Bounded-Independence Sampling of Edges for Combinatorial Graph Properties
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 that in graphs with $m$ edges: 1. $O(\log m)$-wise independence suffices for preserving connectivity when sampling at rate $1/2$ in a graph with min cut $\geqκ\log(m)$ with probability $1-1/\mathrm{poly}(m)$ (for a sufficiently large constant $κ$). 2. $O(\log m)$-wise $(1/\mathrm{poly}(m))$-almost independence suffices for ensuring cycle-freeness when sampling at rate $1/2$ in a graph with minimum cycle length $\geqκ\log(m)$ with probability $1-1/\mathrm{poly}(m)$ (for a sufficiently large constant $κ$). 3. If we relax to arbitrary distributions, we show there is an explicit distribution $X$ on $\{0, 1\}^m$ with marginals $\leq 1/2$ generated using $O(\log(m)\log\log(m))$ random bits such that in a graph with min cut $\geqκ\log(m)$, a sample from $X$ is still connected with probability $1-1/\mathrm{poly}(m)$. To demonstrate the utility of our results, we revisit the problem of using parallel algorithms to find graphic matroid bases, first studied by Karp, Upfal, and Wigderson (FOCS 1985). We show that the optimal algorithms of Khanna, Putterman, and Song (arxiv 2025) can be explicitly derandomized while maintaining near-optimality.