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From arXiv

Hypergraph Unreliability in Quasi-Polynomial Time

The hypergraph unreliability problem asks for the probability that a hypergraph gets disconnected when every hyperedge fails independently with a given probability. For graphs, the unreliability problem has been studied over many decades, and multiple fully polynomial-time approximation schemes are known starting with the work of Karger (STOC 1995). In contrast, prior to this work, no non-trivial result was known for hypergraphs (of arbitrary rank). In this paper, we give quasi-polynomial time approximation schemes for the hypergraph unreliability problem. For any fixed \(\varepsilon \in (0, 1)\), we first give a \((1+\varepsilon)\)-approximation algorithm that runs in \(m^{O(\log n)}\) time on an \(m\)-hyperedge, \(n\)-vertex hypergraph. Then, we improve the running time to \(m\cdot n^{O(\log^2 n)}\) with an additional exponentially small additive term in the approximation.
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Published on March 27, 2024
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