[论文解读] On Estimating Maximum Matching Size in Graph Streams

Sepehr Assadi, Sanjeev Khanna+1 more|arXiv (Cornell University)|Jan 16, 2017

Complexity and Algorithms in Graphs被引用 64

一句话总结

论文研究在图流中估计最大匹配规模，给出插入-only和动态流的新上界和下界，并将结果与矩阵秩联系起来。

ABSTRACT

We study the problem of estimating the maximum matching size in graphs whose edges are revealed in a streaming manner. We consider both insertion-only streams and dynamic streams and present new upper and lower bound results for both models. On the upper bound front, we show that an $α$-approximate estimate of the matching size can be computed in dynamic streams using $\widetilde{O}({n^2/α^4})$ space, and in insertion-only streams using $\widetilde{O}(n/α^2)$-space. On the lower bound front, we prove that any $α$-approximation algorithm for estimating matching size in dynamic graph streams requires $Ω(\sqrt{n}/α^{2.5})$ bits of space, even if the underlying graph is both sparse and has arboricity bounded by $O(α)$. We further improve our lower bound to $Ω(n/α^2)$ in the case of dense graphs. Furthermore, we prove that a $(1+ε)$-approximation to matching size in insertion-only streams requires RS$(n) \cdot n^{1-O(ε)}$ space; here, RS${n}$ denotes the maximum number of edge-disjoint induced matchings of size $Θ(n)$ in an $n$-vertex graph. It is a major open problem to determine the value of RS$(n)$, and current results leave open the possibility that RS$(n)$ may be as large as $n/\log n$. We also show how to avoid the dependency on the parameter RS$(n)$ in proving lower bound for dynamic streams and present a near-optimal lower bound of $n^{2-O(ε)}$ for $(1+ε)$-approximation in this model. Using a well-known connection between matching size and matrix rank, all our lower bounds also hold for the problem of estimating matrix rank. In particular our results imply a near-optimal $n^{2-O(ε)}$ bit lower bound for $(1+ε)$-approximation of matrix ranks for dense matrices in dynamic streams, answering an open question of Li and Woodruff (STOC 2016).

研究动机与目标

• Motivate the study of estimating maximum matching size in streaming graphs and distinguish estimation from exact approximation.
• Provide upper bounds for alpha-approximate size estimation in both dynamic and insertion-only streams.
• Prove lower bounds for alpha-approximate size estimation under different graph sparsity and density regimes.
• Establish near-quadratic lower bounds for near-optimal (1+epsilon) estimation and connect results to matrix rank via Tutte matrices.

提出的方法

• Develop single-pass streaming algorithms that compute alpha-approximate estimates of the matching size in dynamic streams with space ~n^2/alpha^4 and in insertion-only streams with space ~n/alpha^2.
• Show that these estimation bounds separate estimation from full approximation of matchings in streams.
• Leverage reductions from the Boolean Hidden Hypermatching problem to derive lower bounds on space for estimation.
• Apply information theory tools (entropy, mutual information, Fano’s inequality) to obtain lower bounds.
• Use the connection between maximum matching size and matrix rank (Tutte matrix) to extend lower bounds to rank estimation.
• Discuss RS(n) based lower bounds and their implications for near-linear space algorithms.

实验结果

研究问题

• RQ1 What is the space-accuracy tradeoff for estimating the size of a maximum matching in graph streams?
• RQ2 Do alpha-approximate size estimators require less space than alpha-approximate matchers in insertion-only and dynamic streams?
• RQ3 What are the lower bounds on space for (1+epsilon)-approximation of matching size in insertion-only and dynamic streams?
• RQ4 How do RS(n) based lower bounds influence the feasibility of near-optimal estimation in dense graphs?
• RQ5 How do these results translate to estimating the rank of a matrix via the Tutte matrix connection?

主要发现

• An alpha-approximate estimate of the maximum matching size in dynamic streams can be computed in ~O(n^2/alpha^4) space and in insertion-only streams in ~O(n/alpha^2) space.
• Estimation of matching size is provably easier than finding an alpha-approximate matching, yet their space requirements converge for near-optimal accuracy.
• Any alpha-approximation algorithm for estimating dynamic-stream matching size requires at least Omega(sqrt(n)/alpha^2.5) bits, with stronger bounds for dense graphs (Omega(n/alpha^2)).
• For near-optimal (1+epsilon) estimation in insertion-only streams, the lower bound is Omega(RS(n) * n^{1-O(epsilon)}) and Omega(n^{2-O(epsilon)}) in dynamic streams, depending on the model.
• The results imply near-quadratic space lower bounds for (1+epsilon)-approximation of matrix ranks in dense matrices via the Tutte-matrix connection.