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arXiv · 2403.18781

Hypergraph Unreliability in Quasi-Polynomial Time

Ruoxu Cen·Jason Li·Debmalya Panigrahi
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Abstract

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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Ruoxu Cen, Jason Li, Debmalya Panigrahi. 2024-03-27. Hypergraph Unreliability in Quasi-Polynomial Time. https://arxiv.org/abs/2403.18781

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