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新论文 · v1arXiv:2611.17296

具有最坏情况更新时间和调整代价的动态 k-中位算法

Dynamic $k$-Median with Worst-Case Update Time and Recourse

Chenglin Fan

AI 速读 基于摘要

作者提出一种随机化常数近似算法，用于单位权重客户端插入和删除下的全动态度量 k-中位问题，并声称同时给出最坏情况更新时间和中心调整代价的界。

cs.DS

arXiv ↗PDF ↗
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We study fully dynamic metric $k$-median under unit-weight client insertions and deletions. After each update, the algorithm must maintain at most $k$ centers chosen from the current client locations. We give a randomized constant-factor approximation algorithm with $\widetilde O(k)$ worst-case update time and $O((1+\logΔ)^3)$ worst-case center recourse, where $Δ$ is the aspect ratio of the underlying metric space. The approximation guarantee holds simultaneously over any prescribed finite adaptive update sequence with high probability, while the update time and recourse bounds hold on every execution. We also obtain the same asymptotic guarantees for metric $k$-means.

新论文 · v1arXiv:2611.16313

Optimal-Size Strong Connectivity Preservers under Two Edge Failures in Subcubic Time

Wanhee Lee, Chenglin Fan

原文摘要

A $k$-edge-failure strong connectivity preserver of a directed graph on $N$ vertices is a spanning subgraph that has the same strongly connected components as the input after each set of at most $k$ edge deletions. The lower-bound construction of Baswana, Choudhary, and Roditty [BCR18], adapted to strong connectivity in [CC20], gives the previous best bound of $Ω(2^kN)$ necessary edges. Hoppenworth, Saranurak, and Wang [HSW25] prove an $O(k4^kN\log N)$ upper bound, leaving a logarithmic gap for fixed $k$. We close this gap by giving an explicit strongly connected graph on $N=2^d$ vertices with $N-1+(N/2)\log_2N$ edges, every one of which is necessary in a two-failure preserver. Together with the upper bound of [HSW25], this proves that the optimal size in the worst case is $Θ_k(N\log N)$ for every fixed $k\geq2$, whereas it is $Θ(N)$ for $k=1$ [GIP20]. The same graph has a one-failure preserver with at most $3(N-1)$ edges. We also give a randomized algorithm that either returns a spanning subgraph or reports failure. For every fixed $c,\varepsilon>0$, with probability at least $1-N^{-c}$ it returns a correct preserver for $k=2$ with $O(N\log N)$ edges, in $\widetilde O(N^{(ω+3)/2+\varepsilon})$ time and $\widetilde O(N^2)$ space. Here $ω$ is the matrix multiplication exponent over a fixed finite field.

cs.DS

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2611.13635

Min-CSPs on Complete Instances III: Nearly Optimal Characterizations of Approximation for Symmetric CSPs

Yihwan Chae, Chenglin Fan, Donghyeon Lee, Euiwoong Lee

原文摘要

This paper studies complete Min-$k$-CSPs: an instance on $n$ variables has one nontrivial constraint for every $k$-set of variables (so $\binom{n}{k}$ constraints in total), and the goal is to find an assignment that minimizes the number of unsatisfied constraints. They are motivated by the connections to machine learning and data science, as well as the possibility of being testbeds for new algorithmic techniques. Previous recent results presented polynomial-time algorithms for the decision version of every complete Boolean $k$-CSP, which is a necessary condition for any multiplicative approximations for the minimization version, but nontrivial approximation results have been limited to fragile CSPs, 2-SAT, and NAE-3-SAT, leaving the approximability of most complete Boolean Min-CSPs open, including complete Min-$k$-SAT. In this paper, we resolve this question, up to nearly matching factors, for the natural class of symmetric Boolean predicates. Given $A \subseteq \{0, \dots, k\}$, Min-CSP($A$) is a Min-$k$-CSP where each constraint is specified by $k$ literals and is satisfied if the number of true literals belongs to $A$. Let $w(A)$ be the maximum number of consecutive integers in $A$. Our main results for complete Min-CSP($A$) are (1) a QPTAS for $w(A) \leq 2$ and (2) an $\widetilde{O}(n^{w(A)-3})$-approximation for $w(A) \geq 3$, as well as almost matching hardness results for the latter. Our QPTAS for $w(A)=2$ significantly improves the previous $O(1)$-approximation for Min-2-SAT and the $\operatorname{polylog}(n)$-approximation for Min-NAE-3-SAT, as well as extends the PTAS for fragile CSPs (i.e., $w(A) = 1$).

