On the complex zeros and the computational complexity of approximating the reliability polynomial

In this paper we relate the location of the complex zeros of the reliability polynomial to parameters at which a certain family of rational functions derived from the reliability polynomial exhibits chaotic behaviour. We use this connection to prove new results about the location of reliability zeros. In particular we show that there are zeros with modulus larger than $1$ with essentially any possible argument. We moreover use this connection to show that approximately evaluating the reliability polynomial for planar graphs at a non-positive algebraic number in the unit disk is #P-hard.

💡 Research Summary

The paper investigates two intertwined problems concerning the all‑terminal reliability polynomial (R(G;p)) of a graph (G): (i) the location of its complex zeros (reliability zeros) and (ii) the computational difficulty of approximating (R(G;p)) for a fixed algebraic parameter (p).
The authors introduce, for a two‑terminal graph ((G,s,t)), the split‑reliability polynomial (S(G;p)) (the probability that the graph becomes an (s!-!t) split) and define two rational “edge‑interaction” functions:
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