Tight Bounds for Single-Pass Streaming Complexity of the Set Cover Problem

TL;DR

This paper establishes tight bounds on the space complexity of single-pass streaming algorithms for approximating the set cover problem, showing that certain approximation factors require nearly linear space, resolving open questions in streaming complexity.

Contribution

It provides tight space bounds for approximating set cover and its size in the streaming model, including for the more general covering integer programs, answering open questions.

Findings

Space complexity for $ heta(mn/\alpha)$ for $\\alpha$-approximate set cover.

Space complexity for estimating minimum set cover size is $ heta(mn/\alpha^2)$.

Lower bounds hold even for random order set instances.

Abstract

We resolve the space complexity of single-pass streaming algorithms for approximating the classic set cover problem. For finding an $\alpha$-approximate set cover (for any $\alpha= o(\sqrt{n})$) using a single-pass streaming algorithm, we show that $\Theta(mn/\alpha)$ space is both sufficient and necessary (up to an $O(\log{n})$ factor); here $m$ denotes number of the sets and $n$ denotes size of the universe. This provides a strong negative answer to the open question posed by Indyk et al. (2015) regarding the possibility of having a single-pass algorithm with a small approximation factor that uses sub-linear space. We further study the problem of estimating the size of a minimum set cover (as opposed to finding the actual sets), and establish that an additional factor of $\alpha$ saving in the space is achievable in this case and that this is the best possible. In other words, we show that $\Theta(mn/\alpha^2)$ space is both sufficient and necessary (up to logarithmic factors) for estimating the size of a minimum set cover to within a factor of $\alpha$. Our algorithm in fact works for the more general problem of estimating the optimal value of a covering integer program. On the other hand, our lower bound holds even for set cover instances where the sets are presented in a random order.