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 the maximal sample variance over interval data, motivated by robust statist...
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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 the maximal sample variance over interval data, motivated by robust statistics (in fact, the formulation can be written as a nonconvexquadratic program with a specific structure). First, we propose a new version of the algorithm improving its original time bound O(n(2).2(omega)) to O(nlogn+n & sdot;2(omega)), where n is number of input data and omega is the clique number in a certain intersection graph. Then we treat input data as random variables as it is usual in statistics) and introduce a natural probabilistic data generating model. We get 2(omega)=O(n(1/log logn)) on average. This results in average computing time O(n(1+& varepsilon;)) for & varepsilon;>0 arbitrarily small, which may be considered as "surprisingly good" average time complexity for solving an NP-hard problem. Moreover, we prove the following tail bound on the distribution of computation time: hard instances, forcing the algorithm to compute in time 2 Omega(n), occur rarely, with probability tending to zero at the rate e(-n log log n). The main result admits a smoothed-complexity interpretation: the average computing time can be bounded by n(1+O(1/sigma))/(log log n), where sigma measures the dispersion of the distribution of data perturbation.
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