Finding a Small Vertex Cut on Distributed Networks
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We present an algorithm for distributed networks to efficiently find a small vertex cut in the CONGEST model. Given a positive integer κ, our algorithm can, with high probability, either find κ vertices whose removal disconnects the network or return that such κ vertices do not exist. Our algorithm takes κ^3·Õ(D+√(n)) rounds, where n is the number of vertices in the network and D denotes the network's diameter. This implies Õ(D+√(n)) round complexity whenever κ=polylog(n). Prior to our result, a bound of Õ(D) is known only when κ=1,2 [Parter, Petruschka DISC'22]. For κ≥ 3, this bound can be obtained only by an O(log n)-approximation algorithm [Censor-Hillel, Ghaffari, Kuhn PODC'14], and the only known exact algorithm takes O((κΔ D)^O(κ)) rounds, where Δ is the maximum degree [Parter DISC'19]. Our result answers an open problem by Nanongkai, Saranurak, and Yingchareonthawornchai [STOC'19].
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