# The algorithm by Ferson et al. is surprisingly fast: An NP-hard optimization problem solvable in almost linear time with high probability

Ferson et al. (Reliable computing 11(3), p. 207--233, 2005) introduced an algorithm for the NP-hard nonconvex quadratic programming problem called MaxVariance motivated by robust statistics. They proposed an implementation with worst-case time complexity O(n^2 · 2^ω), where ω is the largest clique in a certain intersection graph. First we show that with a careful implementation the complexity can be improved to O(n n + n· 2^ω). Then we treat input data as random variables (as it is usual in statistics) and introduce a natural probabilistic data generating model. We show that ω = O( n/ n) on average under this model. As a result we get average computing time O(n^1+ε) for ε > 0 arbitrarily small. We also prove the following tail bound on computation time: the instances, forcing the algorithm to compute in exponential time, occur rarely, with probability tending to zero faster than exponentially with n →∞.

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