Sublinear-Space Streaming Algorithms for Estimating Graph Parameters on Sparse Graphs

Abstract

In this paper, we design sub-linear space streaming algorithms for estimating three fundamental parameters – maximum independent set, minimum dominating set and maximum matching – on sparse graph classes, i.e., graphs which satisfy $m=O(n)$ where m, n is the number of edges, vertices respectively. Each graph parameter we consider can have size $\varOmega (n)$ even on sparse graph classes, and hence for sublinear-space algorithms we are restricted to parameter estimation instead of attempting to find a solution. We obtain these results:

• Estimating Max Independent Set via the Caro-Wei bound: Caro and Wei each showed $\lambda = \sum _{v} {1}/(d(v) + 1)$ is a lower bound on max independent set size, where vertex v has degree d(v). If average degree, $\bar{d}$, is $\mathcal {O}(1)$, and max degree $\varDelta = \mathcal {O}({\varepsilon }^{2} \bar{d}^{-3} n)$, our algorithms, with at least $1 - \delta$ success probability:

  • In online streaming, return an actual independent set of size $1 \pm {\varepsilon }$ times $\lambda$. This improves on Halldórsson et al. [Algorithmica ’16]: we have less working space, i.e., $\mathcal {O}(\log {\varepsilon }^{-1} \cdot \log n \cdot \log \delta ^{-1})$, faster updates, i.e., $\mathcal {O}(\log {\varepsilon }^{-1})$, and bounded success probability.

  • In insertion-only streams, approximate $\lambda$ within factor $1 \pm {\varepsilon }$, in one pass, in $\mathcal {O}(\bar{d} {\varepsilon }^{-2} \log n \cdot \log \delta ^{-1})$ space. This aligns with the result of Cormode et al. [ISCO ’18], though our method also works for online streaming. In a vertex-arrival and random-order stream, space reduces to $\mathcal {O}(\log (\bar{d} {\varepsilon }^{-1}))$. With extra space and post-processing step, we remove the max-degree constraint.

• Sublinear-Space Algorithms on Forests: On a forest, Esfandiari et al. [SODA ’15, TALG ’18] showed space lower bounds for 1-pass randomized algorithms that approximately estimate these graph parameters. We narrow the gap between upper and lower bounds:

  • Max independent set size within $3/2 \cdot (1 \pm {\varepsilon })$ in one pass and in $\log ^{\mathcal {O}(1)} n$ space, and within $4/3\cdot (1 \pm {\varepsilon })$ in two passes and in $\tilde{\mathcal {O}}(\sqrt{n})$ space; the lower bound is for approx. $\le 4/3$.

  • Min dominating set size within $3 \cdot (1 \pm {\varepsilon })$ in one pass and in $\log ^{\mathcal {O}(1)} n$ space, and within $2\cdot (1 \pm {\varepsilon })$ in two passes and in $\tilde{\mathcal {O}}(\sqrt{n})$ space; the lower bound is for approx. $\le 3/2$.

  • Max matching size within $2 \cdot (1 \pm {\varepsilon })$ in one pass and in $\log ^{\mathcal {O}(1)} n$ space, and within $3/2\cdot (1 \pm {\varepsilon })$ in two passes and in $\tilde{\mathcal {O}}(\sqrt{n})$ space; the lower bound is for approx. $\le 3/2$.