Independent Sets in Vertex-Arrival Streams

TL;DR

This paper investigates the space complexity of finding maximal and maximum independent sets in different graph streaming models, revealing that these problems remain space-intensive even under models that simplify input access.

Contribution

The paper establishes tight space bounds for MIS in explicit and implicit vertex stream models, showing minimal ease over the edge-arrival model, and introduces a new multi-party communication problem for proofs.

Findings

Maximal independent set requires quadratic space in edge streams.

Approximate MIS in explicit vertex streams still demands near-quadratic space.

New bounds for intersection graph problems in geometric and implicit models.

Abstract

We consider the classic maximal and maximum independent set problems in three models of graph streams: In the edge-arrival model we see a stream of edges which collectively define a graph, this model has been well-studied for a variety of problems. We first show that the space complexity for a one-pass streaming algorithm to find a maximal independent set is quadratic (i.e. we must store all edges). We further show that the problem does not become much easier if we only require approximate maximality. In the "explicit" vertex stream model, the input stream is a sequence of vertices making up the graph, where every vertex arrives along with its incident edges that connect to previously arrived vertices. Various graph problems require substantially less space to solve in this setting than for edge-arrival streams. We show that every one-pass $c$-approximation algorithm for maximum independent set (MIS) on explicit vertex streams requires space $\Omega(\frac{n^2}{c^7})$, where $n$ is the number of vertices of the input graph, and it is already known that space $\tilde{\Theta}(\frac{n^2}{c^2})$ is necessary and sufficient in the edge arrival model (Halld\'orsson et al. 2012). The MIS problem is thus not significantly easier to solve under the explicit vertex arrival order assumption. Our result is proved via a reduction to a new multi-party communication problem closely related to pointer jumping. In the "implicit" vertex stream model, the input stream consists of a sequence of objects, one per vertex. The algorithm is equipped with a function that can map a pair of objects to the presence or absence of an edge, thus defining the graph. This model captures, for example, geometric intersection graphs such as unit disc graphs. Our final set of results consists of several improved upper and lower bounds for ball intersection graphs, in both explicit and implicit streams.