组合Optimization的极值概率界
Extremal Probability Bounds in Combinatorial Optimization
摘要：
在本文中，我们计算的概率，最佳值的组合最优化问题的最大化形式与随机目标超过一个给定的数字，假设只有知识的边缘分布的目标系数向量的最优值的最严格可能的界限。边界是"极端的“，因为它们在具有给定边缘的所有联合分布上都有效。我们分析了计算的复杂性，假设离散的边际和确定的情况下，在多项式时间内计算的界限。对于紧致0/1 V-多面体，我们通过一个伪多项式时间算法证明了最紧上界是弱NP-难计算的。另一方面，通过限制对Bernoulli随机变量的关注，证明了紧致0/1 V-多面体的最紧下界是强NP-难计算的。对于紧致0/1 H-多面体，对于项目管理中PERT网络的特殊情况，我们通过给出一个伪多项式时间算法，证明了最紧上界是弱NP-难计算的.本文的结果补充了文献中关于独立随机变量的概率计算的已有结果。
英文摘要：
In this paper, we compute the tightest possible bounds on the probability that the optimal value of a combinatorial optimization problem in maximization form with a random objective exceeds a given number, assuming only knowledge of the marginal distributions of the objective coefficient vector. The bounds are ``extremal'' since they are valid across all joint distributions with the given marginals. We analyze the complexity of computing the bounds assuming discrete marginals and identify instances when the bounds are computable in polynomial time. For compact 0/1 V-polytopes, we show that the tightest upper bound is weakly NP-hard to compute by providing a pseudopolynomial time algorithm. On the other hand, the tightest lower bound is shown to be strongly NP-hard to compute for compact 0/1 V-polytopes by restricting attention to Bernoulli random variables. For compact 0/1 H-polytopes, for the special case of PERT networks arising in project management, we show that the tightest upper bound is weakly NP-hard to compute by providing a pseudopolynomial time algorithm. The results in the paper complement existing results in the literature for computing the probability with independent random variables.
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