Stochastic combinatorial optimization problems are usually defined as planning problems, which involve purchasing and allocating resources in order to meet uncertain needs. For example, network designers need to make their best guess about the future needs of the network and purchase capabilities accordingly. Facing uncertain in the future, we either “wait and see” changes, or postpone decisions about resource allocation until the requirements or constraints become realized. Specifically, in the field of stochastic combinatorial optimization, some inputs of the problems are uncertain, but follow known probability distributions. Our goal is to find a strategy that minimizes the expected cost. In this paper, we consider the two-stage finite-scenario stochastic set cover problem and the single sink rent-or-buy problem by presenting primal-dual based approximation algorithms for these two problems with approximation ratio \(2\eta \) and 4.39, respectively, where \(\eta \) is the maximum frequency of the element of the ground set in the set cover problem.

随机组合优化问题通常被定义为规划问题，涉及购买和分配资源以满足不确定的需求。例如，网络设计人员需要对网络的未来需求做出最好的猜测，并相应地购买能力。面对未来的不确定性，我们要么“观望”变化，要么推迟关于资源分配的决定，直到需求或限制成为现实。具体来说，在随机组合优化领域，问题的一些输入是不确定的，但遵循已知的概率分布。我们的目标是找到一种使预期成本最小化的策略。在本文中，\(2\eta \)和 4.39，其中\(\eta \)是集合覆盖问题中地面集合元素的最大频率。