Polynomial Pass Lower Bounds for Graph Streaming Algorithms

TL;DR

This paper establishes that solving certain fundamental graph problems in the streaming model requires polynomial passes, introducing a new communication problem to derive these lower bounds and demonstrating its broader applicability.

Contribution

The paper introduces the hidden-pointer chasing problem and uses it to derive polynomial lower bounds for graph streaming algorithms, advancing understanding of their computational limitations.

Findings

Weighted minimum s-t cut requires n^{2-o(1)} space without polynomial passes

Hidden-pointer chasing problem is a versatile tool for lower bounds

Submodular function minimization needs n^{2-o(1)} queries unless highly adaptive

Abstract

We present new lower bounds that show that a polynomial number of passes are necessary for solving some fundamental graph problems in the streaming model of computation. For instance, we show that any streaming algorithm that finds a weighted minimum $s$-$t$ cut in an $n$-vertex undirected graph requires $n^{2-o(1)}$ space unless it makes $n^{\Omega(1)}$ passes over the stream. To prove our lower bounds, we introduce and analyze a new four-player communication problem that we refer to as the hidden-pointer chasing problem. This is a problem in spirit of the standard pointer chasing problem with the key difference that the pointers in this problem are hidden to players and finding each one of them requires solving another communication problem, namely the set intersection problem. Our lower bounds for graph problems are then obtained by reductions from the hidden-pointer chasing problem. Our hidden-pointer chasing problem appears flexible enough to find other applications and is therefore interesting in its own right. To showcase this, we further present an interesting application of this problem beyond streaming algorithms. Using a reduction from hidden-pointer chasing, we prove that any algorithm for submodular function minimization needs to make $n^{2-o(1)}$ value queries to the function unless it has a polynomial degree of adaptivity.