Settling the Pass Complexity of Streaming Set Cover

2026-09-30

AI Summary

This study addresses the complexity bottleneck in the streaming set cover problem arising from the trade-off among space, number of passes, and approximation ratio. Methodologically, it constructs a randomized reduction based on the pointer chasing problem from communication complexity and leverages properties of random sets for theoretical analysis. The primary contribution is establishing the first matching lower bound, demonstrating that the space requirement for any α-approximation algorithm using p passes is optimal up to a constant factor. This result yields a tight three-way optimal trade-off, fully characterizing the complexity boundary of this fundamental problem and revealing the underlying reason for the stagnation of progress in related areas over the past decade.

Abstract

In the streaming set cover problem, $m$ sets from a universe of size $n$ are arriving one by one in a stream, and the algorithm is allowed to process the stream using one or a few passes and a space of $o(mn)$, which is sublinear in the input size. The goal is to determine the minimal (or approximately minimal) number of sets that cover the universe at the end of the last pass. This problem has been studied extensively over the years with rapid progress that led to several $O(\log{n})$-approximation algorithms in $\tilde{O}(mn^{1/p})$ space and $O(p)$ passes. However, progress on this front has largely stagnated over the past decade, despite the absence of any lower bounds that rule out even an $O(\log{n})$-approximation in $O(m)$ space and just two passes. We provide a simple explanation for this lack of progress by establishing an optimal three-way space-pass-approximation tradeoff for this problem: any $α$-approximation algorithm for streaming set cover requires $$ \widetildeΩ\Big(\frac{m}α \cdot \big(\frac{n}α\big)^{1/p}\Big) $$ space in $p$ passes whenever $α\ll n^{1/(p+1)}$. In light of prior work, this result is optimal up to constant factors in $p$ and logarithmic factors in $n,m$ for any $α\geq p$. Our bound is optimal with respect to the range of $α$ also, and fully settles the complexity of this fundamental problem in the streaming model. The proof of this result is (surprisingly) simple and non-technical and relies on a randomized reduction from a variant of the standard pointer chasing problem in communication complexity, using elementary properties of random sets.

Problem

• streaming set cover
• pass complexity
• space-approximation tradeoff
• lower bounds

Innovation

• Streaming Set Cover
• Pass Complexity
• Space-Pass-Approximation Tradeoff
• Communication Complexity
• Pointer Chasing