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We consider the algorithm by Ferson et al. (Reliable computing 11(3), p.207-233, 2005) designed for solving the NP-hard problem of computing themaximal sample variance over interval data, motivated by robust statistics (infact, the formulation can be written as a nonconvex quadratic program with aspecific structure). First, we propose a new version of the algorithm improvingits original time bound $O(n^2 2^\omega)$ to $O(n \log n+n\cdot 2^\omega)$,where $n$ is number of input data and $\omega$ is the clique number in acertain intersection graph. Then we treat input data as random variables as itis usual in statistics) and introduce a natural probabilistic data generatingmodel. We get $2^\omega = O(n^{1/\log\log n})$ and $\omega = O(\log n /\log\log n)$ on average. This results in average computing time$O(n^{1+\epsilon})$ for $\epsilon > 0$ arbitrarily small, which may beconsidered as "surprisingly good" average time complexity for solving anNP-hard problem. Moreover, we prove the following tail bound on thedistribution of computation time: hard instances, forcing the algorithm tocompute in time $2^{\Omega(n)}$, occur rarely, with probability tending to zeroat the rate $e^{-n\log\log n}$.

The NP-hard problem of computing the maximal sample variance over interval data is solvable in almost linear time with high probability