Hypergraph Unreliability in Quasi-Polynomial Time
arxiv(2024)
摘要
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 ε∈ (0, 1), we
first give a (1+ε)-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· n^O(log^2 n) with an additional exponentially small
additive term in the approximation.
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