Bounded Independence Edge Sampling for Combinatorial Graph Properties

TL;DR

This paper explores bounded-independence edge sampling in graphs, showing it can preserve connectivity and cycle properties with limited independence, and derandomize algorithms for finding graphic matroid bases.

Contribution

It generalizes previous results on bounded-independence in random graphs and provides explicit derandomization techniques for graph algorithms.

Findings

O(log(m))-wise independence preserves connectivity in certain graphs.

O(log(m))-wise almost independence ensures cycle-freeness under specific conditions.

Derandomization of algorithms for finding graphic matroid bases is achieved.

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 $\geq \kappa \log(m)$ with probability $1 - \frac{1}{\mathrm{poly}(m)}$ (for a sufficiently large constant $\kappa$). 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 $\geq \kappa \log(m)$ with probability $1 - \frac{1}{\mathrm{poly}(m)}$ (for a sufficiently large constant $\kappa$). 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.