Improved approximation guarantees for weighted matching in the semi-streaming model

We study the maximum weight matching problem in the semi-streaming model, and improve on the currently best one-pass algorithm due to Zelke (Proc. of STACS2008, pages 669-680) by devising a deterministic approach whose performance guarantee is 4.91+epsilon. In addition, we study preemptive online algorithms, a sub-class of one-pass algorithms where we are only allowed to maintain a feasible matching in memory at any point in time. All known results prior to Zelke's belong to this sub-class. We provide a lower bound of 4.967 on the competitive ratio of any such deterministic algorithm, and hence show that future improvements will have to store in memory a set of edges which is not necessarily a feasible matching

Semi-Streaming Algorithms for Submodular Function Maximization Under b-Matching Constraint

We consider the problem of maximizing a submodular function under the b-matching constraint, in the semi-streaming model. Our main results can be summarized as follows. - When the function is linear, i.e. for the maximum weight b-matching problem, we obtain a 2+? approximation. This improves the previous best bound of 3+? [Roie Levin and David Wajc, 2021]. - When the function is a non-negative monotone submodular function, we obtain a 3 + 2 ?2 ? 5.828 approximation. This matches the currently best ratio [Roie Levin and David Wajc, 2021]. - When the function is a non-negative non-monotone submodular function, we obtain a 4 + 2 ?3 ? 7.464 approximation. This ratio is also achieved in [Roie Levin and David Wajc, 2021], but only under the simple matching constraint, while we can deal with the more general b-matching constraint. We also consider a generalized problem, where a k-uniform hypergraph is given with an extra matroid constraint imposed on the edges, with the same goal of finding a b-matching that maximizes a submodular function. We extend our technique to this case to obtain an algorithm with an approximation of 8/3k+O(1). Our algorithms build on the ideas of the recent works of Levin and Wajc [Roie Levin and David Wajc, 2021] and of Garg, Jordan, and Svensson [Paritosh Garg et al., 2021]. Our main technical innovation is to introduce a data structure and associate it with each vertex and the matroid, to record the extra information of the stored edges. After the streaming phase, these data structures guide the greedy algorithm to make better choices

Tight Bounds for Vertex Connectivity in Dynamic Streams

We present a streaming algorithm for the vertex connectivity problem in dynamic streams with a (nearly) optimal space bound: for any $n$-vertex graph $G$ and any integer $k \geq 1$, our algorithm with high probability outputs whether or not $G$ is $k$-vertex-connected in a single pass using $\widetilde{O}(k n)$ space. Our upper bound matches the known $\Omega(k n)$ lower bound for this problem even in insertion-only streams -- which we extend to multi-pass algorithms in this paper -- and closes one of the last remaining gaps in our understanding of dynamic versus insertion-only streams. Our result is obtained via a novel analysis of the previous best dynamic streaming algorithm of Guha, McGregor, and Tench [PODS 2015] who obtained an $\widetilde{O}(k^2 n)$ space algorithm for this problem. This also gives a model-independent algorithm for computing a "certificate" of $k$-vertex-connectivity as a union of $O(k^2\log{n})$ spanning forests, each on a random subset of $O(n/k)$ vertices, which may be of independent interest.Comment: Full version of the paper accepted to SOSA 2023. 15 pages, 3 Figure

An Optimal Algorithm for Triangle Counting in the Stream

We present a new algorithm for approximating the number of triangles in a graph G whose edges arrive as an arbitrary order stream. If m is the number of edges in G, T the number of triangles, ?_E the maximum number of triangles which share a single edge, and ?_V the maximum number of triangles which share a single vertex, then our algorithm requires space: O?(m/T?(?_E + ?{?_V})) Taken with the ?((m ?_E)/T) lower bound of Braverman, Ostrovsky, and Vilenchik (ICALP 2013), and the ?((m ?{?_V})/T) lower bound of Kallaugher and Price (SODA 2017), our algorithm is optimal up to log factors, resolving the complexity of a classic problem in graph streaming