[Paper Review] Independent Set Size Approximation in Graph Streams

Graham Cormode, Jacques Dark+1 more|arXiv (Cornell University)|Feb 27, 2017

Complexity and Algorithms in Graphs17 references3 citations

TL;DR

This paper presents space-efficient streaming algorithms for approximating the Caro-Wei bound β(G), a lower bound on the maximum independent set size in a graph. It achieves accurate approximation in polylogarithmic space under vertex-arrival order, while proving an Ω(n/(βc²)) space lower bound for constant-factor c-approximation in arbitrary edge order, establishing tight limits on what is computationally feasible in the streaming model.

ABSTRACT

We study the problem of estimating the size of independent sets in a graph $G$ defined by a stream of edges. Our approach relies on the Caro-Wei bound, which expresses the desired quantity in terms of a sum over nodes of the reciprocal of their degrees, denoted by $β(G)$. Our results show that $β(G)$ can be approximated accurately, based on a provided lower bound on $β$. Stronger results are possible when the edges are promised to arrive grouped by an incident node. In this setting, we obtain a value that is at most a logarithmic factor below the true value of $β$ and no more than the true independent set size. To justify the form of this bound, we also show an $Ω(n/β)$ lower bound on any algorithm that approximates $β$ up to a constant factor.

Motivation & Objective

• To design space-efficient streaming algorithms for estimating the size of maximum independent sets in large graphs where edges arrive sequentially.
• To leverage the Caro-Wei bound β(G) = Σ_v 1/(deg(v)+1) as a tractable proxy for the independence number α(G), which is NP-hard to compute exactly.
• To characterize the trade-offs between approximation accuracy, space usage, and arrival order models (arbitrary vs. vertex-grouped edge arrival).
• To establish tight space lower bounds for approximating β(G) within a constant factor, showing inherent limitations of streaming algorithms.
• To demonstrate that better approximation is possible under vertex-arrival order, where edges from each node arrive together, enabling polylogarithmic space algorithms.

Proposed method

• The paper uses the Caro-Wei bound β(G) as a combinatorial proxy for the independence number α(G), which is known to satisfy α(G) ≥ β(G).
• For arbitrary edge arrival, the algorithm uses reservoir sampling and frequency moment estimation to approximate β(G) with space depending on a lower bound on β(G).
• For vertex-arrival order, the algorithm processes each node and its incident edges together, enabling a more accurate approximation with polylogarithmic space.
• The method relies on a reduction from the communication complexity problem Disjointness (DISJ) to prove space lower bounds, showing that any c-approximation requires Ω(n/(βc²)) space.
• The construction uses a family of graphs with structured cliques and isolated vertices to amplify differences in β(G) based on set intersection, enabling the DISJ reduction.
• The analysis distinguishes between cases where X ∩ Y = ∅ (high β(G)) and X ∩ Y ≠ ∅ (low β(G)), creating a gap that enables distinguishing the DISJ input.

Experimental results

Research questions

• RQ1 Can the Caro-Wei bound β(G) be approximated efficiently in the graph stream model, particularly in polylogarithmic space?
• RQ2 How does the arrival order of edges (arbitrary vs. vertex-grouped) affect the space-approximation trade-off for estimating β(G)?
• RQ3 What is the minimum space required to achieve a constant-factor approximation of β(G) in the arbitrary edge arrival model?
• RQ4 Can the Caro-Wei bound be used to approximate the maximum independent set size with sublinear space in large graphs?
• RQ5 Is there a fundamental limit to how well β(G) can be approximated in the streaming model, and if so, what is it?

Key findings

• An algorithm is presented that approximates β(G) within a factor of O(log n) in the vertex-arrival model using polylogarithmic space, with the approximation being at most a logarithmic factor below the true β(G).
• For arbitrary edge arrival, the algorithm achieves a (1+ε)-approximation of β(G) using space O(β/ε²) when a lower bound on β is known, improving with larger β.
• A tight Ω(n/(βc²)) space lower bound is proven for any algorithm that approximates β(G) within a constant factor c, showing that the space dependence on β is optimal up to logarithmic factors.
• The lower bound is established via a reduction from the Disjointness problem, showing that distinguishing between graphs with β(G) ≈ z and β(G) ≈ z(2c²+1) requires Ω(n/(zc²)) bits of space.
• The results show that while β(G) can be approximated well under vertex-arrival order, the arbitrary edge arrival model imposes stronger space constraints, especially when β(G) is small.
• The paper demonstrates that the Caro-Wei bound is a viable and analyzable proxy for independent set size in streaming graphs, enabling both efficient algorithms and strong lower bounds.