We propose a dynamic version of the classical node packing problem, also called the stable set or independent set problem. The problem is defined by a node set, a node weight vector, and an edge probability vector. For every pair of nodes, an edge is present or not according to an independent Bernoulli random variable defined by the corresponding entry in the probability vector. At each step, the decision maker selects an available node that maximizes the expected weight of the final packing, and then observes edges adjacent to this node. We formulate the problem as a Markov decision process and show that it is NP-Hard even on star graphs. Next, we introduce relaxations of the problem’s achievable probabilities polytope, analogous to the linear and bilinear edge-based formulations in the deterministic case; we show that these relaxations can be weak, motivating a polyhedral study. We derive classes of valid inequalities arising from cliques, paths, and cycles. For cliques, we completely characterize the polytope and show that it is a submodular polyhedron. For both paths and cycles, we give an implicit representation of the polytope via a cut-generating linear program of polynomial size based on a compact dynamic programming formulation. Our computational results show that our inequalities can greatly reduce the upper bound and improve the linear relaxation’s gap, particularly when the instance’s expected density is high.

我们提出经典节点打包问题的动态版本，也称为稳定集或独立集问题。该问题由节点集，节点权重向量和边缘概率向量定义。对于每对节点，根据概率矢量中相应条目定义的独立伯努利随机变量，是否存在边缘。在每个步骤中，决策者都选择一个使最终包装的预期重量最大化的可用节点，然后观察与该节点相邻的边缘。我们将该问题公式化为马尔可夫决策过程，并表明即使在星图上它也是NP-Hard。接下来，我们引入对问题可实现概率多态性的放宽，类似于确定性情况下基于线性和双线性边的公式。我们表明，这些松弛可能是微弱的，从而激发了多面体研究。我们推导了由集团，路径和周期引起的有效不等式的类别。对于团体，我们完全表征了多面体，并表明它是亚模多面体。对于路径和循环，我们都基于紧凑的动态编程公式，通过多项式大小的割生成线性程序，隐式表示多面体。我们的计算结果表明，我们的不等式可以大大减小上限并改善线性弛豫的间隙，尤其是当实例的预期密度很高时。对于路径和循环，我们都基于紧凑的动态编程公式，通过多项式大小的割生成线性程序，隐式表示多面体。我们的计算结果表明，我们的不等式可以大大减小上限并改善线性弛豫的间隙，尤其是当实例的预期密度很高时。对于路径和循环，我们都基于紧凑的动态编程公式，通过多项式大小的割生成线性程序，隐式表示多面体。我们的计算结果表明，我们的不等式可以大大减小上限并改善线性弛豫的间隙，尤其是当实例的预期密度很高时。