Given $n$ positive integers $a_1,a_2,\dots,a_n$, and a positive integer right hand side $\beta$, we consider the feasibility version of the subset sum problem which is the problem of determining whether a subset of $a_1,a_2,\dots,a_n$ adds up to $\beta$. We show that if the right hand side $\beta$ is chosen as $\lfloor r\sum_{j=1}^n a_j \rfloor$ for a constant $0 < r < 1$ and if the $a_j$'s are independentand identically distributed from a discrete uniform distribution taking values ${1,2,\dots,\lfloor 10^{n/2} \rfloor }$, 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.

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关于普通分支定界几乎所有子集和问题的难度

给定 $n$ 个正整数 $a_1,a_2,\dots,a_n$ 和一个正整数右侧 $\beta$，我们考虑子集求和问题的可行性版本，即确定 $ a_1,a_2,\dots,a_n$ 加起来为 $\beta$。我们表明，如果右侧 $\beta$ 被选为 $\lfloor r\sum_{j=1}^n a_j \rfloor$ 以得到一个常数 $0 < r < 1$ 并且如果 $a_j$ 是从取值 ${1,2,\dots,\lfloor 10^{n/2} \rfloor }$ 的离散均匀分布中独立同分布，那么子集和问题的实例产生的概率需要创建一个当 $n$ 趋于无穷大时，当单个变量上的一个分支以任何顺序变为 $1$ 时，分支定界节点的指数数量。