Smoothed analysis shows the expected SOS rank of unweighted minimum knapsack is O(sqrt(n) log(n/sigma)) after Gaussian perturbation of q, though linear rank occurs when q is exponentially close to an integer.
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On the Distribution of Unweighted Minimum Knapsack Instances with Large SOS Rank
Smoothed analysis shows the expected SOS rank of unweighted minimum knapsack is O(sqrt(n) log(n/sigma)) after Gaussian perturbation of q, though linear rank occurs when q is exponentially close to an integer.