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We present dynamic algorithms withpolylogarithmicupdate time for estimating the size of the maximum matching of a graph undergoing edge insertions and deletions with approximation ratiostrictly better than 2. Specifically, we obtain a\(1+\frac{1}{\sqrt {2}}+\epsilon \approx 1.707+\epsilon \)approximation in bipartite graphs and a 1.973 + ϵ approximation in general graphs. We thus answer in the affirmative the value version of the major open question repeatedly asked in the dynamic graph algorithms literature. Our randomized algorithms’ approximation and worst-case update time bounds both hold w.h.p. against adaptive adversaries. Our algorithms are based on simulating new two-pass streaming matching algorithms in the dynamic setting. Our key new idea is to invoke the recent sublinear-time matching algorithm of Behnezhad (FOCS’21) in a white-box manner to efficiently simulate the second pass of our streaming algorithms, while bypassing the well-known vertex-update barrier. 

Computing a dense subgraph is a fundamental problem in graph mining, with a diverse set of applications ranging from electronic commerce to community detection in social networks. In many of these applications, the underlying context is better modelled as a weighted hypergraph that keeps evolving with time. This motivates the problem of maintaining the densest subhypergraph of a weighted hypergraph in a dynamic setting, where the input keeps changing via a sequence of updates (hyperedge insertions/deletions). Previously, the only known algorithm for this problem was due to Hu et al. [19]. This algorithm worked only on unweighted hypergraphs, and had an approximation ratio of (1 +ϵ)r2 and an update time of O(poly(r, log n)), where r denotes the maximum rank of the input across all the updates. We obtain a new algorithm for this problem, which works even when the input hypergraph is weighted. Our algorithm has a significantly improved (near-optimal) approximation ratio of (1 +ϵ) that is independent of r, and a similar update time of O(poly(r, log n)). It is the first (1 +ϵ)-approximation algorithm even for the special case of weighted simple graphs. To complement our theoretical analysis, we perform experiments with our dynamic algorithm on large-scale, real-world data-sets. Our algorithm significantly outperforms the state of the art [19] both in terms of accuracy and efficiency.