[Paper Review] Independent Sets in Vertex-Arrival Streams

Graham Cormode, Jacques Dark+1 more|arXiv (Cornell University)|Jul 22, 2018

Complexity and Algorithms in Graphs12 references10 citations

TL;DR

This paper investigates the space complexity of approximating maximum independent sets (MIS) in vertex-arrival graph streams, showing that explicit vertex streams require Ω(n²/c⁷) space for c-approximation, making MIS significantly harder than in edge-arrival models. It introduces a novel multi-party communication problem related to pointer jumping to prove strong lower bounds, and establishes improved upper and lower bounds for geometric intersection graphs like unit squares and intervals in both explicit and implicit vertex stream models.

ABSTRACT

We consider the maximal and maximum independent set problems in three models of graph streams: - In the edge model we see a stream of edges which collectively define a graph; this model is well-studied for a variety of problems. We 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 it is not much easier if we only require approximate maximality. This contrasts strongly with the other two vertex-based models, where one can greedily find an exact solution in only the space needed to store the independent set. - In the "explicit" vertex model, the input stream is a sequence of vertices making up the graph. 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 in edge-arrival streams. We show that every one-pass c-approximation streaming algorithm for maximum independent set (MIS) on explicit vertex streams requires Omega({n^2}/{c^6}) bits of space, where n is the number of vertices of the input graph. It is already known that Theta~({n^2}/{c^2}) bits of space are necessary and sufficient in the edge arrival model (Halldórsson et al. 2012), thus the MIS problem is not significantly easier to solve under the explicit vertex arrival order assumption. Our result is proved via a reduction from a new multi-party communication problem closely related to pointer jumping. - In the "implicit" vertex model, the input stream consists of a sequence of objects, one per vertex. The algorithm is equipped with a function that maps pairs of objects to the presence or absence of edges, 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 interval and square intersection graphs, in both explicit and implicit streams. In particular, we show a gap between the hardness of the explicit and implicit vertex models for interval graphs.

Motivation & Objective

• To understand the space complexity of finding approximate maximum independent sets (MIS) in vertex-arrival graph stream models, particularly in explicit and implicit vertex streams.
• To compare the difficulty of MIS in vertex-arrival models versus the classical edge-arrival model, where vertex models allow greedy algorithms with subquadratic space.
• To close the gap between known upper and lower bounds for geometric intersection graphs (e.g., unit squares, intervals) in vertex stream settings.
• To develop new communication complexity tools—specifically a multi-party generalization of the INDEX problem—to prove hardness results for MIS approximation.
• To examine whether geometric structure (e.g., unit discs, intervals) enables better space efficiency in MIS approximation compared to general graphs.

Proposed method

• Reduces the MIS approximation problem to a novel multi-party communication problem closely related to pointer jumping, using a construction based on Ruzsa-Szemerédi graphs for approximate maximality.
• Employs a geometric construction using balls (or intervals) in 1D and 2D to encode bit vectors across multiple parties, where independent sets correspond to solutions of a communication problem.
• Uses erasure codes in the hard instance graphs to bound clique sizes and ensure that only specific combinations of bits yield large independent sets.
• Applies packing arguments in 2D to show that MIS size can be used to recover encoded information, enabling lower bounds via communication complexity.
• Designs explicit streaming algorithms for geometric graphs (e.g., unit squares) using a strip decomposition and weight class rounding to achieve (3+ǫ)-approximation with ˜O(α(G)/ǫ) space.
• Adapts the construction for weighted MIS by rounding weights to powers of (1+ǫ) and tracking only ˜O(ǫ⁻¹) weight classes per strip, achieving (3+ǫ)-approximation with ˜O(log W + α(G)/ǫ) space.

Experimental results

Research questions

• RQ1 How does the space complexity of c-approximating MIS in explicit vertex streams compare to that in edge-arrival streams?
• RQ2 Can the hardness of MIS in explicit vertex streams be proven via a communication complexity framework, and what new primitives are needed?
• RQ3 What are the tightest possible upper and lower bounds for approximating MIS in streams of unit squares or unit intervals in explicit and implicit vertex models?
• RQ4 Does the geometric structure of intersection graphs (e.g., unit discs, intervals) allow for o(α(G)) space algorithms for constant-factor MIS approximation?
• RQ5 Can the approximation factor for MIS in unit square streams be improved beyond 3, and what is the minimal space required for better approximations?

Key findings

• Finding a maximal independent set in edge-arrival streams requires Ω(n²) space, even for approximate maximality, showing that the edge model is fundamentally harder than vertex models.
• For explicit vertex streams, c-approximating α(G) requires Ω(n²/c⁷) space, which is significantly more than ˜O(α(G)) space used in vertex models for exact MIS, indicating that MIS is not significantly easier in explicit vertex streams.
• For implicit vertex streams of unit squares, a (3+ǫ)-approximation to MIS can be achieved in ˜O(α(G)/ǫ) space, while any better than 5/2-approximation requires Ω(n) space.
• For unit interval graphs in explicit vertex streams, achieving better than 5/3-approximation requires Ω(n) space, making it harder than the implicit stream variant.
• The paper constructs a 3-party communication problem based on ball arrangements where the MIS size encodes the outcome of a joint computation, proving that any algorithm with approximation factor better than 5/2 must use Ω(n) space.
• For 2D geometric graphs with arbitrary-sized squares, the paper leaves open whether a constant-factor approximation exists, and whether the gap between upper and lower bounds for unit squares can be closed.