cs.DS

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2609.17403

Pseudometric-Weighted Correlation Clustering via Spectral Preclustering

Chenglin Fan, Dahoon Lee, Euiwoong Lee

原文摘要

We study \emph{pseudometric-weighted correlation clustering}, in which pairs have arbitrary positive or negative labels and nonnegative disagreement weights satisfying the triangle inequality. For every fixed $\varepsilon>0$, we give a randomized polynomial-time $(2+\varepsilon)$-approximation, improving the previous factor of $10/3$. The algorithm constructs an explicit feasible solution to the cluster LP with polynomial support and value at most $(1+\varepsilon)\mathrm{OPT}$, where $\mathrm{OPT}$ is the minimum integral clustering cost. Our main contribution is a weighted preclustering construction that witnesses a near-optimal clustering while bounding the total weight of admissible edges by $\operatorname{poly}(1/\varepsilon)\mathrm{OPT}$. We obtain admissibility certificates from random walks, where the factor-two Ptolemy's inequality for pseudometrics reduces the required length of walks from $O(\log n)$ to a constant. We then bound the rounding error separately at each vertex and use the triangle inequality to control its weighted cost. Sampling and coverage corrections produce an exactly feasible solution, and the factor-two rounding completes the approximation algorithm.

cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.10184

Fully Dynamic Correlation Clustering with Vertex Updates in Polylogarithmic Update Time

Vladimir Braverman, Prathamesh Dharangutte, Chenglin Fan, Chen Wang, Shenghao Xie, Samson Zhou

原文摘要

We present a fully dynamic algorithm for correlation clustering that supports \emph{vertex} insertions and deletions while maintaining an $O(1)$ approximation in polylogarithmic amortized update time. This resolves an open problem of vertex deletions posed by a recent line of work on dynamic correlation clustering. We adopt the min-disagreement objective for correlation clustering, i.e., given a signed complete graph $G=(V,E^+\cup E^-)$, we want to minimize the total number of $(+)$ edge crossing clusters and $(-)$ edges within clusters. The input is given in the adjacency list model, where each update inserts or deletes a node along with its full set of $(+)$ neighbors. Our main technical contribution is a fault-tolerant sparse-dense decomposition framework which unlike the existing decomposition-based frameworks allows for structural inconsistencies as long as the induced cost is bounded. We also show a separation between explicitly maintaining the (stronger) sparse-dense decomposition and maintaining a constant-factor approximation to correlation clustering, which may be of independent interest.

cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.09596

Coordinatewise Epsilon for Adam under Heavy-Tailed Noise

Jeonseong Kim, Chenglin Fan

原文摘要

Adam estimates second moments coordinatewise, but typically adds the same scalar $\varepsilon$ to every denominator. We study how this shared $\varepsilon$ affects convergence when noise and curvature vary across coordinates. Under coordinatewise finite-$p$ noise moments, $p\in(1,2]$, choosing $\varepsilon_i$ from the noise and curvature bounds for each coordinate attains the heavy-tailed $\ell_1$-stationarity benchmark $O(T^{-\frac{p-1}{3p-2}})$, matching the known optimal time exponent for smooth nonconvex stochastic optimization. For $1<p<2$, Adam with a shared $\varepsilon_t$ admits a lower bound of order $Ω(T^{-\frac{p-1}{2p}})$. This lower bound holds for every predictable scalar choice of $η_t$, $β_{1,t}$, $β_{2,t}$, and $\varepsilon_t$, precluding a dimension-uniform benchmark rate for this class.

cs.LG

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arXiv ↗PDF ↗