Bounded Independence Edge Sampling for Combinatorial Graph Properties

2026-03-26

AI Summary

This work addresses the derandomization challenge in graph algorithms arising from random edge sampling, particularly when dealing with union bounds over exponentially many events. Focusing on bounded independence, the paper extends the results of Alon and Nussboim on bounded-independence random graphs to general graphs, showing that $O(\log m)$-wise independence suffices to preserve key graph properties—such as connectivity and acyclicity—with high probability. The approach combines bounded-independent hashing, probabilistic methods, and structural graph analysis to retain these properties with failure probability $1/\mathrm{poly}(m)$ in graphs whose minimum cut or girth is $\Omega(\log m)$. Notably, this yields the first explicit derandomization of parallel matroid basis algorithms for graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

Random subsampling of edges is a commonly employed technique in graph algorithms, underlying a vast array of modern algorithmic breakthroughs. Unfortunately, using this technique often leads to randomized algorithms with no clear path to derandomization because the analyses rely on a union bound on exponentially many events. In this work, we revisit this goal of derandomizing randomized sampling in graphs. We give several results related to bounded-independence edge subsampling, and in the process of doing so, generalize several of the results of Alon and Nussboim (FOCS 2008), who studied bounded-independence analogues of random graphs (which can be viewed as edge subsamples of the complete graph). Most notably, we show: 1. $O(\log(m))$-wise independence suffices for preserving connectivity when sampling at rate $1/2$ in a graph with minimum cut $\geq κ\log(m)$ with probability $1 - \frac{1}{\mathrm{poly}(m)}$ (for a sufficiently large constant $κ$). 2. $O(\log(m))$-wise $\frac{1}{\mathrm{poly}(m)}$-almost independence suffices for ensuring cycle-freeness when sampling at rate $1/2$ in a graph with minimum cycle length $\geq κ\log(m)$ with probability $1 - \frac{1}{\mathrm{poly}(m)}$ (for a sufficiently large constant $κ$). To demonstrate the utility of our results, we revisit the classic problem of using parallel algorithms to find graphic matroid bases, first studied in the work of Karp, Upfal, and Wigderson (FOCS 1985). In this regime, we show that the optimal algorithms of Khanna, Putterman, and Song (arxiv 2025) can be explicitly derandomized while maintaining near-optimality.

Problem

• derandomization
• bounded independence
• edge sampling
• graph properties
• combinatorial graphs

Innovation

• bounded independence
• edge sampling
• derandomization
• graph connectivity
• matroid basis