On the Hardness of Almost All Subset Sum Problems by Ordinary Branch-and-Bound

01/04/2020

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by   Mustafa Kemal Tural, et al.

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Given n positive integers a_1,a_2,...,a_n, and a positive integer right hand side β, we consider the feasibility version of the subset sum problem which is the problem of determining whether a subset of a_1,a_2,...,a_n adds up to β. We show that if the right hand side β is chosen as r∑_j=1^n a_j for a constant 0 < r < 1 and if the a_j's are independentand identically distributed from a discrete uniform distribution taking values 1,2,..., 10^n/2, then the probability that the instance of the subset sum problem generated requires the creation of an exponential number of branch-and-bound nodes when one branches on the individual variables in any order goes to 1 as n goes to infinity.