Scalable $k$-clique Densest Subgraph Search

Xiaowei Ye; Miao Qiao; Rong-Hua Li; Qi Zhang; Guoren Wang

Home/Archive/arXiv

arXiv · 2403.05775

Scalable $k$-clique Densest Subgraph Search

Xiaowei Ye·Miao Qiao·Rong-Hua Li·Qi Zhang·Guoren Wang

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we present a collection of novel and scalable algorithms designed to tackle the challenges inherent in the $k$-clique densest subgraph problem (\kcdsp) within network analysis. We propose \psctl, a novel algorithm based on the Frank-Wolfe approach for addressing \kcdsp, effectively solving a distinct convex programming problem. \textcolor{black}{\psctl is able to approximate \kcdsp with near optimal guarantees.} The notable advantage of \psctl lies in its time complexity, which is independent of the count of $k$-cliques, resulting in remarkable efficiency in practical applications. Additionally, we present \spath, a sampling-based algorithm with the capability to handle networks on an unprecedented scale, reaching up to $1.8\times 10^9$ edges. By leveraging the \ccpath algorithm as a uniform $k$-clique sampler, \spath ensures the efficient processing of large-scale network data, accompanied by a detailed analysis of accuracy guarantees. Together, these contributions represent a significant advancement in the field of $k$-clique densest subgraph discovery. In experimental evaluations, our algorithms demonstrate orders of magnitude faster performance compared to the current state-of-the-art solutions.

Explore related subjects

cs.DS

Keep this discovery

BibTeX RIS

Xiaowei Ye, Miao Qiao, Rong-Hua Li, Qi Zhang, Guoren Wang. 2024-03-09. Scalable $k$-clique Densest Subgraph Search. https://arxiv.org/abs/2403.05775

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗

arXiv2026

A deterministic $(1+\varepsilon)^n$ approximation for the permanent of a nonnegative matrix

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

cs.DS↗

arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