Incremental (1−ϵ)(1-\epsilon)-approximate dynamic matching in O(poly(1/ϵ))O(poly(1/\epsilon)) update time

In the dynamic approximate maximum bipartite matching problem we are given bipartite graph $G$ undergoing updates and our goal is to maintain a matching of $G$ which is large compared the maximum matching size $\mu(G)$. We define a dynamic matching algorithm to be $\alpha$ (respectively $(\alpha, \beta)$)-approximate if it maintains matching $M$ such that at all times $|M | \geq \mu(G) \cdot \alpha$ (respectively $|M| \geq \mu(G) \cdot \alpha - \beta$). We present the first deterministic $(1-\epsilon )$-approximate dynamic matching algorithm with $O(poly(\epsilon ^{-1}))$ amortized update time for graphs undergoing edge insertions. Previous solutions either required super-constant [Gupta FSTTCS'14, Bhattacharya-Kiss-Saranurak SODA'23] or exponential in $1/\epsilon$ [Grandoni-Leonardi-Sankowski-Schwiegelshohn-Solomon SODA'19] update time. Our implementation is arguably simpler than the mentioned algorithms and its description is self contained. Moreover, we show that if we allow for additive $(1, \epsilon \cdot n)$-approximation our algorithm seamlessly extends to also handle vertex deletions, on top of edge insertions. This makes our algorithm one of the few small update time algorithms for $(1-\epsilon )$-approximate dynamic matching allowing for updates both increasing and decreasing the maximum matching size of $G$ in a fully dynamic manner

Incremental (1-?)-Approximate Dynamic Matching in O(poly(1/?)) Update Time

In the dynamic approximate maximum bipartite matching problem we are given bipartite graph G undergoing updates and our goal is to maintain a matching of G which is large compared the maximum matching size ?(G). We define a dynamic matching algorithm to be ? (respectively (?, ?))-approximate if it maintains matching M such that at all times |M | ? ?(G) ? ? (respectively |M| ? ?(G) ? ? - ?). We present the first deterministic (1-?)-approximate dynamic matching algorithm with O(poly(?^{-1})) amortized update time for graphs undergoing edge insertions. Previous solutions either required super-constant [Gupta FSTTCS\u2714, Bhattacharya-Kiss-Saranurak SODA\u2723] or exponential in 1/? [Grandoni-Leonardi-Sankowski-Schwiegelshohn-Solomon SODA\u2719] update time. Our implementation is arguably simpler than the mentioned algorithms and its description is self contained. Moreover, we show that if we allow for additive (1, ??n)-approximation our algorithm seamlessly extends to also handle vertex deletions, on top of edge insertions. This makes our algorithm one of the few small update time algorithms for (1-?)-approximate dynamic matching allowing for updates both increasing and decreasing the maximum matching size of G in a fully dynamic manner. Our algorithm relies on the weighted variant of the celebrated Edge-Degree-Constrained-Subgraph (EDCS) datastructure introduced by [Bernstein-Stein ICALP\u2715]. As far as we are aware we introduce the first application of the weighted-EDCS for arbitrarily dense graphs. We also present a significantly simplified proof for the approximation ratio of weighed-EDCS as a matching sparsifier compared to [Bernstein-Stein], as well as simple descriptions of a fractional matching and fractional vertex cover defined on top of the EDCS. Considering the wide range of applications EDCS has found in settings such as streaming, sub-linear, stochastic and more we hope our techniques will be of independent research interest outside of the dynamic setting

Nearly Optimal Communication and Query Complexity of Bipartite Matching

We settle the complexities of the maximum-cardinality bipartite matching problem (BMM) up to poly-logarithmic factors in five models of computation: the two-party communication, AND query, OR query, XOR query, and quantum edge query models. Our results answer open problems that have been raised repeatedly since at least three decades ago [Hajnal, Maass, and Turan STOC'88; Ivanyos, Klauck, Lee, Santha, and de Wolf FSTTCS'12; Dobzinski, Nisan, and Oren STOC'14; Nisan SODA'21] and tighten the lower bounds shown by Beniamini and Nisan [STOC'21] and Zhang [ICALP'04]. We also settle the communication complexity of the generalizations of BMM, such as maximum-cost bipartite $b$-matching and transshipment; and the query complexity of unique bipartite perfect matching (answering an open question by Beniamini [2022]). Our algorithms and lower bounds follow from simple applications of known techniques such as cutting planes methods and set disjointness.Comment: Accepted in FOCS 202

Sublinear-Round Parallel Matroid Intersection

Despite a lot of recent progress in obtaining faster sequential matroid intersection algorithms, the fastest parallel poly(n)-query algorithm was still the straightforward O(n)-round parallel implementation of Edmonds\u27 augmenting paths algorithm from the 1960s. Very recently, Chakrabarty-Chen-Khanna [FOCS\u2721] showed the lower bound that any, possibly randomized, parallel matroid intersection algorithm making poly(n) rank-queries requires ??(n^{1/3}) rounds of adaptivity. They ask, as an open question, if the lower bound can be improved to ??(n), or if there can be sublinear-round, poly(n)-query algorithms for matroid intersection. We resolve this open problem by presenting the first sublinear-round parallel matroid intersection algorithms. Perhaps surprisingly, we do not only break the O?(n)-barrier in the rank-oracle model, but also in the weaker independence-oracle model. Our rank-query algorithm guarantees O(n^{3/4}) rounds of adaptivity, while the independence-query algorithm uses O(n^{7/8}) rounds of adaptivity, both making a total of poly(n) queries