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- W3100115795 abstract "The connectivity of a hypergraph is the minimum number of hyperedges whose deletion disconnects the hypergraph. We design an $$hat{O}_r(p + min {lambda ^{frac{r-3}{r-1}} n^2, n^r/lambda ^{frac{r}{r-1}}, lambda ^{frac{5r-7}{4r-4}}n^{frac{7}{4}}})$$ (The $$hat{O}_r(cdot )$$ notation hides terms that are subpolynomial in the main parameter and terms that depend only on r) time algorithm for computing hypergraph connectivity, where $$p := sum _{e in E} |e|$$ is the input size of the hypergraph, n is the number of vertices, r is the rank (size of the largest hyperedge), and $$lambda $$ is the connectivity of the input hypergraph. Our algorithm also finds a minimum cut in the hypergraph. Our algorithm is faster than existing algorithms if $$r = O(1)$$ and $$lambda = n^{varOmega (1)}$$ . The heart of our algorithm is a structural result showing a trade-off between the number of hyperedges taking part in all minimum cuts and the size of the smaller side of any minimum cut. This structural result can be viewed as a generalization of an acclaimed structural theorem for simple graphs [Kawarabayashi-Thorup, JACM 19 (Fulkerson Prize 2021)]. We extend the framework of expander decomposition to hypergraphs to prove this structural result. In addition to the expander decomposition framework, our faster algorithm also relies on a new near-linear time procedure to compute connectivity when one of the sides in a minimum cut is small." @default.
- W3100115795 created "2020-11-23" @default.
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- W3100115795 date "2022-01-01" @default.
- W3100115795 modified "2023-10-07" @default.
- W3100115795 title "Faster Connectivity in Low-Rank Hypergraphs via Expander Decomposition" @default.
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- W3100115795 doi "https://doi.org/10.1007/978-3-031-06901-7_6" @default.
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