Settling the Pass Complexity of Approximate Matchings in Dynamic Graph Streams

TL;DR

The paper resolves the pass complexity for approximating maximum matchings in dynamic graph streams under semi-streaming space, achieving an $O( ext{log log } n)$-pass upper bound that scales to a $(1+ ε)$-approximation on weighted graphs, and proving a matching Ω$( ext{log log } n)$-pass lower bound. The upper bound hinges on a model-independent fractional-matching to MIS reduction, executed via sampling and sparse-recovery within a dynamic streaming MIS framework. The lower bound leverages a novel Augmented Hidden Matrices construction and a multi-round hierarchical embedding with a refined round-elimination argument to translate communication lower bounds into streaming space requirements. Together, these results precisely pin the pass complexity of $O(1)$-approximate dynamic-matchings at Θ$( ext{log log } n)$ and demonstrate the power and limits of graph sketching in dynamic streaming. The work also highlights implications for MPC and related models, illustrating that graph sketching suffices to achieve efficient multi-round approximations with near-optimal memory.

Abstract

A semi-streaming algorithm in dynamic graph streams processes any $n$-vertex graph by making one or multiple passes over a stream of insertions and deletions to edges of the graph and using $O(n \cdot \mbox{polylog}(n))$ space. Semi-streaming algorithms for dynamic streams were first obtained in the seminal work of Ahn, Guha, and McGregor in 2012, alongside the introduction of the graph sketching technique, which remains the de facto way of designing algorithms in this model and a highly popular technique for designing graph algorithms in general. We settle the pass complexity of approximating maximum matchings in dynamic streams via semi-streaming algorithms by improving the state-of-the-art in both upper and lower bounds. We present a randomized sketching based semi-streaming algorithm for $O(1)$-approximation of maximum matching in dynamic streams using $O(\log\log{n})$ passes. The approximation ratio of this algorithm can be improved to $(1+ε)$ for any fixed $ε> 0$ even on weighted graphs using standard techniques. This exponentially improves upon several $O(\log{n})$ pass algorithms developed for this problem since the introduction of the dynamic graph streaming model. In addition, we prove that any semi-streaming algorithm (not only sketching based) for $O(1)$-approximation of maximum matching in dynamic streams requires $Ω(\log\log{n})$ passes. This presents the first multi-pass lower bound for this problem, which is already also optimal, settling a longstanding open question in this area.

Figures (13)

• Figure 1: A clique in $(a)$ and two different sets of blocking edges in $(b)$ and $(c)$ based on different ordering of vertices in the randomized greedy MIS (the red vertex joins the MIS, green vertices join the vertex cover, and red edges are blocking). While the probabilities of edges becoming blocking are $2/n$ in a clique and can form a $2$-approximate fractional matching, blocking edges in each single run form stars and are very far from matchings themselves.
• Figure 2: An illustration of our new fractional matching assignment $x$ in a single run of the randomized greedy MIS on the graph in $(a)$. Figure $(b)$ shows the blocking edges, plus the vertex that joins the MIS in this iteration (red), and its neighbors that join the vertex cover (green). In Figure $(c)$, we have the assignment of $1/5$ over all edges of a vertex that joins the vertex cover, but in Figure $(d)$, the assignment of $1/3$ misses one edge, since its other endpoint has a higher degree (unlike \ref{['fig:clique-alg']}, this figure shows a single iteration of the algorithm and not multiple runs).
• Figure 3: Consider running the randomized greedy MIS algorithm on the graph in $(a)$ using the specified ordering. The semi-streaming algorithm of AhnCGMW15 processes vertices in large batches, say, a batch of $3$ vertices in part $(b)$. This allows the algorithm to determine which vertices in the batch are in the MIS (red) or the vertex cover (green) in a single pass. Then, in part $(c)$, using another pass, the algorithm identifies all the other vertices in the graph that also join the vertex cover (again, green vertices). However, for the fractional matching reduction, there are more considerations: for instance, the edge $(4,6)$ receives a fractional matching from vertex $4$ but not vertex $6$, as vertex $4$ is already removed by the time $6$ is added to the vertex cover (even though, the algorithm of AhnCGMW15 treats both vertices $4$ and $6$ the same way). In terms of time stamps, time stamp of $4$ is $t(4)=1$ while for $6$ it is $t(6)=3$.
• Figure 4: An illustration of the hierarchical embedding with a collection of $(4 \times 4)$ many $(p-1)$-pass hard instances inside a single $p$-pass hard instance which is the graph $G$. The special hidden instances here correspond to $i^{\star}=3$ which need to be solved to solve the entire problem. We emphasize that, unlike this figure, in the actual construction the sub-instances for different $\mathcal{G}_i$ and $\mathcal{G}_j$ need to necessarily share some vertices so they all fit in the same graph.
• Figure 5: An illustration of the inputs $A$ and $B$ to Alice and Bob in the lower bound of DarkK20. Here, $n=6$ and $i^{\star}=4$ and both $\sigma_r$ and $\sigma_c$ are identity permutations. In general, the submatrix of $B$ corresponds to a combinatorial rectangle in $A$, the induced matching is of size $n/2-o(n)$ with high probability, and the vertex cover is of size $o(n)$.

Theorems & Definitions (90)

• Remark 1
• Proposition 3.2: Chernoff Bound; cf. DubhashiP09
• Definition 3.3: Limited Independence Hash Functions
• Proposition 3.4: MotwaniR95
• Proposition 3.5: SchmidtSS95
• Proposition 3.6: Sparse Recovery; cf. DasS13AssadiKM23
• Proposition 3.7: AssadiKL16
• Proposition 3.8: McGregor05AhnG11GamlathKMS19AssadiLT21
• Definition 3.9
• Proposition 3.10: cf. AlonMS96