on a new way of analyzing graph structures, by viewing graphs as \emph{paths and their intersections}. We believe this new understanding is of independent interest and will prove useful in other problems in graph theory and graph algorithms. We also design an efficient algorithm for finding such a circular ordering that makes $D$ satisfy Monge property, if one exists. Combined with our result above, this gives an efficient algorithm for the distance realization problem.

cs.DS↗

arXiv2024

Cut-Preserving Vertex Sparsifiers for Planar and Quasi-bipartite Graphs

We study vertex sparsification for preserving cuts. Given a graph $G$ with a subset $|T|=k$ of its vertices called terminals, a \emph{quality-$q$ cut sparsifier} is a graph $G'$ that contains $T$, such that, for any partition $(T_1,T_2)$ of $T$ into non-empty subsets, the value of the min-cut in $G'$ separating $T_1$ from $T_2$ is within factor $q$ from the value of the min-cut in $G$ separating $T_1$ from $T_2$. The construction of cut sparsifiers with good (small) quality and size has been a central problem in graph compression for years. Planar graphs and quasi-bipartite graphs are two important special families studied in this research direction. The main results in this paper are new cut sparsifier constructions for them in the high-quality regime (where $q=1$ or $1+\varepsilon$ for small $\varepsilon>0$). We first show that every planar graph admits a planar quality-$(1+\varepsilon)$ cut sparsifier of size $\tilde O(k/\text{poly}(\varepsilon))$, which is in sharp contrast with the lower bound of $2^{\Omega(k)}$ for the quality-$1$ case. We then show that every quasi-bipartite graph admits a quality-$1$ cut sparsifier of size $2^{\tilde O(k^2)}$. This is the second to improve over the doubly-exponential bound for general graphs (previously only planar graphs have been shown to have single-exponential size quality-$1$ cut sparsifiers). Lastly, we show that contraction, a common approach for constructing cut sparsifiers adopted in most previous works, does not always give optimal bounds for cut sparsifiers. We demonstrate this by showing that the optimal size bound for quality-$(1+\varepsilon)$ contraction-based cut sparsifiers for quasi-bipartite graphs lies in the range $[k^{\tilde\Omega(1/\varepsilon)},k^{O(1/\varepsilon^2)}]$, while in previous work an upper bound of $\tilde O(k/\varepsilon^2)$ was achieved via a non-contraction approach.

cs.DS↗

arXiv2024

New Structures and Algorithms for Length-Constrained Expander Decompositions

Expander decompositions form the basis of one of the most flexible paradigms for close-to-linear-time graph algorithms. Length-constrained expander decompositions generalize this paradigm to better work for problems with lengths, distances and costs. Roughly, an $(h,s)$-length $ϕ$-expander decomposition is a small collection of length increases to a graph so that nodes within distance $h$ can route flow over paths of length $hs$ with congestion at most $1/ϕ$. In this work, we give a close-to-linear time algorithm for computing length-constrained expander decompositions in graphs with general lengths and capacities. Notably, and unlike previous works, our algorithm allows for one to trade off off between the size of the decomposition and the length of routing paths: for any $ε> 0$ not too small, our algorithm computes in close-to-linear time an $(h,s)$-length $ϕ$-expander decomposition of size $m \cdot ϕ\cdot n^ε$ where $s = \exp(\text{poly}(1/ε))$. The key foundations of our algorithm are: (1) a simple yet powerful structural theorem which states that the union of a sequence of sparse length-constrained cuts is itself sparse and (2) new algorithms for efficiently computing sparse length-constrained flows.

cs.DS↗

arXiv2024

Towards the Characterization of Terminal Cut Functions: a Condition for Laminar Families

We study the following characterization problem. Given a set $T$ of terminals and a $(2^{|T|}-2)$-dimensional vector $π$ whose coordinates are indexed by proper subsets of $T$, is there a graph $G$ that contains $T$, such that for all subsets $\emptyset\subsetneq S\subsetneq T$, $π_S$ equals the value of the min-cut in $G$ separating $S$ from $T\setminus S$? The only known necessary conditions are submodularity and a special class of linear inequalities given by Chaudhuri, Subrahmanyam, Wagner and Zaroliagis. Our main result is a new class of linear inequalities concerning laminar families, that generalize all previous ones. Using our new class of inequalities, we can generalize Karger's approximate min-cut counting result to graphs with terminals.

cs.DS↗